Categorical Webs and $S$-duality in 4d $\mathcal{N}=2$ QFT
Matteo Caorsi Sergio Cecotti
hep-th math-ph math.MP math.RT


Chiral symmetry constraints on resonant amplitudes
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    December 30, 2023
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We  review the categorical approach to the BPS sector of a 4d 𝒩=2 QFT, clarifying many tricky issues and presenting a few  novel results.

 To a given 𝒩=2 QFT one associates  several triangle categories: they describe various kinds of BPS objects from different physical viewpoints (e.g. IR versus UV). These diverse categories are related by a web of exact functors expressing physical relations between the various objects/pictures. A basic theme of this review is the emphasis on the full web of categories, rather than on  what we can learn from a single description. A second general theme is viewing the cluster category as a sort of `categorification' of 't Hooft's theory of quantum phases for a 4d non-Abelian gauge theory.
  
The S-duality group is best described as the auto-equivalences of the full web of categories. This viewpoint leads to a combinatorial algorithm to search for S-dualities of the  given 𝒩=2 theory. If the ranks of the gauge and flavor groups are not too big, the algorithm may be effectively run on a laptop. This viewpoint also leads to a clearer view of 3d mirror symmetry.

For class 𝒮 theories, all the relevant triangle categories may also be constructed in terms of geometric objects on the Gaiotto curve, and we present the dictionary between triangle categories and  the WKB approach of GMN. We also review how the VEV's of UV line operators are related to cluster characters.


July, 2017





§ INTRODUCTION AND OVERVIEW


The BPS objects of a supersymmetric theory are naturally described in terms of (-linear) triangle categories <cit.> and their stability conditions <cit.>.
The BPS sector of a given physical theory 𝒯 is described by a plurality of different triangle categories 𝔗_(a) depending on:[ The index a take values in some index set I.]


  a) the class of BPS objects (particles, branes, local or non-local operators,...) we are interested in;

  b) the physical picture (fundamental UV theory, IR effective theory,...);

  c) the particular engineering of 𝒯 in QFT/string/M-/F-theory. 
 
The diverse BPS categories 𝔗_(a) are related by a web of exact functors, 𝔗_(a)𝔗_(b), which express physical consistency conditions between the different physical pictures and BPS objects. The simplest instance is given by two different engineerings of the same theory: the duality 𝒯↔𝒯^' induces equivalences of triangle categories
𝔗_(a)𝔗^'_(a) for all objects and all physical descriptions a∈ I. An example  is mirror symmetry between 
IIA and IIB string theories compactified on a pair of mirror Calabi-Yau 3-folds, ℳ, ℳ^∨ which induces on the BPS branes homological mirror symmetry, that is, the equivalences of triangle categories <cit.>

    D^b(𝖢𝗈𝗁 ℳ)≅ D^b(𝖥𝗎𝗄 ℳ^∨),    
    D^b(𝖢𝗈𝗁 ℳ^∨)≅ D^b(𝖥𝗎𝗄 ℳ).

In the same way, the functor relating the IR and UV descriptions of the BPS sector may be seen as homological Renormalization Group, while the functor relating particles and branes may be seen as describing properties of the combined system. 

A duality induces a family of equivalences 𝖽_(a), one for each category 𝔗_(a), and these equivalences should be compatible with the functors 𝖼_(a,b), that is, they should give an equivalence of the full web of categories and functors. Our philosophy is that the study of equivalences of the full functorial web is a very efficient tool  to detect  dualities.
We shall focus on the case of 4d 𝒩=2 QFTs, but the strategy has general validity. We are particularly concerned with S-dualities, i.e. auto-dualities of the theory 𝒯 which act non trivially on the UV degrees of freedom. 


Building on previous work by several people[ References to previous work are provided in the appropriate sections of the paper.], we present our proposal for the triangle categories describing different BPS objects,  both from the UV and IR points of view,
and study the functors relating them. This leads, in particular, to a categorical understanding of the S-duality
 groups and of the vev of UV line operators. The categorical language unifies in a systematic way 
all aspects of the BPS physics, and leads to new powerful techniques to compute susy protected quantities in 𝒩=2 4d
theories. We check in many explicit examples that the results obtained from this more abstract viewpoint reproduce the ones obtained by more traditional techniques.
However the categorical approach may also be used to tackle problems which look too hard for other techniques. 



  
Main triangle categories and functors.
The basic example of a web of functors relating distinct BPS categories for 4d 𝒩=4 QFT is the 
 following exact sequence of triangle categories (Theorem 5.6 of <cit.>):

    0 → D^bΓ𝔓𝔢𝔯 Γ𝒞(Γ)→ 0,

where (see . <ref> for precise definitions and details):


  * Γ is the Ginzburg algebra <cit.> of a  quiver with superpotential <cit.> 
associated to the 𝒩=2 theory at hand;

  * D^bΓ is the bounded derived category of Γ. D^bΓ may be seen as a “universal envelope” of 
the categories describing,
in the deep IR, the BPS
particle spectrum in the several BPS chambers. To discuss states in the IR  we need to fix a Coulomb vacuum u; this datum
defines a stability condition Z_u on D^bΓ. The category which describes the BPS particles in the u vacuum is 
the subcategory of D^bΓ consisting of objects which are semi-stable for Z_u. The BPS particles
 arise from (the quantization of the moduli
 of) the simple objects in this subcategory. Its Grothendieck group K_0(D^bΓ) is identified with 
 the Abelian group of the IR additive 
conserved quantum numbers (electric, magnetic, and flavor charges) which take value in the lattice Λ≅ K_0(D^bΓ). D^bΓ 
is a 3-Calabi-Yau (3-CY)[ See . <ref> for precise definitions. Informally, a triangle category is k-CY iff it behaves as the derived category of coherent sheaves, D^b 𝖼𝗈𝗁 ℳ_k, on a Calabi-Yau k-fold ℳ_k. ] triangle category,
which implies that 
its Euler form 

    χ(X,Y)≡∑_k∈ (-1)^k Hom_D^bΓ(X,Y[k]),     X, Y∈ D^bΓ

is a skew-symmetric form Λ×Λ→ whose physical meaning is the Dirac electro-magnetic pairing between the
 charges [X], [Y]∈Λ carried by the states associated to the stable objects X, Y∈ D^bΓ;

  * 𝒞(Γ) is the cluster category of Γ  which describes[ This is slightly imprecise. Properly 
speaking, the line operators correspond to the generic objects on the
 irreducible components of
the moduli spaces of isoclasses of objects of 𝒞(Γ).] the BPS UV line operators. This identification 
is deeply related to the Kontsevitch-Soibelmann wall-crossing formula <cit.>, see <cit.>. The Grothendieck group K_0(𝒞(Γ)) 
then corresponds to the, additive as well as multiplicative,
 UV quantum numbers of the line operators. These quantum numbers, in particular the multiplicative ones follow from the analysis by 't Hooft of the quantum phases of a 4d non-Abelian gauge theory being determined by the topology of the gauge group
<cit.>. 't Hooft arguments are briefly reviewed in . <ref>: the UV line quantum numbers take value in a finitely generated Abelian group whose torsion part consists of two copies of the fundamental group of the gauge group while its free part describes flavor. 
 The fact that K_0(𝒞(Γ)) is automatically equal to the correct UV group, as predicted  by 't Hooft (detecting the precise topology of the gauge group!), yields convincing evidence for the proposed identification, see . <ref>.  𝒞(Γ) is a 2-CY category, and hence its Euler form induces
a symmetric form on the additive UV charges, which roughly speaking has the form

    K_0(𝒞(Γ))/K_0(𝒞(Γ))_torsion ⊗  
     K_0(𝒞(Γ))/K_0(𝒞(Γ))_torsion→,

but whose precise definition is slightly more involved[ The subtleties in the definition are immaterial when the QFT is UV superconformal (as contrasted to  asymptotically-free) and all chiral operators have integral dimensions.] (see . <ref>). We call this pairing the Tits form of 𝒞(Γ). Its physical meaning is simple: while in the IR the masses break (generically) the flavor group to its maximal torus U(1)^f, in the deep UV the masses become irrelevant and the flavor group gets  enhanced to its maximal non-Abelian form F. Then the UV category should see the full F and not just its Cartan torus. The datum of the group F may be given as its weight lattice together with its Tits form; the cluster Tits form is equal to the Tits form of the non-Abelian flavor group F, and we may read F directly from the cluster category. In fact the cluster category also detects the global topology of the flavor group, distinguishing (say) SO(N) and Spin(N) flavor groups.[ In facts, the cluster Grothendieck group K_0(𝒞(Γ)) should contain even more detailed informations on the flavor. For instance, in SU(2) gauge theory with N_f flavors the states of even magnetic charge are in tensor representations of the flavor SO(2N_f) while states of odd magnetic charge are in spinorial representation of Spin(8); K_0(𝒞(Γ)) should know the correlation between the parity of the magnetic charge and SO(2N_f) vs. Spin(2N_f) flavor symmetries (and it does). ] 
For objects of 𝒞(Γ) there is also a weaker notion of `charge', taking value in the lattice Λ of electric/magnetic/flavor charges, namely the index, which is the quantity referred to as `charge' in many treatments.
Since 𝒞(Γ) yields an UV description of the theory, there must exist relations between its mathematical properties and the physical conditions assuring UV completeness of the associated QFT. We shall point of some of them in . <ref>;

  * 𝔓𝔢𝔯 Γ is the perfect derived category of Γ. From eqn.(<ref>) we see that, morally speaking,
the triangle category 𝔓𝔢𝔯 Γ describes all 
possible BPS IR object generated by the insertion of UV line operators, dressed (screened)
by particles, in all possible vacua. This rough idea is basically correct. Perhaps the most convincing argument comes from consideration of  class 𝒮 theories,
where we have a geometric construction of the perfect category 𝔓𝔢𝔯 Γ <cit.>
as well as a detailed  understanding of the BPS physics <cit.>.
In agreement with this identification, the Grothendieck group K_0(𝔓𝔢𝔯 Γ) is isomorphic to the IR group Λ.
𝔓𝔢𝔯 Γ is not CY, instead the Euler form defines a perfect pairing

    K_0(D^bΓ)⊗ K_0(𝔓𝔢𝔯 Γ)→;


  * the exact functor 𝗋 in eqn.(<ref>) may be seen as the homological (inverse) RG flow.
 




  
Dualities. The (self)-dualities of an 𝒩=2 theory should relate BPS objects to BPS objects of the same kind, and
hence should be (triangle) auto-equivalences of the above categories which are consistent with the functors relating them (e.g. 𝗌, 𝗋 in eqn.(<ref>)). We may
describe the physical situation from different viewpoints.
In the IR picture one would have the putative `duality' group
Aut D^bΓ; however a subgroup acts trivially on all observables <cit.>,
and the physical IR `duality' group is[ For the precise definition of Auteq D^bΓ, see . 5. Aut(Q) is the group of automorphisms of the quiver Q modulo the subgroup which fixes all nodes.]

    𝒮_IR≡Aut D^bΓ/{physically trivial autoequivalences}
    = Auteq D^bΓ⋊Aut(Q).

In the UV (that is, at the operator level) the natural candidate `duality' group is

    𝒮_UV≡Aut 𝒞(Γ)/{physically trivial}


From the explicit description of Aut  D^bΓ (see .5) we learn that 𝒮_IR extends to a group
of autoequivalences of 𝔓𝔢𝔯 Γ which preserve
D^bΓ (by definition). Hence the exact functor 𝗋𝔓𝔢𝔯 Γ→𝒞(Γ) 
in eqn.(<ref>) induces a group homomorphism

    𝒮_IR𝒮_UV,

whose image is

    𝕊= r(Auteq D^bΓ)⋊Aut Q.

𝕊 is a group of auto-equivalences whose action is defined at the operator level, that is, independently of a choice of vacuum.
They are equivalences of the full web of BPS categories in eqn.(<ref>). Thus 𝕊 is the natural candidate for the role of
the (extended) S-duality group of our 𝒩=2 model. Indeed, in the examples where we know the  S-duality group from more 
conventional considerations,
it coincides with our categorical group 𝕊. In this survey we take equation (<ref>) as the definition of the 
S-duality group.

Clearly, the essential part of 𝕊 is the group r(Auteq D^bΓ). It turns out that
precisely this group is an object of central interest in the mathematical literature which provides an 
explicit combinatorial
description of it <cit.>. This combinatorial description is the basis of an algorithm for computer search of S-dualities, see . <ref>. If our 𝒩=2 theory is not too complicated (that is, the ranks of the gauge and flavor group are not too big) the algorithm may be effectively implemented on a laptop, see . <ref> for explicit examples. 

For class 𝒮 theories, the above combinatorial description of S-duality has a nice geometric
intepretation as the (tagged) mapping class group of the Gaiotto surface, in agreement with the predictions of <cit.>(see also <cit.>), see . 5.2. More generally,
for class 𝒮 theories all categorical constructions have a simple geometric realization which makes manifest their physical meaning. 

The IR group 𝒮_IR may be understood in terms of duality walls, see .<ref>. 




  
Cluster characters and vev of line operators. The datum of a Coulomb vacuum u defines a map

    ⟨  - ⟩_u GenOb(𝒞(Γ)) →,

given by taking the vev in the vacuum u of the UV line operator associated to a given generic object of the cluster category
𝒞(Γ).
Physically, the renormalization group implies that the
 map ⟨ - ⟩_u factors through the (Laurent) ring [ℒ] of line operators 
in the effective (Abelian) IR theory. The associated map

    GenOb(𝒞(Γ)) →[ℒ]
 
 is called a cluster character and is well understood in the mathematical literature. Thus the theory of cluster character solves 
(in principle) the problem of computing the vev of arbitrary BPS line operators (see .<ref>).





  
Organization of the paper. The rest of this paper is organized as follows. In section <ref> we review the mathematics of derived DG categories,
cluster categories, and related topics in order to provide the reader with a
language which allows to unify and generalize several previous analysis of the BPS sector of a 𝒩=2 4d theory.
In section <ref> we recall some general physical properties that our categories should enjoy in order to be valid descriptions of the various BPS objects. Here we stress the various notions of charge, with particular reference to the 't Hooft charges in the UV description.
Section <ref> is the core of the paper, where we present our physical interpretation of the categories described mathematically in section <ref> and show that they satisfy all physical requirements listed in section <ref>. In section <ref> we discuss S-duality from the point of view of triangle categories, present first examples and describe the relation to 3d mirror symmetry. In section <ref> we introduce our combinatorial algorithm to find S-dualities and give a number of examples. In section <ref> we consider class 𝒮[A_1] theories and describe all the relevant triangle categories in geometric terms à la Gaiotto. In particular, this shows that our categorical definition of S-duality is indeed equivalent to more conventional physical definitions.
In section <ref> we discuss vev's of line operators from the point of view of cluster characters. Explicit computer codes are presented in the appendices.



§ MATHEMATICAL BACKGROUND


In this section we recall the basic definitions of DG categories <cit.>, 
cluster categories <cit.>, stability conditions for Abelian 
and triangulated categories <cit.> and show some concrete examples. 
We then specialize these definitions to the Ginzburg algebra <cit.> Γ associated
 to a BPS quiver with (super)potential (Q,W) <cit.>. 
 
 Some readers may prefer to skip this section  and refer back to it when looking for definitions and/or details on some mathematical tool used in the main body of this survey.
 


 §.§ Differential graded categories


The main reference for this section is <cit.>.
Let k be a commutative ring,[ In all our physical applications k will be the (algebraically closed) field of complex numbers ℂ.] for example a field or the ring of
integers . We will write ⊗ for the tensor product over k. 
A k-algebra
is a k-module A endowed with a k-linear associative multiplication A ⊗_k A → A
admitting a two-sided unit 1 ∈ A. 

For example, a -algebra is just a (possibly non-commutative)
ring. A k-category  is a “k-algebra with several objects”. Thus, it is the datum of a class of objects obj(), of a k-module
(X, Y ) for all objects X, Y of , and of k-linear associative composition maps

    (Y, Z) ⊗(X, Y ) →(X, Z),     (f, g) ↦ fg

admitting units 1_X ∈(X, X). For example, we can interpret k-algebras as k-categories
with only one object. The category 𝗆𝗈𝖽 A of finitely generated right A-modules over a k-algebra A is
an example of a k-category with many objects. It is also an example of a k-linear
category (i.e. a k-category which admits all finite direct sums).
A graded k-module is a k-module V together with a decomposition indexed by
the positive and the negative integers:

    V =⊕_p∈V^p.

The shifted module V [1] is defined by V [1]^p = V^ p+1, p ∈. A morphism f : V →
V of graded k-modules of degree n is a k-linear morphism such that f (V^p) ⊂ V^p+n
for all p ∈. 


The tensor product V ⊗ W of two graded k-modules V and W is the
graded k-module with components

    (V ⊗ W )^n =⊕_p+q=nV^p ⊗ W^q ,  n ∈.

The tensor product f ⊗ g of two maps f : V → V and g : W → W of graded
k-modules is defined using the Koszul sign rule: we have

    (f ⊗ g)(v ⊗ w) = (-1)^pq f (v) ⊗ g(w)

if g is of degree p and v belongs to V^q.


A graded k-algebra is a graded k-module A
endowed with a multiplication morphism A ⊗ A → A which is graded of degree 0,
associative and admits a unit 1 ∈ A^0.

An “ordinary” k-algebra may be identified with a graded k-algebra concentrated in degree 0. 

A differential graded (=DG) k-module is a -graded k-module V endowed with a
differential d_V, i.e. a map d_V V → V of degree 1 such that d^2_V = 0. Equivalently, V
is a complex of k-modules. The shifted DG module V [1] is the shifted graded module
endowed with the differential -d_V . 

The tensor product of two DG k-modules is the
graded module V ⊗ W endowed with the differential d_V ⊗ 1_W + 1_V ⊗ d_W.

A differential graded k-algebra A is a DG k-module endowed with a multiplication morphism A ⊗ A → A graded 
of degree 0 and associative. Moreover, the differential satisfies the graded Leibnitz rule:

    d(ab)=(da) b+(-1)^(a) a(db),   ∀  a,b∈ A  and  a  homogeneous.


The cohomology of a DG algebra is defined as H^*(A):=ker  d/im d.
Let 𝗆𝗈𝖽 A denote the category of finitely generated  DG modules over the DG algebra A.

The derived category D(A):=D(𝗆𝗈𝖽 A) is the localization
of the category 𝗆𝗈𝖽 A with respect to the class of quasi-isomorphisms. 

Thus, the objects of D(A) are the DG modules and its morphisms are obtained from morphisms of DG
modules by formally inverting all quasi-isomorphisms. The bounded derived category of 𝗆𝗈𝖽 A, denoted D^bA, 
is the triangulated subcategory of D(A) whose objects are quasi-isomorphic to objects with bounded cohomology.

The perfect derived category of a DG algebra A, 𝔓𝔢𝔯 A, is the smallest full triangulated subcategory of D(A) containing
A which is stable under taking shifts, extensions and direct summands.



 §.§ Quivers and mutations


In this section we follow <cit.>. Let k be an algebraically closed field. 

A (finite) quiver Q is a (finite) oriented graph (possibly with loops and 2-cycles). We denote its set of vertices by Q_0 and its
set of arrows by Q_1. For an arrow a of Q, let s(a) denote its source node and t(a) denote
its target node. The lazy path corresponding to a vertex i will be denoted by e_i.


The path algebra kQ̂ is the associative unital algebra whose 
elements are finite compositions of arrows of Q, where the composition of a,b ∈ Q_1 is denoted ab and 
it is nonzero iff s(b)=t(a). The complete path algebra kQ is the completion of the path algebra with 
respect to the ideal I generated by the arrows of Q. 

Let I be the ideal of kQ generated by
the arrows of Q. A potential W on Q is an element of the closure of the space
generated by all non trivial cyclic paths of Q. We say two potentials are cyclically
equivalent if their difference is in the closure of the space generated by all differences
a_1 ... a_s - a_2 ... a_sa_1, where a_1 ... a_s is a cycle.

Let u,p and v be nontrivial paths of Q such that c=upv is a nontrivial cycle. For the path p of Q, we define 
    ∂_p : kQ → kQ

as the unique continuous linear
map which takes a cycle c to the sum ∑_
c=upv vu taken over all decompositions of the
cycle c (where u and v are possibly lazy paths).

Obviously two cyclically equivalent
potentials have the same image under ∂_p. If p = a is an arrow of Q, we call ∂_a the
cyclic derivative with respect to a.
Let W be a potential on Q such that W is in I^2 and no two cyclically equivalent
cyclic paths appear in the decomposition of W. Then the pair (Q, W)
is called a quiver with potential. 

Two quivers with potential (Q, W) and (Q', W')
are right-equivalent if Q and Q' have the same set of vertices and there exists an
algebra isomorphism ϕ : kQ → kQ' whose restriction on vertices is the identity
map and ϕ(W) and W'
are cyclically equivalent. Such an isomorphism ϕ is called a
right-equivalence.


The Jacobian algebra of a quiver with potential (Q, W), denoted by J(Q, W), is
the quotient of the complete path algebra kQ by the closure of the ideal generated by
∂_a W, where a runs over all arrows of Q:

    J(Q,W):=kQ/∂_aW.

We say that the quiver with potential (Q,W) is Jacobi-finite if the Jacobian algebra 
J(Q,W) is finite-dimensional over k.

It is clear that two right-equivalent quivers
with potential have isomorphic Jacobian algebras. A quiver with potential is called
trivial if its potential is a linear combination of cycles of length 2 and its Jacobian
algebra is the product of copies of the base field k. 




  §.§.§ Quiver mutations

Let (Q, W) be a quiver with potential. Let i∈ Q_0 a vertex. Assume the following conditions:


  * the quiver Q has no loops;

  * the quiver Q does not have 2-cycles at i;

We define a new quiver with potential μ̃_i(Q, W) = (Q', W') as
follows. The new quiver Q' is obtained from Q by


  * For each arrow β with target i and each arrow α with source i, add a new
arrow [αβ] from the source of β to the target of α.

  * Replace each arrow α with source or target i with an arrow α^* in the opposite
direction.

If we represent the quiver with its exchange matrix B_ij, i.e.
 the matrix such that 
 
    B_ij=#{ arrows from  i  to  j}-#{ arrows from  j  to  i}
 
then the transformation that B_ij undergoes is

    B'_ij=
    - B_ij,     i=k  or j=k
    
    B_ij+max[-B_ik,0] B_kj+B_ikmax[B_kj,0]    otherwise.

The new potential W' is the sum of two potentials W'_1 and W'_2. The potential W'_1
is obtained from W by replacing each composition αβ by [αβ], where β is an arrow
with target i. The potential W'_2 is given by <cit.>

    W'_2 =∑_α,β[αβ]β^*α^*,

the sum ranging over all pairs of arrows α and β such that β ends at i and α
starts at i.

Let I be the ideal in kQ generated by all arrows. Then, a quiver
with potential is called reduced if ∂_aW is contained in I^2 for all arrows a of Q.

One shows that all quivers with potential (Q,W) are right-equivalent to the direct sum of a 
reduced quiver with potential and a trivial one.[ In terms of the corresponding SQM system, the process
of replacing the pair (Q,W) by its reduced part (Q_red.,W_red.) corresponds to integrate away
the massive Higgs bifundamentals.]

We can now give the definition of the mutated quiver: we define μ_i(Q, W) as the reduced part 
of μ̃_i(Q, W), and call μ_i the mutation at the vertex i. An example will clarify all these concepts.
[A_3 quiver]
Consider the quiver A_3 given by Q: ∙_1 α←∙_2 β→∙_3 with W=0. 
Let us consider the quiver μ_1(Q):  ∙_1 α^*→∙_2 β→∙_3 with W=0. 
Now apply the mutation at vertex 2: we get

           ∙_2  [dl]_α      

            ∙_1  [rr]^[αβ]          ∙_3 [ul]_β^*  

     μ_2(μ_1(Q))    

with potential W=αβ^*[αβ].

We conclude this subsection with the following 
-12pt


  * The right-equivalence class of μ̃_i(Q, W) is
determined by the right-equivalence class of (Q, W).

  * The quiver with potential μ̃^2_i(Q, W) is right-equivalent to the direct sum
of (Q, W) with a trivial quiver with potential.

  * The correspondence μ_i acts as an involution on the right equivalence
classes of reduced quivers with potential.







 §.§ Cluster algebras

We follow <cit.>. Let Q be a 2-acyclic quiver
with vertices 1, 2, ..., n, and let F = (x_1, ... , x_n) be the function field in n indeterminates
over . Consider the pair (x⃗, Q), where x⃗ = {x_1,... , x_n}. The cluster
algebra C(x⃗, Q) will be defined to be a subring of F.


The pair (x⃗, Q) consisting of a transcendence basis x⃗ for F over the rational numbers
, together with a quiver with n vertices, is called a seed. For i = 1, ... , n we
define a mutation μ_i taking the seed (x⃗, Q) to a new seed (x⃗', Q'), where Q' = μ_i(Q)
as discussed in <ref>, and x⃗' is obtained from x⃗ by replacing x_i by a new element x'_i in
F. Here x'_i is defined by 
    x_ix'_i = m_1 + m_2,

where m_1 is a monomial in the variables
x_1, ... , x_n, where the power of x_j
is the number of arrows from j to i in Q, and m_2 is the
monomial where the power of x_j
is the number of arrows from i to j. (If there is no
arrow from j to i, then m_1 = 1, and if there is no arrow from i to j, then m_2 = 1.)
Note that while in the new seed the quiver Q' only depends on the quiver Q, then
x' depends on both x and Q. We have 
    μ^2_i(x⃗, Q) = (x⃗, Q).

The procedure to get the full cluster algebra is iterative. We perform this mutation operation for all i = 1, ... , n, then we perform it on the new seeds and so on. Either we get new seeds or we get back one of the seeds already computed. The n-element subsets
x⃗, x⃗', x⃗”, ... occurring are by definition the clusters, the elements in the clusters are
the cluster variables, and the seeds are all pairs (x⃗', Q') occurring in the iterative procedure. The corresponding
cluster algebra C(x⃗, Q), which as an algebra only depends on Q, is the subring of F
generated by the cluster variables.
 Let Q be the quiver 1 → 2 → 3 and x⃗ = {x_1, x_2, x_3}, where
x_1, x_2, x_3 are indeterminates, and F = (x_1, x_2, x_3). We have μ_1(x⃗, Q) = (x', Q'),
where Q' = μ_1(Q) is the quiver 1 ← 2 → 3 and x⃗' = {x'_1, x_2, x_3}, where x_1x'_1 = 1+x_2,
so that x'_1 =1+x_2/x_1. And so on.
The clusters are: 
    {x_1, x_2, x_3},{1+x_2/x_1, x_2, x_3},{ x_1, x_1+x_3/x_2, x_3},{x_1, x_2,1+x_2/x_3},
 

    {1+x_2/x_1,x_1+(1+x_2)x_3/x_1x_2, x_3},{1+x_2/x_1, x_2,1+x_2/x_3},{x_1+(1+x_2)x_3/x_1x_2,x_1+x_3/x_2, x_3},


    {x_1,x_1+x_3/x_2, (1+x_2)x_1+x_3/x_2x_3}, {x_1,(1+x_2)x_1+x_3/x_2x_3, 1+x_2/x_3},


    {1+x_2/x_1, x_1+(1+x_2)x_3/x_1x_2, (1+x_2)x_1+(1+x_2)x_3/x_1x_2x_3},{1+x_2/x_1, (1+x_2)x_1+(1+x_2)x_3/x_1x_2x_3,1+x_2/x_3},
 
    {x_1+(1+x_2)x_3/x_1x_2, x_1+x_3/x_2, (1+x_2)x_1+(1+x_2)x_3/x_1x_2x_3},
 
    {(1+x_2)x_1+(1+x_2)x_3/x_1x_2x_3,x_1+x_3/x_2,(1+x_2)x_1+x_3/x_2x_3},


    {(1+x_2)x_1+(1+x_2)x_3/x_1x_2x_3,(1+x_2)x_1+x_3/x_2x_3, 1+x_2/x_3},
 
and the cluster variables are: 
    x_1,x_2, x_3,1+x_2/x_1,x_1+x_3/x_2,1+x_2/x_3,x_1+(1+x_2)x_3/x_1x_2,(1+x_2)x_1+x_3/x_2x_3,(1+x_2)x_1+(1+x_2)x_3/x_1x_2x_3.




  §.§.§ The cluster exchange graph (CEG)

If Q'
is a quiver mutation equivalent to Q, then the cluster algebras C(Q') and
C(Q) are isomorphic. The n-regular connected graph whose vertices are the seeds of C(x⃗,Q) (up to simultaneous renumbering of rows, columns and variables) and whose edges connect the seeds related by a single mutation is called cluster exchange graph (=CEG). The CEG for Example <ref> is represented in figure <ref>.





 §.§ Ginzburg DG algebras

Given a quiver Q with potential W, we can associate to it the Jacobian algebra J(Q,W):=kQ/∂ W, where kQ 
is the quiver path algebra (see section <ref>). It is also possible to extend the path algebra kQ to a DG algebra: the Ginzburg algebra.
 (Ginzburg <cit.>). Let (Q, W) be a quiver with potential. Let Q̂ be the graded quiver
with the same set of vertices as Q and whose arrows are:


  * the arrows of Q (of degree 0);

  * an arrow a^*: j → i of degree -1 for each arrow a : i → j of Q;

  * a loop t_i: i → i of degree -2 for each vertex i ∈ Q_0.

The completed Ginzburg DG algebra Γ(Q, W) is the DG algebra whose underlying
graded algebra is the completion[ The completion is taken with respect to the I-adic topology, 
where I is the ideal of the path algebra generated by all arrows of the quiver.] of the graded
path algebra kQ̂. The differential of Γ(  Q, W) is the unique continuous linear
endomorphism homogeneous of degree 1 which satisfies the Leibniz rule (i.e.
d(uv) = (du)v + (-1)^p udv for all homogeneous u of degree p and all v), and takes
the following values on the arrows of Q̂:

    d(a)    = 0
     d(a^*)    = ∂_aW,       ∀ a ∈ Q_1;
    
     d(t_i)    = e_i(∑_a∈ Q_1[a, a^*])e_i,       ∀ i ∈ Q_0.

We shall write Γ(Q,W) simply as Γ, unless we wish to stress its dependence on (Q,W).

From the definition of Γ and d, one sees that H^0Γ≅ J(Q,W).


To the DG algebra Γ we  associate three important triangle categories which we are now going to define and analyze in detail.



 §.§ The bounded and perfect derived categories


The DG-category 𝗆𝗈𝖽 Γ is the category whose objects are finitely 
generated graded Γ-modules and the morphisms spaces have the structure of DG modules 
(cfr. section <ref>).
 The derived category DΓ:=D(𝗆𝗈𝖽 Γ) 
<cit.> is the localization of 𝗆𝗈𝖽 Γ 
at quasi-isomorphisms (the cohomology structure is given by the differential d of the Ginzburg algebra). 
Thus, the objects of DΓ are DG modules. There are two fundamental subcategories associated to DΓ:


  * The bounded derived category D^b Γ: it is the full subcategory of DΓ such that its objects 
are graded modules M for which, given a certain N>0, H^n (M)=0 for all |n| > N.  This category is 3-CY (see below).

  * The perfect derived category 𝔓𝔢𝔯 Γ, i.e. the smallest full triangulated subcategory 
of DΓ which contains Γ and is closed under extensions, shifts in degree and taking direct summands. 
 
Both 𝔓𝔢𝔯 Γ and D^b Γ are triangulated subcategories of DΓ and in particular,
 𝔓𝔢𝔯 Γ⊃ D^b Γ as a full subcategory (as explained in <cit.>). 
Furthermore,[ See <cit.> for more details.] the category D^bΓ 
has finite-dimensional morphism spaces (even its graded
morphism spaces are of finite total dimension) and is 3-Calabi-Yau (3-CY), by which we
mean that we have bifunctorial isomorphisms[ More generally, we say that a triangle category is ℓ-CY (for ℓ∈ℕ) iff we have the bifunctorial isomorphism
DHom(X,Y)≅Hom(Y,X[ℓ]).]

    D Hom(X, Y ) ≅Hom(Y, X[3]) ,

where D is the duality functor Hom_k(-, k) and [1] the shift functor. The simple
J(Q, W)-modules S_i can be viewed as Γ-modules via the canonical morphism

    Γ→ H^0(Γ).
 
[A_2 quiver]
Consider the A_2 quiver 1→ 2. The following is a graded indecomposable Γ-module:

    t_1^*  (k[-1]⊕ k[-3]) a^*a⇌ k  t_2^*,

where a=0, a^*:k1↦k[-1], t_2^*=0, and t_1^*:k[-1]1↦ k[-3]. 
This object can be generated from S_1[-1], S_1[-3] and S_2 by successive extensions. Moreover, the modules S_i, i =1,2 
and their shifts are enough to generate[ In the triangulated category 𝒯, a set of objects S_i ∈𝒯 
is a generating set if all objects of 𝒯 can be obtained from the generating set via an iterated cone construction.] all
 (homologically finite) graded modules.




  §.§.§ Seidel-Thomas twists and braid group actions
 Simple Γ-modules S_i become 3-spherical objects in D^bΓ (hence also in DΓ), that is,
 
    Hom(S,S[j])≅ k (δ_j,0+δ_j,3).

 They yield the
Seidel-Thomas <cit.> twist functors T_S_i. These are autoequivalences of DΓ such that 
each object X fits into a triangle

    Hom^∙_D(S_i, X) ⊗_k S_i → X → T_S_i(X) → .

By construction, T_S_i restricts to an autoequivalence of the subcategory D^bΓ⊂ DΓ. From the explicit realization of T_S_i as a cone
in DΓ, eqn.(<ref>), it is also clear that
it restricts to an auto-equivalence of 𝔓𝔢𝔯 Γ.



As shown in <cit.>, the twist functors give rise to a (weak) action on DΓ of the braid group
associated with Q, i.e. the group with generators σ_i, i ∈ Q_0, and relations 
    σ_iσ_j =
    σ_jσ_i

if i and j are not linked by an arrow in Q and

    σ_iσ_jσ_i = σ_jσ_iσ_j

if there is exactly one arrow between i and j (no relation if there are two or more
arrows). 


We write Sph(D^bΓ)⊂Aut(D^bΓ) for the subgroup of autoequivalences
 generated by the Seidel-Thomas twists associated to all simple objects S_i∈ D^bΓ.

 


  §.§.§ The natural t-structure and the canonical heart
 
The category DΓ admits a natural t-structure whose truncation functors are
those of the natural t-structure on the category of complexes of vector spaces
(because Γ is concentrated in degrees ≤ 0). Thus, we have an induced natural
t-structure on D^bΓ. Its heart  is canonically equivalent to the category
𝗇𝗂𝗅 J(Q, W) of nilpotent modules[ If (Q,W) is Jacobi-finite (as in our applications), 𝗇𝗂𝗅 J(Q, W)≡𝗆𝗈𝖽  J(Q, W).] <cit.>. In particular, the inclusion of  into D^bΓ induces an isomorphism
of Grothendieck groups

    K_0() ≅ K_0(D^bΓ)≅⊕_i  [S_i].





  
The skew-symmetric form.
Notice that the lattice K_0(D^bΓ) carries the canonical Euler form defined by

    X,Y=∑_i=0^3(-1)^iHom_D(Γ)(X,Y[i]).

It is skew-symmetric thanks to the 3-Calabi-Yau property (<ref>). Indeed it follows from the
Calabi-Yau property and from the fact that Ext^i_(L, M) = Hom_D^bΓ(L, M[i]) for i = 0
and i = 1 (but not i > 1 in general) that for two objects L and M of ⊂ D^bΓ, we have

    L, M = Hom(L, M) - Ext^1(L, M) + Ext^1(M, L) - Hom(M, L).

Since the dimension of Ext^1(S_i, S_j ) equals the number of arrows in Q from j to
i (Gabriel theorem <cit.>), we obtain that the matrix of -,- in the basis of the simples of  has its 
(i, j)-coefficient equal to the number of arrows from i to j minus the number of arrows
from j to i in Q, that is,
 (cfr. eqn.(<ref>))
 
    ⟨ S_i,S_j⟩= B_ij.




  §.§.§ Mutations at category level

The main reference for this subsection is <cit.>. Let k be a vertex of the quiver Q not 
lying on a 2-cycle and let (Q', W') be the
mutation of (Q, W) at k. Let Γ' be the Ginzburg algebra
associated with (Q', W'). Let ' be the canonical heart in D^bΓ'. There are
two canonical equivalences 

    DΓ' → DΓ

given by functors Φ^± related by

    T_S_k∘Φ^- →Φ^+.

where, again, T_S_k is the Seidel-Thomas twist generated by the spherical object S_k.
If we put P_i = Γ e_i, i ∈ Q_0, and similarly for Γ', then both Φ^+ and Φ^- send P'_i to
P_i for i ≠ k; the images of P'_k under the two functors fit into triangles

    P_k →⊕_k → i P_i →Φ^-(P'_k) →

and

    Φ^+(P'_k) →⊕_j→ k P_j → P_k .

The functors Φ^± send ' onto the hearts μ_k^±() of two new t-structures. These
can be described in terms of  and the subcategory[ Here and below, given a (collection of) object(s) 𝒪 of a linear category 𝔏, by 𝖺𝖽𝖽 𝒪 we mean the additive closure of 𝒪 in 𝔏, that is, the full subcategory over the direct summands of finite direct sums of copies of 𝒪.] 
𝖺𝖽𝖽 S_k as follows:
Let S^⊥_k be the right orthogonal subcategory of S_k in [ Its objects are those M's
with Hom(S_k, M) = 0. It is a full subcategory of .]. Then μ^+_k() is formed by the objects X of D^bΓ such that
the object H^0(X) belongs to S^⊥_k, the object H^1(X) belongs to 𝖺𝖽𝖽 S_k and H^p(X)
vanishes for all p ≠  0, 1. Similarly, the subcategory μ^-_k() is formed by the objects
X such that the object H^0(X) belongs to the left orthogonal subcategory ^⊥ S_k, the
object H^-1(X) belongs to 𝖺𝖽𝖽 S_k and H^p(X) vanishes for all p ≠ -1, 0. The
subcategory μ^+_k() is the right mutation of  and μ^-_k() is its left mutation.
By construction, we have

    T_S_k(μ^-_k()) = μ^+_k().

Since the categories  and μ^±() are hearts of bounded, non degenerate t-structures
on D^bΓ, their Grothendieck groups identify canonically with that of D^bΓ. They
are endowed with canonical basis given by the simples. Those of  identify with
the simples S_i, i ∈ Q_0, of 𝗇𝗂𝗅 J(Q, W). The simples of μ^+_k() are S_k[-1], the
simples S_i of  such that Ext^1(S_k, S_i) vanishes and the objects T_S_k(S_i) where
Ext^1(S_k, S_i) is of dimension ≥ 1. By applying T^-1_S_k to these objects we obtain the
simples of μ^-_k().


We saw that D^bΓ⊂𝔓𝔢𝔯 Γ as a full subcategory <cit.>: 
what is then the meaning of the Verdier quotient <cit.> of these two triangulated categories? 




 §.§ The cluster category


The next result is the main step in the construction
of new 2-CY categories with cluster-tilting object which generalize the acyclic
cluster categories introduced by Buan-Marsh-Reineke-Todorov to categorify the cluster agebras of Fomin and Zelevinski.
 Let A be a DG-algebra with the following properties:


  * A is homologically smooth (i.e. A∈𝔓𝔢𝔯(A⊗ A^op)),

  * H^p(A) = 0 for all p ≥ 1,

  * H^0(A) is finite dimensional as a k-vector space,

  * A is bimodule 3-CY, i.e. 

    Hom_D(A)(X, Y ) ≅ DHom_D(A)(Y, X[3]),

for any X ∈ D(A) and Y ∈ D^b(A).

Then the triangulated category 

    𝒞(A) = 𝔓𝔢𝔯 A/D^b A

 is Hom-finite, 2-CY, i.e.
 
    Hom_𝒞(A)(X, Y ) ≅ DHom_𝒞(A)(Y, X[2]),     X,Y∈𝒞(A).

 and the
object A is a cluster-tilting object[ See Definition <ref>.] with 

    End_𝒞(A)(A) ≅ H^0(A).
















The category 𝒞(A) is called the generalized cluster category and 
it reduces to the standard cluster category <cit.> in the acyclic case. 
It is triangulated since it is the Verdier quotient of triangulated 
categories.[ The main references for these categorical facts are <cit.>.
We recall the definition of Verdier quotient of triangle categories:

Lemma.
Let D be a triangulated category. Let D' ⊂ D be a full triangulated subcategory. Let S⊂𝖬𝗈𝗋(D)
be the subset of morphisms such that there exists a distinguished triangle (X,Y,Z,f,g,h)∈ D with Z
isomorphic to an object of D^'. 
Then S is a multiplicative system compatible with the triangulated structure on D.

Definition.
 Let D be a triangulated category. Let B be a full triangulated subcategory.
 We define the (Verdier) quotient category D/B by the formula D/B=S^-1D, where S is the multiplicative 
system of D associated to B via the previous lemma.]




  §.§.§ The case of the Ginzburg algebra of (Q,W)
 In particular, we may specialize to the case where A=Γ, i.e. the Ginzburg algebra of a quiver with 
potential (Q,W), and write the following sequence:

    0 → D^bΓ𝔓𝔢𝔯 Γ𝒞(Γ) → 0

the above theorem states that this sequence is exact and 𝗋(Γ)=T, where T is the canonical cluster-tilting 
object[ See Definition <ref>.] of 𝒞(Γ). The first map in eqn.(<ref>) is the
 inclusion map: see <cit.> for details. 

Moreover, an object M∈𝔓𝔢𝔯 Γ belongs to the subcategory D^bΓ if and only if
the space Hom_𝔓𝔢𝔯 Γ(P, M) is finite-dimensional for each P∈𝔓𝔢𝔯 Γ. 
In particular, this implies that there is a duality between the simple objects S_i ∈ D^bΓ 
and the projective objects Γ e_i ∈𝔓𝔢𝔯 Γ

    ⟨Γ e_i, S_j⟩=δ_ij.



 The completed Ginzburg DG algebra Γ(  Q, W) is
homologically smooth and bimodule 3-Calabi-Yau.

We have already shown that Γ(Q, W) is non zero only in negative degrees, and
that H^0(Γ(  Q, W)) ≅  J(Q, W). Therefore by the theorem above we get the following

Let (Q, W) be a Jacobi-finite quiver with potential. Then the category

    𝒞(Γ(Q,W)):= 𝔓𝔢𝔯 Γ(Q,W)/D^b(Γ(  Q, W))

is Hom-finite, 2-Calabi-Yau, and has a canonical cluster-tilting object[ See Definition <ref>.] whose endomorphism
algebra is isomorphic to J(Q, W).


We shall write 𝒞(Γ(Q,W)) simply as 𝒞(Γ) leaving (Q,W) implicit.



  §.§.§ The cluster category of a hereditary category


The above structure simplifies in the case of a cluster category arising from a hereditary (Abelian) category ℋ (with a Serre functor and a tilting object) <cit.>. Physically this happens for the following list of complete 𝒩=2 QFTs <cit.>: i) Argyres-Douglas of type ADE, ii) asymptotically-free SU(2) gauge theories coupled to fundamental quarks and/or Argyes-Douglas models of type D, and iii) SCFT SU(2) theories with the same kind of matter. In terms of quiver mutations classes, they correspond (respectively) to ADE Dynkin quivers of the finite, affine, and elliptic type[ In the elliptic type we are restricted to the four types D_4, E_6, E_7 and E_8, corresponding to the four tubular weighted projective lines <cit.>.
Elliptic D_4 is SU(2) with N_f=4 <cit.>.] <cit.>. In all these case we have
an hereditary (Abelian) category ℋ, with the Serre functor S=τ [1] where τ is the Auslander-Reiten translation. That is, in their derived category we have

    Hom_D^b(ℋ)(X,Y)≅ DHom_D^b(ℋ)(Y,τ X[1])

τ is an auto-equivalence of
D^b(ℋ). The cluster category can be shown to be equivalent to the orbit category
<cit.>

    𝒞(ℋ)≅ D^b(ℋ)/⟨τ^-1[1]⟩^.
 
For future reference, we list
the relevant categories ℋ
(further details may be found in <cit.>):


  * For Argyres-Douglas of type ADE, we have ℋ≅𝗆𝗈𝖽 k 𝔤⃗, where 𝔤⃗ is a quiver obtained by choosing an orientation to the Dynkin graph of type 𝔤∈ ADE (all orientations being derived-equivalent). τ satisfies the equation (for more refined results see <cit.>)

    τ^h=[-2],

where h is the Coxeter number of the associated Lie algebra 𝔤;

  * for SU(2) gauge theories coupled to Argyres-Douglas systems of types[ In our conventions, p_i=1 means the empty matter system, while p_i=2 is a free quark doublet.] D_p_1,⋯,D_p_s,
we have ℋ=𝖼𝗈𝗁 𝕏(p_1,…,p_s), the coherent sheaves over a weighted projective line of weights (p_1,…, p_s) <cit.>[ For a review of the category of coherent sheaves on weighted projective lines and corresponding cluster categories  from a physicist prospective, see <cit.>.]. τ acts by multiplication by the canonical sheaf ω, and hence is periodic iff ω=0; in general, ω is minus the Euler characteristic of 𝕏(p_1,…,p_s), χ=2-∑_i(1-1/p_i).
However, τ is always periodic of period lcm(p_i) when restricted to the zero rank sheaves (`skyskrapers' sheaves).












 §.§ Mutation invariance

We have already stated that mutations correspond to Seiberg-like dualities. Therefore, our categorical construction makes sense only if it is invariant by mutations: indeed, we do not want the categories representing the physics to change when we change the mathematical description of the same dynamics.


The following two results give a connection between the DG categories we just analyzed and quivers with potentials linked by mutations.
 Let (Q, W) be a quiver with potential without loops and i ∈ Q_0 not on a 2-cycle in
Q. Denote by Γ := Γ(  Q, W) and Γ':= Γ(  μ_i(Q, W)) the completed Ginzburg DG algebras.


  * <cit.> There are triangle equivalences

    𝔓𝔢𝔯  Γ[r]^∼    𝔓𝔢𝔯 Γ^'

                     D^bΓ[r]^∼@^(->[u]    D^bΓ^'@^(->[u] 

Hence we have a triangle equivalence 𝒞(Γ) ≅𝒞(Γ').

  * <cit.>  We have a diagram

    𝔓𝔢𝔯  Γ[rrr]^∼[d]^H^0            𝔓𝔢𝔯  Γ^'[d]^H^0

                    𝗆𝗈𝖽  J(Q,W) @<.>[rrr]^mutation           𝗆𝗈𝖽  J(μ_i(Q,W)) 






 Let 𝒞 be a Hom-finite triangulated category. An object T ∈𝒞 is
called cluster-tilting (or 2-cluster-tilting) if T is basic (i.e. with pairwise non-isomorphic direct summands) and if we have

    𝖺𝖽𝖽 T = {X ∈𝒞 | Hom_𝒞(X, T [1]) = 0} = {X ∈𝒞 | Hom_𝒞(T, X[1]) = 0}.

Note that a cluster-tilting object is maximal rigid (the converse is not always true, see
<cit.>), and that the second equality in the definition always holds when 𝒞 is
2-Calabi-Yau.

If there exists a cluster-tilting object in a 2-CY category 𝒞, then it is possible
to construct others by a recursive process resumed in the following:
 Let 𝒞 be a Hom-finite 2-CY triangulated category
with a cluster-tilting object T. Let T_i be an indecomposable direct summand
of T ≅ T_i ⊕ T_0. Then there exists a unique indecomposable T^*_i non isomorphic
to T_i such that T_0 ⊕ T^*_i is cluster-tilting. Moreover T_i and T^*_i are linked by the
existence of triangles

    T_i u→Bv→T^*_i w→ T_i[1]   and   T^*_iu'→B'v'→T_iw'→T^*_i[1]

where u and u' are minimal left 𝖺𝖽𝖽 T_0-approximations and v and v' are minimal
right 𝖺𝖽𝖽 T_0-approximations.

These triangles allow to make a mutation of the cluster-tilting object: they are called IY-mutations.
 Let 𝒞 be a 2-CY triangulated category with
a cluster-tilting object T. Then the functor

    F_T = Hom_𝒞(T, -)𝒞→𝗆𝗈𝖽 End_𝒞(T)

induces an equivalence 

    𝒞/𝖺𝖽𝖽 T [1] ≅𝗆𝗈𝖽 End_𝒞(T ).

If the objects T and T^' are linked by an IY-mutation, then the categories
𝗆𝗈𝖽 End_𝒞(T ) and 𝗆𝗈𝖽 End_𝒞(T^') are nearly Morita equivalent, that is, there exists
a simple End_𝒞(T)-module S, and a simple End_𝒞(T^')-module S^', and an equivalence
of categories 

    𝗆𝗈𝖽 End_𝒞(T)/𝖺𝖽𝖽 S ≅𝗆𝗈𝖽 End_𝒞(T^')/𝖺𝖽𝖽  S^'.

Moreover, if X has no direct summands in 𝖺𝖽𝖽 T[1], then F_TX is projective (resp. injective) if and only if X lies in 𝖺𝖽𝖽 T (resp. in 𝖺𝖽𝖽 T[2]).

Thus, from Theorem <ref> and the above Proposition, we get that in the Jacobi-finite
case, for any cluster-tilting object T ∈𝒞(Γ) which is IY-mutation equivalent to
the canonical one, we have:

         𝒞(Γ) [dl]_F_T^'[dr]^F_T    

                    𝗆𝗈𝖽  End_𝒞(Γ)(T) @<.>[rr]^mutation        𝗆𝗈𝖽 End_𝒞(Γ)(T^')




 §.§ Grothendieck groups,
skew-symmetric pairing, and the index




  §.§.§ Motivations from physics



In a quantum theory there are two distinct notions of `quantum numbers': the  quantities which are conserved in all physical processes and, on the other hand, the numbers which are used to label (i.e. to distinguish) states and operators.  
If a class of BPS objects is described (in a certain physical set-up) by the triangle category 𝔗, 
these two notions of `quantum numbers' get identified as follows:


  * conserved quantities:  numerical invariants of objects X∈𝔗 which only depend on their Grothendieck class [X]∈ K_0(𝔗).[ In general, the conserved quantum numbers take value in the numeric Grothendieck group K_0(𝔗)_num. For the categories we consider in this paper, the Grothendieck group is a finitely generated Abelian group and the two groups coincide.] This is the free Abelian group over the isoclasses of objects of 𝔗 modulo the relations given by distinguished triangles of 𝔗;

  * labeling  numbers: correspond to  numerical invariants of the objects X∈𝔗 which are well-defined, that is, depend only on its isoclass (technically, on their class in the split-Grothendieck group).


Of course, conserved quantities are in particular labeling  numbers.
Depending on the category 𝔗 there may be or not be   enough conserved quantities K_0(𝔗) to label all the relevant BPS objects.


In the categorical approach to the BPS sector of a supersymmetric  theory, the basic problem takes the form:

Given a class of BPS objects A in a specified physical set-up, determine the corresponding triangulated category 𝔗_A.
 

The Grothendieck group is a very handy tool to solve this Problem. Indeed, the BPS objects of A carry certain conserved quantum numbers which satisfy a number of physical consistency requirements. The allowed quantum numbers take value in an Abelian group 𝖠𝖻_A, and the consistency requirements endow the group with some extra mathematical structures. Both the group 𝖠𝖻_A and the extra structures on it are known from physical considerations (we shall review the ones of interest in . <ref>). Then suppose we have a putative solution 𝔗_A of the above problem. We compute its Grothendieck group; if K_0(𝔗_A)≇𝖠𝖻_A, we can rule out
𝔗_A as a solution of the above Problem. Even if 
K_0(𝔗_A)≅𝖠𝖻_A, but K_0(𝔗_A) is not naturally endowed with the required extra structures, we may rule out 𝔗_A. On the other hand, if we find that K_0(𝔗_A)≅𝖠𝖻_A and the Grothendieck group is canonically equipped with the physically expected structures, we gain confidence on the proposed solution, especially if the requirements on K_0(𝔗_A) are quite restrictive.  

Therefore, as a preparation for the discussion of their physical interpretation in section 4, we need to analyze in detail the Grothendieck groups of the three triangle categories D^bΓ, 𝔓𝔢𝔯 Γ, or 𝒞(Γ). These  categories are related by the functors 𝗌, 𝗋 which, being exact, induce group homomorphisms between the corresponding Grothendieck groups.  In all three cases K_0(𝔗) is a finitely generated Abelian group carrying additional structures;
later in the paper we shall compare this structures with the one required by quantum physics. 



  §.§.§ The lattice K_0(D^bΓ) and the skew-symmetric form

The group K_0(D^bΓ) is easy to compute using the following
 The Abelian category 𝗇𝗂𝗅 J(Q,W) is the heart of a bounded t-structure in D^bΓ.

Hence, since we assume (Q,W) to be Jacobi-finite, 𝗇𝗂𝗅 J(Q,W)≅𝗆𝗈𝖽 J(Q,W) and

    K_0(D^bΓ)≃ K_0( J(Q,W))

 is isomorphic to the free Abelian group over the isoclasses [S_i] of the simple Jacobian modules S_i, that is,  K_0(D^bΓ)≅^n
(n being the number of nodes of Q).

 D^bΓ is 3-CY, and then the lattice K_0(D^bΓ) is equipped with an intrinsic skew-symmetric pairing given by the Euler characteristics, see discussion around eqn.(<ref>). This pairing has an intepretation in terms of modules of the Jacobian algebra B≡ J(Q,W)≅End_𝒞(Γ)(Γ).


 Let  X,Y∈𝗆𝗈𝖽 B. Then the  form

    ⟨ X, Y⟩_a= 
    Hom(X,Y)-Ext^1(X,Y)-Hom(Y,X)+Ext^1(Y,X)

descends to an antisymmetric form on K_0(𝗆𝗈𝖽 B). Its matrix in the basis of simples {S_i} is the exchange matrix B of the quiver Q (cfr. eqn.(<ref>)).



In conclusion, for the 3-CY category D^bΓ, the Grothendieck group is a rank n lattice equipped with a skew-symmetric bilinear form ⟨-,-⟩.
We shall refer to the radical of this form as the flavor lattice
Λ_flav=rad ⟨-,-⟩.

 



  §.§.§ K_0(𝔓𝔢𝔯 Γ)≅ K_0(D^bΓ)^∨


More or less by definition, K_0(𝔓𝔢𝔯 Γ) is the free Abelian group over the classes [Γ_i] of indecomposable summands of Γ. Since the general perfect object has infinite homology, there is no well-defined Euler bilinear form. However,
eqn.(<ref>) implies that for X∈𝔓𝔢𝔯 Γ, Y∈ D^bΓ,

    Hom_𝔓𝔢𝔯(X,Y[k])=Hom_𝔓𝔢𝔯(Y,X[k])=0  for k<0 or k>3

and hence we have a Euler pairing

    K_0(𝔓𝔢𝔯 Γ)× K_0(D^bΓ)→,

under which

    ⟨Γ_i, S_j⟩=-⟨ S_j, Γ_i⟩=δ_ij.

Thus [S_i] and [Γ_i] are dual basis and both Grothendieck groups are free (i.e. lattices) of rank n. Then we have two group isomorphisms

    ^n→ K_0(D^bΓ)       (m_1,m_2,…,m_n) ⟼⊕_i=1^n m_i [S_i]
       K_0(𝔓𝔢𝔯 Γ)→^n
           [X]⟼(⟨ X, S_1⟩,⟨ X, S_2⟩,…,
    ⟨ X, S_n⟩).
 
The image of K_0(D^bΓ) inside K_0(𝔓𝔢𝔯 Γ)≅ Z^n is isomorphic to the image of B^n→^n where B is the exchange matrix of the quiver Q.[ Note that this image is invariant under quiver mutation.]
We have the obvious isomorphism

    K_0(𝔓𝔢𝔯 Γ)≅ K_0(𝖺𝖽𝖽 Γ).






  §.§.§ The structure of K_0(𝒞(Γ))


From the basic exact sequence of categories (<ref>) we get

    0 [r]     K_0(D^b Γ) [r]^s     K_0(𝔓𝔢𝔯 Γ) [r]^r     K_0(𝒞(Γ))[r]     0

hence 

    K_0(𝒞(Γ))≅^n/B·^n.

K_0(𝒞(Γ)) is not a free Abelian group (in general) but has a torsion part which we denote as 𝗍𝖧 (and call the 't Hooft group)

    K_0(𝒞(Γ))= K_0(𝒞(Γ))_free⊕𝗍𝖧≅^f⊕𝖠⊕𝖠

where f=corank B and 𝖠 is the torsion group

    𝖠= ⊕_s /d_s ,     d_s| d_s+1

where the d_s are the positive integers in the normal form of B <cit.>

    B  0⊕ 0⊕⋯⊕ 0 ^f summands ⊕[    0  d_1; -d_1    0 ]⊕[    0  d_2; -d_2    0 ]⊕⋯⊕[    0  d_ℓ; -d_ℓ    0 ]




  §.§.§ The index of a cluster object


Since the rank of the Abelian group K_0(𝒞(Γ)) is (in general) smaller than n, the Grothendieck class is not sufficient to label different objects (modulo deformation). We need to introduce other `labeling quantum numbers' which do the job. This corresponds to the math concept of index (or, dually, coindex).


 For each object L∈𝒞(Γ) there is a triangle

    T_1→ T_0→ L→    with  T_1,T_0∈𝖺𝖽𝖽 Γ.

The difference

    [T_0]-[T_1]∈ K_0(𝖺𝖽𝖽 Γ)

does not depend on the choice of this triangle.


The quantity

    𝗂𝗇𝖽(L)≡ [T_0]-[T_1]∈
    K_0(𝖺𝖽𝖽 Γ)≡
    K_0(𝗉𝗋𝗈𝗃 J(Q,W))≅ K_0(𝔓𝔢𝔯 Γ)≅Λ^∨

is called the index of the object L∈𝒞(Γ).


It is clear from the Lemma that the class [L]∈ K_0(𝒞(Γ)) is the image of 𝗂𝗇𝖽(L) under the projection

    Λ^∨→Λ^∨/B·Λ.


As always, we use the canonical cluster-tilting object Γ; the modules F_ΓΓ_i∈𝗆𝗈𝖽 J(Q,W);
 are the indecomposable projective modules (cfr. Proposition <ref>). We write S_i≡𝖳𝗈𝗉 F_ΓΓ_i∈𝗆𝗈𝖽 J(Q,W)  for the simple with support at the i-th node.

  Let X∈𝒞(Γ) be indecomposable. Then

    𝗂𝗇𝖽  X=-[Γ_i]    X≅Γ_i[1]
    ∑_i=1^n ⟨ F_Γ X,S_i⟩ [Γ_i]    otherwise,

where ⟨ -,-⟩ is the Euler form in 𝗆𝗈𝖽 J(Q,W).


 The dual notion to the index is the coindex <cit.>. For X∈𝒞(Γ) one has

    𝗂𝗇𝖽 X=-𝖼𝗈𝗂𝗇𝖽 X[1]
    𝖼𝗈𝗂𝗇𝖽 X-𝗂𝗇𝖽 X=∑_i=1^n ⟨ S_i, F_Γ X⟩_a [Γ_i]
    𝖼𝗈𝗂𝗇𝖽 X-𝗂𝗇𝖽 X  depends only on F_Γ X∈𝗆𝗈𝖽 J(Q,W).

From (<ref>) it is clear that the projections in K_0(𝒞(Γ)) of the index and coindex agree.


The precise mathematical statement corresponding to the rough idea that the `index yields enough quantum numbers to distinguish operator'
is the following
 Two rigid objects of 𝒞(Γ) are isomorphic if and only if their indices are equal.



We shall show in .<ref> how this is related to UV completeness of the corresponding QFT.






 §.§ Periodic subcategories,  the normalized Euler and Tits  forms


We have seen that the group K_0(D^bΓ) has an extra structure namely a skew-symmetric pairing. It is natural to look   
 for additional structures on the group K_0(𝒞(Γ)).
The argument around (<ref>) implies that the Euler form of the 2-CY category 𝒞(Γ) if defined is symmetric:

    ⟨ X,Y⟩_𝒞(Γ)   ≡∑_k∈ (-1)^k Hom_𝒞(Γ)(X,Y[k])=
       =∑_k∈ (-1)^2-k Hom_𝒞(Γ)(Y, X[2-k])=⟨ Y, X⟩_𝒞(Γ).

However the sum in the rhs  is typically not defined, since it is not true (in general) that Hom_𝒞(Γ)(X,Y[k])=0 for  k≪ 0. In order to remediate this, we  introduce an alternative concept.

 We say that a full subcategory ℱ(p)⊂𝒞(Γ), closed under shifts, direct sums and summands, is p-periodic (p∈ℕ) iff
the functor [p] restricts to an equivalence in ℱ(p), and ℱ(p) is maximal with respect to these properties.
Note that we do not require p to be the minimal period.


 A p-periodic sub-category, ℱ(p)⊂𝒞(Γ), is triangulated and 2-CY[ ℱ(p) is linear, Hom-finite, and 2-CY. However, it is not necessarily a generalized cluster category since it may or may not have a tilting object.  The prime examples of such a category without a tilting object are the cluster tubes, see <cit.>. Sometimes the term `cluster categories' is extended also to such categories.] and the inclusion functor
ℱ(p)𝒞(Γ) is exact.


 Since ℱ(p) is closed under shifts, direct sums, and summands in 𝒞(Γ), it suffices to verify that X,Y∈ℱ(p) implies Z∈ℱ(p) for all triangles X→ Y→ Z→ in 𝒞(Γ). Applying [p] to the triangle, one gets Z[p]≃ Z.



Let ℱ(p)⊂𝒞(Γ) be p-periodic. We define the normalized Euler form as

    ⟨⟨ X,Y⟩⟩=⟨⟨ Y,X⟩⟩ =1/p∑_k=0^p-1(-1)^kHom_𝒞(Γ)(X,Y[k]),    X,Y∈ℱ(p).

Note that it is independent of the chosen p as long as Y[p]≅ Y.


 If p is odd, ⟨⟨-,-⟩⟩≡0.



The normalized Euler form ⟨⟨ -,-⟩⟩ induces a symmetric form on the group

    K_0(ℱ(p))/K_0(ℱ(p))_torsion,

which we call the Tits form of ℱ(p).



We shall see in . <ref> the physical meaning of the periodic sub-categories and their Tits form.
 

 



  §.§.§ Example: cluster category of the projective line of weights (2,2,2,2)


As an example of Tits form in the sense of the above Proposition, we consider the cluster category (see .<ref>)

    𝒞= D^b(ℋ)/⟨τ^-1[1]⟩^,    where ℋ=𝖼𝗈𝗁 𝕏(2,2,2,2)

which corresponds to SU(2) SQCD with N_f=4 <cit.>. We may think of this cluster category as having the same objects as 𝖼𝗈𝗁 𝕏(2,2,2,2) and extra arrows <cit.>. In this case ω=0, and hence the category 𝒞 is triangulated and periodic of period p=2 in the sense of Definition <ref>, so ℱ(2) is the full cluster category 𝒞.
We write 𝒪 for the structure sheaf and 𝒮_i,0 for the unique simple sheaf with support at the i-th special point such that Hom_𝖼𝗈𝗁 𝕏(𝒪,𝒮_i,0)≅ k. The cluster Grothedieck group
K_0(𝒞) is generated by
[𝒪] and [𝒮_i,0] (i=1,2,3,4) subjected to the relation <cit.>

    2[𝒪]=∑_i=1^4[𝒮_i,0].

Thus we may identify

    K_0(𝒞)≅{ (w_1,w_2,w_3,w_4)∈(12)^2 |  w_i=w_j 1}≡Γ_weight, 𝔰𝔭𝔦𝔫(8).

by writing a class as ∑_iw_i[𝒮_i,0].
The Tits pairing is

    ⟨⟨ [𝒮_i,0],
    [𝒮_j,0]⟩⟩=δ_i,j,

Then K_0(𝒞) equipped with this pairing is isomorphic to the 𝔰𝔭𝔦𝔫(8) weight lattice equipped with its standard inner product 
(valued in 12) dual to the even one given on the root lattice by the Cartan matrix. We remark that a class in K_0(𝒞) is a spinorial 𝔰𝔭𝔦𝔫(8) weight iff it is of the form
k[𝒪]+∑_i m_i[𝒮_i,0] (m_i∈)
with k odd. The physical meaning of this statement and eqn.(<ref>) will be clear in . <ref>.





 §.§ Stability conditions for Abelian and triangulated categories


We start with the Abelian category case, since it all boils down to it. The main reference for this part is <cit.>.
Let  be an Abelian category and K_0(  ) its Grothendieck group.

 
A Bridgeland stability condition on an Abelian category 𝒜 is a group homomorphism

    Z : K_0( ) →,

satisfying certain properties:[ If [X]∈ K_0() is the class of X∈, we write simply Z(X) for Z([X]). ]


  * Z() ⊂ℍ∖ℝ_>0, the closed upper half plane minus the positive reals;

  * If Z(E) = 0, then E = 0. This allows to define the map 
    Z(-) K_0()∖{0}→ (0,π];


  * The Harder-Narasimhan (HN) property. Every object E ∈ admits a filtration

    0 = E_0 ⊂ E_1 ⊂ E_2 ⊂···⊂ E_n = E,

such that, for each i:


  * E_i+1/E_i is Z-semistable;[ See below Definition <ref> of  semistability of objects in an abelian category.]

  * Z(E_i+1/E_i) >  Z(E_i+2/E_i+1).



We also have the following
 An object E∈ is called Z-stable if for all nonzero proper subobjects
E_0 ⊂ E,

    Z(E_0) <  Z(E).

If ≤ replaces <, then we get the definition of Z-semistable.

We are now going to give the corresponding definitions for the triangulated categories. The definition is more involved since there is no concept of subobject.


 A slicing  of a triangulated category  is a collection of full
additive subcategories (ϕ) for each ϕ∈ satisfying


  * (ϕ + 1) =  (ϕ)[1];

  * For all ϕ_1 > ϕ_2 we have Hom((ϕ_1), (ϕ_2)) = 0;

  * For each 0 ≠ E ∈ there is a sequence ϕ_1 > ϕ_2 > ··· > ϕ_n of real
numbers and a sequence of exact triangles

    0=E_0 [rr]        E_1 [r][dl]     ⋯⋯[r]    E_n-1 [rr]        E_n=E [dl] 

                   A_1 @.>[ul]         ⋯       A_n @.>[ul]    

with A_i∈(ϕ_i) (which we call the Harder-Narasimhan filtration of E).



 We call the objects in (ϕ) semistable of phase ϕ.


And finally, the definition of stability conditions in a triangulated category.
 A stability condition on a triangulated category  is a pair (Z, )
where Z : K_0() → is a group homomorphism (called central charge) and  is
a slicing, so that for every 0 ≠ E ∈(ϕ) we have

    Z(E) = m(E)   e^iπϕ

for some m(E) ∈>0.

Indeed, the following proposition shows that to some extent (once we identify a
t-structure), stability is intrinsically defined. It also describes how stability conditions
are actually constructed:
 Giving a stability condition (Z, )
on a triangulated category  is equivalent to giving a heart  of a bounded t-structure with a stability
function Z_  : K_0( ) → such that ( Z_ , ) have the
Harder-Narasimhan property, i.e. any object in  has a HN-filtration by Z_-stable
objects.

We will focus on how to obtain a stability condition from the datum ( Z_ , ), as
this is how stability conditions are actually constructed:
 If  is the heart of a bounded t-structure on , then we have K_0() =
K_0( ), so clearly Z and Z_ determine each other.
Given ( Z_ , ), we define (ϕ) for ϕ∈ (0, 1] to be the Z_-semistable objects
in  of phase ϕ(E) = ϕ. This is extended to all real numbers by (ϕ + n) =
 (ϕ)[n] ⊂[n] for ϕ∈ (0, 1] and 0 ≠ n ∈ℤ. The compatibility condition 

    1/π  Z(E)=ϕ

is satisfied by construction,
so we just need show that  satisfies the remaining properties in our definition of slicing.
The Hom-vanishing condition in definition <ref> follows from the definition of heart of a bounded t-structure. Finally, given
E ∈, its filtration by cohomology objects A_i ∈[k_i], and the
HN-filtrations 0 → A_i1→ A_i2→…→ A_im_i = A_i given by the HN-property
inside  can be combined into a HN-filtration of E: it begins with

    0 → F_1 = A_11[k_1]→ F_2 = A_12[k_1]→⋯→ F_m_1 = A_1[k_1] = E_1,

i.e. with the HN-filtration of A_1. Then the following filtration steps F_m_1+i are an
extensions of A_2i[k_2] by E_1 that can be constructed as the cone of the composition
A_2i[k_2]→ A_2[k_2] → ^[1] E_1 (the octahedral axiom shows that these have the same
filtration quotients as 0 → A_21[k_2] → A_22[k_2]⋯ ); continuing this way we obtain a
filtration of E as desired.
Conversely, given the stability condition, we set =  ((0, 1]) as before; by
the compatibility condition, the central charge Z(E) of any -semistable
object E lies in ℍ∖_>0; since any object in  is an extension of semistable
ones, this follows for all objects in  by the additivity. Finally, it is fairly straightforward
to show that Z-semistable objects in  are exactly the semistable objects
with respect to .




§ SOME PHYSICAL PRELIMINARIES


In the next section we shall relate the various triangle categories introduced in the previous section to the BPS objects of a 4d
𝒩=2 QFT as described  from two different points of view: i) the microscopic UV description (i.e. in terms of a UV complete Lagrangian description or a UV  fixed-point SCFT), and ii) the effective Seiberg-Witten IR description. Before doing that, we discuss some general properties of these physical systems. As discussed in . <ref>,
the categories 𝔗_A which describe the BPS objects  should enjoy the categorical versions of these physical properties in order to be valid solutions to the Problem in . <ref>.



 §.§ IR viewpoint




  §.§.§ IR conserved charges


The Seiberg-Witten theory <cit.> describes, in a quantum exact way, the low-energy physics of our 4d 𝒩=2 model in a given vacuum u along its Coulomb branch. Assuming u and the mass deformations to be generic, the effective theory is an Abelian gauge theory U(1)^r coupled to states carrying both electric and magnetic charges. The flavor group is also Abelian U(1)^f, so that the IR conserved charges consist of r electric, r magnetic, and f flavor charges. In a non-trivial theory the gauge group is compact, and the flavor group is always compact, so these charges are quantized. Then the conserved charges take value in a lattice Λ (a free Abelian group) of rank

    n=2r+f.

The lattice Λ is equipped with an extra structure, namely a skew-symmetric quadratic form

    ⟨-,-⟩Λ×Λ→,

given by the Dirac electro-magnetic pairing. The radical of this form,

    Λ_flav=rad ⟨-,-⟩≡{λ∈Λ | ⟨μ,λ⟩=0  ∀ μ∈Λ}⊂Λ,
 
is the lattice of flavor charges and has rank f. The effective theory has another bosonic complex-valued conserved charge, namely the central charge of the 4d
𝒩= 2 superalgebra Z := ϵ^αβϵ_AB{Q^A_α, Q^B_β}. Z is not an independent charge but a linear combination of the charges in Λ with complex coefficients which depend on all IR data, and in particular on the vacuum u. Hence,
for a given u, the susy central charge is a linear map
(group homomorphism)

    Z_uΛ→.

Any given state of charge λ∈Λ has mass greater than or equal to |Z_u(λ)|. BPS states are the ones which saturate this bound.




In the case of a 4d 𝒩=2 with a UV Lagrangian formulation, r and f are the ranks of the (non-Abelian) gauge G and flavor F groups, respectively. At extreme weak coupling, the IR electric and flavor charges are the weights under the respective maximal tori.



  §.§.§ The IR landscape vs. the swampland


The IR 𝒩=2 theories we consider are not generic Abelian gauge theories with electric and magnetic charged matter.
They belong to the landscape (as opposed to the swampland), that is, they have a well defined UV completion. Such theories have special properties.

One property which seems to be true in the landscape, is that there are “enough” conserved IR charges to label all BPS states, so we don't need extra quantum numbers to distinguish the BPS objects in the IR description.
This condition is certainly not sufficient to distinguish the landscape from the swampland, but it plays a special role in our discussion.

To support the suggestion that being UV complete is related to Λ being large enough to label IR objects, we mention a simple fact.

 Let the UV theory consists of a 𝒩=2 gauge theory with semi-simple gauge group G and quark half-hypermultiplets in the (reducible) quaternionic representation H. Assume that the beta-functions of all simple factor of G are non-positive.
In the IR theory along the Coulomb branch, consider the BPS hypermultiplets h_i with zero magnetic charge and write [h_i] for their IR charges in Λ. Then 

    [h_i]=[h_j] and h_i≠ h_j  ⇒   [h_i]∈Λ_flav.

That is, the charges in Λ are enough to distinguish (zero magnetic charge) hypermultiplets unless they carry only flavor charge (i.e. are electrically neutral).  
  

 The hypermultiplets with purely flavor charge (called “everywhere light” since their mass is independent of the Coulomb branch parameters) just decouple in the IR, so in a sense they are no part of the IR picture.


To show the above fact, just list for all possible gauge group all representations compatible with non-positivity of the beta-function. Check, using Weyl formula, that the multiplicities of all weights for these representations is 1 except for the zero weight. 





 §.§ UV line operators and the 't Hooft group
 





  §.§.§ 't Hooft theory of quantum phases of gauge theories

We start by recalling the classical arguments by 't Hooft on the quantum phases of a 4d gauge theory <cit.>. 
The basic order operator in a gauge theory is the Wilson line associated to a  (real) curve C
in space time and a representation R of the gauge group G,

    W_R(C)= tr_R e^-∫_C A.

Here C is either a closed loop or is stretched out to infinity. In the second case we don't take the trace and hence the operator
depends on a choice of a weight w of the representation R
modulo che action of Weyl group. In the 𝒩=2 case, the Wilson line (<ref>) is replaced by its half-BPS counterpart <cit.> which, to preserve half supersymmetries should be stretched along a straight line L;
we still denote this operator as W_w(L).[ The half-BPS lines are also parametrized by an angle ϑ which specifies which susy subalgebra leaves them invariant. We suppress ϑ from the notation.]  

What are the quantum numbers carried by W_w(L)?
This class of UV line operators is labelled by (the Weyl orbit of) the weight w, so gauge weights  are useful quantum numbers. However, these numbers do not correspond to conserved quantities in a general gauge theory. For instance, consider pure (super-)Yang-Mills theory and let R be the adjoint representation. Since an adjoint Wilson line may terminate at the location of a colored particle transforming in the adjoint representation, a gluon (gluino) particle-antiparticle pair may be dynamically created out of the vacuum, breaking the line, see figure <ref>.


If our gauge theory is in the confined phase,  breaking the line L is energetically favorable, so the line label w does not correspond to a conserved quantity. 
On the contrary, a Wilson line in the fundamental representation cannot break in pure SU(N)
(S)YM, since there is no dynamical particle which can be created out of the vacuum where it can terminate.
The obstruction to breaking the line is the center Z(SU(N))≅_N of the gauge group under which all local degrees of freedom are inert while the fundamental Wilson line is charged. Stated differently, the gluons may screen all color degrees of freedom of a physical state  but the center of the gauge group. The conclusion is that the conserved quantum numbers of the line operators W_R(L) consist of  the representation R seen as a
representation of the center of the gauge group, Z(G),
which take value in the dual group Z(G)^∨≅Z(G).
On the other hand, in SU(N)
(S)QCD we have quarks transforming in the fundamental representation; hence  
a quark-antiquark pair may be created to break a fundamental Wilson line. Then, in presence of fundamental matter, Wilson lines do not carry any conserved quantum number. In general, the conserved quantum numbers of the Wilson lines of a gauge theory with gauge group G take value in the finite Abelian group
π_1(G_eff)^∨≅π_1(G_eff),
where  G_eff is 
the quotient group of G which acts effectively
on the microscopic UV degrees of freedom.

For clarity of presentation, the above discussion was in the confined phase. This is not the case of the 𝒩=2 theory which we assume to be realized in its Coulomb phase.
In the physically realized phase the line W_w(L) may be stable; then  its labeling quantum number w becomes an emergent conserved quantity of the IR description (see .<ref>). However, from the UV perspective, the only strictly conserved quantum numbers are still the (multiplicative) characters of π_1(G_eff) which take value in the group

    π_1(G_eff)^∨≡Hom(π_1(G), U(1))≅π_1(G_eff).


More generally, we may have Wilson-'t Hooft lines <cit.> which carry both electric and magnetic weights. Their multiplicative conserved quantum numbers take value in the (Abelian) 't Hooft group

    𝗍𝖧= π_1(G_eff)^∨⊕π_1(G_eff),

equipped with the canonical skew-symmetric bilinear pairing (the Weil pairing)[ As always, μ denotes the group of roots of unity. The name `Weil pairing' is due to its analogy with the Weil pairing in the torsion group of a polarized Abelian variety which arises in exactly the same way.]

    𝗍𝖧×𝗍𝖧→μ,     (x,y)× (x^',y^')↦ x(y^') x^'(y)^-1.




The 't Hooft multiplicative quantum numbers of a line operator, written additively, are just its electric/magnetic weights
(w_e,w_m) modulo the weight lattice of G_eff.


The best way to understand the proper UV conserved quantum numbers of line operators is to consider the different sectors in which we may decompose the microscopic path integral of the theory which preserve the symmetries of a line operator stretched in the 3-direction in space (that is, rotations in the orthogonal plane and translations).
In a 4d gauge theory quantized on a periodic 3-box of size L
we may defined the 't Hooft twisted
path integral <cit.> (see <cit.> for nice reviews)

    e^-β F(e⃗, m⃗, θ, μ_s,β)≡Tr_e⃗, m⃗[e^-β H+iθν+μ_s F_s],    e⃗∈(π_1(G_eff)^∨)^3, m⃗∈π_1(G_eff)^3,
    
where e⃗, m⃗ are 't Hooft (multiplicative) electric and magnetic fluxes, θ is the instanton angle, and μ_s are chemical potentials in the Cartan algebra of the flavor group F. Imposing rotational invariance in the 1-2 plane and taking the Fourier transform with respect to the μ_s we remain (at fixed θ)
with the quantum numbers

    (e_3, m_3, w)∈π_1(G_eff)^∨⊕π_1(G_eff)⊕(weight lattice of F).

We shall call the vector (e_3, m_3, w) the 't Hooft charge and the group in the rhs the extended 't Hooft group.

We stress that the structure of the Weil pairing is required in order to relate the Euclidean path integral in given topological sectors  to the free energy F(e⃗, m⃗, θ, μ_s,β) with fixed non-abelian fluxes <cit.>.

 The boundary condition on the Euclidean box which corresponds to a given 't Hooft charge does no break any supercharges, that is, we do not need to specify a BPS angle ϑ to define it. 




  §.§.§ Non-Abelian enhancement of the flavor group in the UV


Consider a UV complete 𝒩=2 gauge theory. In the IR theory
the flavor group is (generically) Abelian of rank f. In the UV the masses are irrelevant and the flavor group enhances from the Abelian group U(1)^f to some possibly non-Abelian rank f Lie group F.
The free part of the 't Hooft group
(<ref>) is then the weight lattice of F. This weight lattice is equipped with a quadratic form dual to the Cartan form on the root lattice.
From the quadratic form we recover the non-Abelian Lie group F. 
In conclusion: 

 The UV conserved quantities are encoded in the extended 't Hooft group, a finitely generated Abelian group  of the form

    π_1(G_eff)^∨⊕π_1(G_eff) ⊕Γ_flav,weight,
 
whose free part has rank f. The extended 't Hooft group (<ref>) is equipped with two additional structures: i) the Weil pairing on the torsion part, ii) the dual Cartan symmetric form on the free part. Moreover, iii) the UV lines carry an adjoint action of the half quantum monodromy 𝕂 (see . <ref>) which acts on the 't Hooft group as -1. 




  
Finer structures on the 't Hooft group. The 't Hooft group  (<ref>) detects the global topology of the gauge group G_eff; it also detects the topology of the flavor group F,
e.g. it distinguishes between the flavor groups SO(N) and Spin(N), since they have different weight lattices

    [ Γ_𝔰𝔭𝔦𝔫(N): Γ_𝔰𝔬(N) ]=2.

But there even finer informations on the flavor symmetry which we should be able to recover from the relevant categories. To illustrate the issue, consider SU(2) SQCD with N_f fundamental hypers. In the perturbative sector (states of zero magnetic charge) the flavor group is SO(2N_f), but non-perturbatively it gets enhanced to Spin(2N_f). More precisely, states of odd (resp. even) magnetic charge are in spinorial (resp. tensorial) representations of the flavor group Spin(2N_f). This is due to the zero modes of the Fermi fields in the magnetic monopole background
<cit.>, which is turn are predicted by the Atiyah-Singer index theorem. The index theorem is an integrated version of the axial anomaly, so the correlation between magnetic charge and flavor representations should emerge from the same aspect of the category which expresses the U(1)_R anomaly (and the β-function). 
    



  §.§.§ The effective `charge' of a UV line operator


We have two kinds of quantum numbers: conserved quantities and labeling numbers. In the IR we expect (see .<ref>) that conserved quantities are (typically) sufficient to label BPS objects. However, the UV group of eqn.(<ref>) is too small to distinguish inequivalent BPS line operators.

We may introduce a different notion of `charge' for UV operators which takes value in a rank n=2r+f lattice. This notion, albeit referred to UV objects, depends on a IR choice, e.g. the choice of a  vacuum u. Suppose that in this vacuum we have
n species of stable lines L_i (i=1,…, n) which are preserved by the the same susy sub-algebra preserving L and  carry emergent IR quantum numbers [L_i] which are ℚ-linearly independent. We may consider the BPS state |{n_i}⟩ in which
we have a configuration of parallel stable lines with n_1 of type L_1, n_2 of type L_2, and so on. Suppose that for our BPS line operator L

    ⟨{n_i^'}| L |{n_i}⟩≠ 0

It would be tempting to assign to the operator L the `charge'

    ∑_i(n_i^'-n_i)[L_i]∈⊕_i [L_i].

Such a charge would be well-defined on UV operators provided two conditions are satisfied: i) for all L we can find a pair of states |{n_i}⟩, |{n_i^'}⟩ such that eqn.(<ref>) holds, and moreover ii) we can show that n^'_i-n_i does not depend on the chosen |{n_i}⟩, |{n_i^'}⟩. The attentive reader may notice that this procedure is an exact parallel to the definition of the index of a cluster object (Definition <ref>). However the i-th `charge' n^'_i-n_i is PCT-odd only if the lines L, L_i carry `mutually local charge', that is, have trivial braiding; the projection of the `charge' so defined in the 't Hooft group (<ref>) is, of course, independent of all choices. This follows from the fact that the action of PCT on  the UV lines is given by the half quantum monodromy (see . <ref>) which does not act as -1 on the present `effective' charges; of course, it acts as -1 on the 't Hooft charges as it should.



 §.§ The quantum monodromy


There is one more crucial structure on the UV BPS operators, namely the quantum monodromy <cit.>. Let us consider first the case in which the UV fixed point is a good regular SCFT. At the UV fixed point the U(1)_r R-symmetry is restored.
Let e^2π i r be the operator implementing a U(1)_r rotation by  
2π (it acts on the supercharges as -1). e^2π i r acts on a chiral primary operator of the UV SCFT as multiplication by e^2π i Δ, where Δ is the scaling dimension of the chiral operator.
Suppose that for all chiral operators Δ∈ℕ, then e^2π i r=(-1)^F
acts as 1 on all UV observables.
More generally, if all Δ∈ m ℕ for some integer m,
the operator (e^2π i r)^m acts as 1 on observables <cit.>.

If the theory is asymptotically-free, meaning that the UV fixed point is approached with logarithmic deviations from scaling, the above relations get
also corrected, in a way that may be described rather explicitly, see  <cit.>. 

Now suppose we deform the SCFT by relevant operators to flow to the original 𝒩=2 theory. We claim that, although the Abelian R-charge r is no longer conserved,  e^2π ir remain a symmetry in this set up[ For the corresponding discussion in 2d, see <cit.>.] <cit.>. This is obvious when the dimensions Δ are integral, since e^2π ir commutes with the deforming operator. The quantum monodromy 𝕄 is the 
operator induced in the massive theory from e^2π i r in this way <cit.>.
It is well defined only up to conjugacy,[ When the UV fixed point SCFT is non degenerated, the operator 𝕄 is semisimple, and its conjugacy class is encoded in its spectrum, that is, the spectrum of dimensions of chiral operators Δ1.]
and may be written as a Kontsevitch-Soibelmann (KS) product of BPS factors ordered according to their phase[ In eqn.(<ref>) we use the notations of <cit.>: the product is over the BPS stable states of charge λ∈Λ and spin s_λ taken in the clockwise order in their phase Z_u(λ); ψ(z;q)=∏_n≥0(1-q^n+1/2z)^-1 is the quantum dilogarithm, and the X_λ are quantum torus operators, i.e. they satisfy the algebra X_λ X_λ^'=q^⟨λ,λ^'⟩/2 X_λ+λ^' with ⟨-,-⟩ the Dirac pairing. ] <cit.>

    𝕄=∏^↺_λ∈BPSΨ(q^s_λX_λ;q)^(-1)^2s_λ.

The KS wall-crossing formula <cit.> is simply the statement that the conjugacy class of 𝕄, being an UV datum, is independent of the particular massive deformation as well as of the particular BPS chamber we use to compute it (see <cit.>).

We may also define the half-monodromy 𝕂, such that 𝕂^2=𝕄 <cit.>. The effect of the adjoint action of 𝕂 on a line operator L is to produce its PCT-conjugate. Then 𝕂 inverts the 't Hooft charges.

We summarize this subsection in the following 

 If our 𝒩=2 has a regular UV fixed-point SCFT and the dimension of all chiral operators satisfy Δ∈ mℕ for a certain integer m, then 𝕂^2m acts as the identity on the line operators. 𝕂 acts as -1 on the 't Hooft charges.



 
  






§ PHYSICAL MEANING OF THE CATEGORIES D^BΓ, 𝔓𝔢𝔯 Γ, 𝒞(Γ)


We start this section by reviewing as quivers with (super)potentials arise in the description of the BPS sector of a (large class of) 4d 𝒩=2 theories, see <cit.>.



 §.§ 𝒩=2 BPS spectra and quivers




We consider the IR physics of a 4d 𝒩 = 2 model at a generic vacuum u along its Coulomb branch. 
We have the IR structures described in 
. <ref>: a charge lattice Λ of rank n=2r+f,
 equipped with an integral skew-symmetric form given by the Dirac electro-magnetic pairing, and a complex linear form given by the 𝒩=2 central charge: 
 
    -,-Λ×Λ→,       Z_uΛ→.
 
 A 4d 𝒩 = 2 model has a BPS quiver at u
iff there exists a set of n hypermultiplets, stable in the vacuum u, such that <cit.>: i) their charges e_i∈Λ generate Λ, i.e. Λ≅⊕_i e_i, and ii) the charge of each BPS states (stable in u), λ∈Λ, satisfies

    λ∈Λ_+   or  -λ∈Λ_+,

where Λ_+ = ⊕_i _+ e_i is the convex cone of `particles' [ As contrasted with `antiparticles' whose charges belong to -Λ_+.]. 
The BPS quiver Q is encoded in the
skew-symmetric n× n exchange matrix 
    B_ij :=e_i, e_j,     i,j=1,⋯,n.
 The nodes of Q are in one-to-one correspondence with the
generators {e_i} of Λ. If B_ij≥ 0 then there are |B_ij| arrows from node i to node j; viceversa for B_ij<0.


To find the spectrum of particles with given charge λ = ∑_i m_i e_i ∈Λ_+ we may study the effective theory on their world-line. This is a SQM model with four supercharges <cit.>, corresponding to the subalgebra of 4d susy which preserves the world-line. A particle is BPS in the 4d sense iff it
is invariant under 4 supersymmetries, that is, if it is a susy vacuum state of the world-line SQM. The 4-supercharge SQM is based on the quiver Q defined in eqn.(<ref>) <cit.>: to the i–th node there correspond a 1d U(m_i) gauge multiplet, while to an arrow i→ j a 1d chiral multiplet  in the (m_i,m_j) bifundamental representation of the groups at its two ends.
To each oriented cycle in Q there is associated a single-trace gauge invariant chiral operator, namely the trace of the product of the Higgs fields along the cycle.
The (gauge invariant) superpotential of the SQM is a complex linear combination of such operators associated to cycles of Q <cit.>. Since we are interested only in the susy vacua, we are free to integrate out all fields entering quadratically in the superpotential.
We remain with a SQM system described by a reduced quiver with (super)potential (Q,W) in the sense of section 2.

Then the solutions of the SQM F-term
equations are exactly the modules X of the Jacobian algebra[ From now on the ground field k is taken to be .] 
J(Q, W) with dimension vector X = λ∈Λ. 


The D-term equation is traded for the stability condition <cit.>. Given the central charge Z_u(-), we can 
choose a phase θ∈ [0,2π) such that Z_u(Λ_+) lies inside[ ℍ denotes the upper half-plane ℍ:={z∈ | Im z>0}.] ℍ_θ:=e^iθℍ. 
Given a
module X∈𝗆𝗈𝖽 J(Q, W), we define its stability function as ζ(X) := e^-iθZ_u(X) ∈ℍ. The module X is stable iff

    ζ(Y)<ζ(X),   ∀  Y⊂ X  proper submodule.

A stable module X is always a brick, i.e.
End_𝗆𝗈𝖽 J(Q,W)X≅ <cit.>.  



Keeping into account gauge equivalence,
the SQM classical vacuum space is the compact
Kähler variety <cit.>

    M_λ:={
    X ∈ J(Q,W)  |  X stable,  X = λ}/∏_iGL(m_i
    , ),

that is, the space of isoclasses of stable Jacobian modules of the given dimension λ.
The space of SQM quantum vacua is then H^∗(M_λ,) which carries a representation R of SU(2) by hard Lefschetz <cit.>, whose maximal spin is M_λ/2; the space-time spin content of the charge λ BPS particle is[ The Cartan generator of SU(2)_R acting on a BPS particle described by a (p,q)-harmonic form on M_λ is (p-q); however, it is conjectured that only trivial representations of SU(2)_R appear <cit.>.] 

    (0⊕2)⊗R.

 For example, the charge λ BPS states consist of a (half) hypermultiplet iff the corresponding moduli
space is a point, i.e. if the module X is rigid.



The splitting between particles
and antiparticles is conventional:
different choices lead to different pairs (Q,W). However all these (Q,W) should lead to equivalent SQM quiver models. Indeed, distinct pairs are related by a chain of 1d Seiberg dualities <cit.>.
The Seiberg dualities act on (Q,W) as the quiver mutations described in section 2. Indeed, the authors of <cit.> modeled their construction on  Seiberg's original work <cit.>.


The conclusion of this subsection
is that to a (continuous family of)
4d 𝒩=2 QFT (with the quiver property) there is associated a full mutation-class of quivers with potentials (Q,W). All (Q,W) known to arise from consistent QFTs are Jacobi-finite, and we assume this condition throughout.


Using the mathematical constructions reviewed in . 2,
to such an 𝒩=2 theory we naturally associate the three triangle categories
D^bΓ, 𝔓𝔢𝔯 Γ, and 𝒞(Γ), together with the functors 𝗌, 𝗋 relating them. We stress that the association is intrinsic, in the sense that  
the  categories are independent of the choice of (Q,W) in the  mutation-class  modulo triangle equivalence (cfr. Theorem <ref>). 
Our next task is to give a physical interpretation to these three naturally defined categories. We start from the simpler one, D^bΓ. 



 §.§ Stable objects of D^bΓ and BPS states

Let Γ be the Ginzburg algebra associated to the pair (Q,W). Keller proved <cit.> that the Abelian category 𝗆𝗈𝖽 J(Q,W) is the heart of a bounded t-structure in D^bΓ. In particular, its Grothendieck group is

    K_0(D^bΓ)≅ K_0(𝗆𝗈𝖽 J(Q,W))≡Λ,

that is the lattice of the IR conserved charges (. <ref>).
Thus, given a stability condition on the Abelian category 𝗆𝗈𝖽 J(Q,W), we can  extend it to the entire triangular category D^bΓ. In particular, since the semi-stable objects of D^bΓ are  the elements of (ϕ) (cfr. the proof of Proposition <ref>), we have two possibilities:


  * ϕ∈ (0,1], then the only semistable objects are the semistable objects of 𝗆𝗈𝖽 J(Q,W) in the sense of “Abelian category stability” plus the zero object of D^bΓ;

  * ϕ∉(0,1], then the only semistable objects are the shifts of the semistable objects of 𝗆𝗈𝖽 J(Q,W)  in the sense of “Abelian category stability”.

In other words, a generic object E ∈ D^bΓ is unstable if it has a nontrivial HN filtration. Thus, up to shift [n], the only possible semistable objects in D^bΓ are those objects belonging to the heart 𝗆𝗈𝖽 J(Q,W) that are “Abelian”-stable in it. 
We have already seen that the category 𝗆𝗈𝖽 J(Q,W) describes the BPS spectrum of our 4d 𝒩=2 QFT: by what we just concluded, the isoclasses of stable objects X of D^bΓ with Grothendieck class [X]=λ∈Λ are parametrized, up to even shifts[ Since the shift by [1] acts on the BPS states as PCT, it is quite natural to identify the BPS states associated to stable objects differing by even shifts.], by the Kähler manifolds M_λ≅ M_-λ in eqn.(<ref>) whose cohomology yields the BPS states.

The category 𝒫(ϕ) is an Abelian category in its own right.
The stable objects with BPS phase
e^iπϕ are the simple objects in this category; in particular they are bricks in 𝒫(ϕ) hence bricks in 𝗆𝗈𝖽 J(Q,W),
that is,

    X stable  ⇒  End_𝗆𝗈𝖽 J(Q,W)(X)≅.





 §.§ Grothendieck groups vs. physical charges


When the triangle category 𝒯 describes a class of BPS objects, the Abelian group
K_0(𝒯) is identified with the conserved quantum numbers carried by those objects. In particular, the group K_0(𝒯) should carry all  the additional structures required by the physics of the corresponding BPS objects, as described in . <ref>.

Let us pause a while to discuss the Grothendieck groups of the three triangle categories K_0(𝒯), where 𝒯=D^bΓ, 𝔓𝔢𝔯 Γ, or 𝒞(Γ), and check that they indeed possess all properties and additional structures as required by their proposed physical interpretation. 



  §.§.§ K_0(D^bΓ)
 

Since D^bΓ describes BPS particles, K_0(D^bΓ) is just the IR charge lattice Λ, see eqn.(<ref>).
Physically, the charge lattice carries the structure of a skew-symmetric integral bilinear form, namely the Dirac electromagnetic pairing.
This matches with the fact that, 
since D^bΓ is 3-CY,  its Euler form (<ref>) is skew-symmetric and is identified with the Dirac pairing (compare eqn.(<ref>) and the last part of Proposition <ref>). We stress that the pairing is intrinsic (independent of all choices) as it should be on physical grounds.





  §.§.§ K_0(𝒞(Γ)): structure

The structure of the group K_0(𝒞(Γ)) was described in . <ref>.
We have

    K_0(𝒞(Γ))= ^f⊕𝖠^∨⊕𝖠

where 𝖠 is the torsion group[ Of course, 𝖠^∨≅𝖠; however it is natural to distinguish the group and its dual.]

    𝖠= ⊕_s /d_s ,     d_s| d_s+1

where the d_s are the positive integers appearing in the normal form of B, see eqn.(<ref>).

The physical meaning of the Grothendieck group (<ref>) is easily understood by considering the case of pure 𝒩=2 super-Yang-Mills with gauge group G. 
Then one shows <cit.>

    𝖠=Z(G)≡the center of the (simply-connected) gauge group G

that is

    K_0(𝒞(Γ_SYM,G))≅Z(G)^∨⊕Z(G).
  
This is exactly the group of multiplicative quantum numbers labeling the UV Wilson-'t Hooft line operators in the pure SYM case <cit.>,
as reviewed in . <ref>.
This strongly suggests the identification of the cluster Grothendieck group K_0(𝒞(Γ)) with
the group of additive and multiplicative quantum numbers carried by the UV line operators. 

This is confirmed by of the example of 𝒩=2 SQCD with (semi-simple) gauge group G and quark hypermultiplets in a (generally reducible) representation R. One finds <cit.>

    K_0(𝒞(Γ_SQCD))≅^rank  F⊕π_1(G_eff)^∨⊕π_1(G_eff),

where F is the flavor group and G_eff is the quotient of G
acting effectively on the UV degrees of freedom. Again, this corresponds to the UV extended 't Hooft group as defined in . <ref>.
More generally one has:

 In all 𝒩=2 theories with a Lagrangian formulation (and a BPS quiver) we have

    K_0(𝒞(Γ))≅(the extended 't Hooft group of . <ref>).


This is already strong evidence that
the cluster category 𝒞(Γ) describes UV line operators. For 𝒩=2 theories without a Lagrangian, we adopt the above Fact as the definition of the extended 't Hooft group.

From Fact <ref> we know that the physical 't Hooft group has three additional structures. Let us show that all three structures are naturally present in K_0(𝒞(Γ)).



  §.§.§ K_0(𝒞(Γ)): action of half-monodromy and periodic subcategories


There is a natural candidate for the half-monodromy: on X∈𝒞(Γ), 𝕂 acts as X↦ X[1] and hence the full monodromy as X↦ X[2]. 
Then 𝕂 acts on K_0(𝒞(Γ)) as -1, as required. Let us check that this action has the correct physical properties e.g. the right periodicity as described in Fact <ref>. 

[Periodicity for Argyres-Douglas models] We use the notations of . <ref>. We know that the quantum monodromy 𝕄 has a periodicity[ For the relation of this fact with the Y-systems, see <cit.>.] equal to (a divisor of) h+2 <cit.>, corresponding to the fact that the dimension of the chiral operators Δ∈1h+2ℕ. Indeed, from the explicit description of the cluster category, eqn.(<ref>),
we have 𝒞(𝔤)= D^b(𝗆𝗈𝖽 𝔤)/⟨τ^-1[1]⟩^, so that τ≅ [1] in 𝒞(𝔤). Hence,

    𝕄^h+2≡ [h+2] ≅τ^h[2]=Id,

where we used eqn.(<ref>). 


Under the identification 𝕂↔ [1], we may rephrase Fact <ref>
in the form:

 Let 𝒞(Γ) be the cluster category associated to a 𝒩=2 theory with a regular UV fixed-point SCFT such that all chiral operators have dimensions Δ∈ mℕ. Then 𝒞(Γ) is periodic with minimal period p| 2m. 
If the theory has flavor charges, p is even (more in general: p is even unless the 't Hooft group is a vector space over 𝔽_2). In particular, for 𝒩=2 theories with a regular UV fixed-point the cluster Tits form ⟨⟨ [X],[Y]⟩⟩ is well-defined.




  
Asymptotically-free theories.
It remain to discuss the  asymptotically-free theories. The associate cluster categories 𝒞(Γ) are not periodic. However, from the properties of the 't Hooft group, we expect that, whenever our theory has a non-trivial flavor symmetry, 𝒞(Γ) still contains a periodic sub-cluster category of even period. We give an informal argument corroborating this idea which may be checked in several explicit examples.

Sending all non-exactly marginal couplings to zero, our asymptotically-free theory reduces to a decoupled system of free glue and UV regular matter SCFTs. Categorically, this means that cluster category of each matter SCFT, 𝒞_mat embeds as an additive sub-category in 𝒞(Γ) closed under shifts (by PCT). The embedding functor ι is not exact (in general), so we take the triangular hull of the full subcategory over the objects in its image
𝖧𝗎_((ι 𝒞_mat)_full)⊂𝒞(Γ). If the model has non-trivial flavor, at least one matter subsector has non trivial flavor, and the corresponding category 𝒞_matter is periodic of even period p. Its objects satisfy X[p]≅ X and this property is preserved by ι.
The triangle category
𝖧𝗎_((ι 𝒞_mat)_full) is generated by these periodic objects and hence is again periodic of period p. Then set

    ℱ(p)=𝖧𝗎_((ι 𝒞_mat)_full).

Again, the flavor Tits form is well defined.


The above discussion shows that the presence of a p-periodic subcategory ℱ(p)⊂𝒞 is related to the presence of a sector in the 𝒩=2 theory described  by susy protected operators of dimension 
    Δ= 2pℕ.
 Let us present some simple examples.

[Pure SU(2) SYM] The cluster category 𝒞_SU(2) is not periodic; this is a manifestation of the fact that the β-function of the theory is non zero <cit.>.
However, let us focus on the perturbative (≡ zero magnetic charge) sector in the g_YM→0 limit. The chiral algebra is generated by a single operator of dimension Δ=2,
namely tr(ϕ^2). Hence we expect that the zero-magnetic charge sector is described by a subcategory of 
𝒞_SU(2) which is 1-periodic. Indeed, this is correct, ℱ(1) being a ℙ^1 family of homogenous cluster tubes.


[SU(2) SYM coupled to D_p Argyres-Douglas]
In this case the matter is an Argyres-Douglas theory of type D_p; the matter half quantum monodromy 𝕂_matter has order (h(D_p)+2)/(2,h(D_p))=p as we may read from the spectrum of chiral ring dimensions of the Argyres-Douglas model <cit.>. Thus the matter corresponds to a periodic subcategory ℱ(p)⊂𝒞. This category is a cluster tube of period p
<cit.>.
See also <cit.>.


 Equivalently, we may understand that the presence of a non-trivial flavor group implies the existence of a 2-periodic subcategory ℱ(p)⊂𝒞 by the fact that the corresponding conserved super-currents have canonical dimension 1 which cannot be corrected by RG. 








  §.§.§ K_0(𝒞(Γ)): non-Abelian enhancement of flavor

As discussed in . <ref>,
the IR flavor symmetry U(1)^f gets enhanced in the UV to a non-Abelian group F. The identification of K_0(𝒞(Γ)) with the extended 't Hooft group requires, in particular, that its free part is equipped with the correct dual Cartan form for the flavor group F.

In . <ref> we defined a Tits form associated to (a periodic subcategory of) 𝒞(Γ). This is a symmetric form on the free part of the Grothendieck group, and is the natural candidate for the dual Cartan form of the physical flavor group F. 
Let us check in a couple of examples that this identification yields the correct flavor group:
the cluster category knows the actual non-Abelian group.


[SU(2) with N_f≥1 fundamentals] We use the same notations[ However we often write simply 𝕏 instead of 𝕏(p_1,…,p_s) leaving the weights implicit.] as in . <ref>. The cluster category is

    𝒞_N_f=D^b(𝖼𝗈𝗁 𝕏( 2,⋯,2^N_f 2's ))/⟨τ^-1[1]⟩^.

For N_f≠4 this category is not periodic since the canonical sheaf has non-zero degree (in the physical language: the β-function is non-zero). We are in the situation discussed at the end of . <ref>, and the present example is also an illustration of that issue.

The 2-periodic triangle 2-CY subcategory ℱ(2)𝒞(N_f) is given by the orbit category of the derived category of finite-length sheaves. 
It consists of a ℙ^1 family of cluster tubes; in ℙ^1 there are N_f special points whose cluster tubes have period 2.
Let 𝒮_i,k, k∈/2, be the simples in the i-th special cluster tube, satisfying

    𝒮_i,k[1]≅τ𝒮_i,k≅𝒮_i,k+1

and let 𝒮_z be the simple over the regular point  z∈ℙ^1, τ𝒮_z≅𝒮_z. Thus [𝒮_z]=0 
and K_0(ℱ(2))
 is generated by the [𝒮_i,0] (i=1,…, N_f).
 The image of K_0(ℱ(2))
 in K_0(𝒞(N_f)) has index 2; indeed in K_0(𝒞(N_f)) we have an extra generator [𝒪] and a relation <cit.>

    2[𝒪]=∑_i=1^N_f[𝒮_i,0]

 Then as in . <ref> (for the special case N_f=4) we have
 
    K_0(𝒞(N_f))≅{(w_1,⋯,w_N_f)∈(12)^N_f |  w_i=w_j1}≡Γ_weight, 𝔰𝔭𝔦𝔫(2N_f)

 with
 
    ⟨⟨ [𝒮_i,0],[𝒮_j,0]⟩⟩=δ_i,j,

 that is, K_0(𝒞(N_f)) is the 𝔰𝔭𝔦𝔫(2N_f) weight lattice equipped with the dual Cartan pairing which is the correct physical extended ' t Hooft group for this model  which has π_1(G_eff)=1 and F=Spin(2N_f), as expected. 
 
 
 The case of N_f=4 was already presented in . <ref>. In that case 𝒦=0 (i.e. β=0), the theory is UV superconformal, and the cluster category is periodic.
The correlation between magnetic charge and Spin(8) representation becomes the fact that the modular group PSL(2,) acts on the flavor by triality <cit.>, see <cit.> for details from the cluster category viewpoint. 


 
 

  §.§.§ Example <ref>: Finer flavor structures,  U(1)_r anomaly, Witten effect
 The cluster category contains even more detailed information on the UV flavor physics of the corresponding 𝒩=2 QFT. Let us illustrate the finer flavor structures in the case  of SU(2) SYM coupled to N_f flavors[ Or, more generally, to several Argyres-Douglas systems of type D.] (Example <ref>).
 
 
 Note that the sublattice
 K_0(ℱ(2))⊂ K_0(𝒞(N_f)) is the weight lattice of SO(2N_f); since 
 ℱ(2) is the cluster sub-category of the `perturbative' (zero magnetic charge) sector, we recover the finer flavor structures mentioned at the end of . <ref>. In facts, eqn.(<ref>) is the image in the Grothendieck group of the equation which is the categorical expression of the U(1)_r anomaly <cit.>. Indeed, in the language of coherent sheaves, the U(1)_r anomaly is measured by the non-triviality of the canonical sheaf 𝒦  (think of a (1,1) σ-model: 𝒦 trivial means the target space is Calabi-Yau, which is the condition of no anomaly). 
 The coefficient of the β-function, b, is (twice) its degree,[ Notice that 𝒦=0 does not mean that 𝒦 is trivial but only that it is a torsion sheaf in the sense that 𝒦^m≅𝒪 for some integer m.] 𝒦=-χ(𝕏) <cit.>. 
 As a preparation to the examples of . 6, we briefly digress to recall how this comes about.

 


  
β-function and Witten effect. The AR translation τ acts on 𝖼𝗈𝗁 𝕏 as multiplication by the canonical sheaf <cit.>
 
    τ𝒜↦𝒜⊗𝒦≡𝒜⊗𝒪(ω⃗).
 
 Hence the U(1)_R anomaly and β-function may be read from the action of τ on the derived category D^b𝖼𝗈𝗁 𝕏 which we may identify as the IR category of BPS particles.[ Indeed, for N_f≤ 3, the triangle category D^b𝖼𝗈𝗁 𝕏 admits 𝗆𝗈𝖽 𝔤̂ as the core of a t-structure (here 𝔤̂ is an acyclic affine quiver in the mutation class of the model <cit.>; see also Example <ref>.] Now, in the cluster category of a weighted projective line, 𝒞(𝖼𝗈𝗁 𝕏)≡ D^b(𝖼𝗈𝗁 𝕏)/⟨τ^-1[1]⟩, one has τ≅[1], while [1] acts in the UV as the half-monodromy, that is, as a UV U(1)_r rotation by π.   
 In the normalization of ref.<cit.> (see their eqn.(4.3)), the complexified SU(2) Yang-Mills coupling at weak coupling, a→∞, is

    θ/π+8π i/g^2= -b/π ilog a+⋯,

Under a U(1)_r rotation by π, a→ e^π ia, the vacuum angle shifts as θ→θ-bπ.
Since a dyon of magnetic charge m carries an electric charge
m θ/2π mod 1 (the Witten effect <cit.>),
under the action of τ the IR electric/magnetic charges (e,m) should undergo the flow 

    τ (e,m)→ (e-m b/2,m).

For an object of D^b(𝖼𝗈𝗁 𝕏) the magnetic (electric) charge correspond to its rank (degree); then comparing eqns.(<ref>),(<ref>) we get b=-2𝒦=2 χ(𝕏). 
 


  
Finer flavor structures (. <ref>).   The Grothendieck group of
 𝖼𝗈𝗁 𝕏(2,…,2) is generated by  [𝒪], [𝒮_0], [𝒮_i,j] (i=1,…,N_f, j∈/2) subjected to the relations
 [𝒮_0]=[𝒮_i,0]+[𝒮_i,1] ∀ i, see Proposition 2.1 of  <cit.>. The action of τ in K_0(𝖼𝗈𝗁 𝕏) is

    [τ𝒮_i,j]=[𝒮_i,j+1],       [τ𝒪]-[𝒪]=(N_f-2)[𝒮_0]-∑_i=1^N_f[𝒮_i,0].

The difference [τ𝒪]-[𝒪] measures the non-triviality of the canonical sheaf, that is, the β-function/U(1)_r anomaly. In the cluster category, for all sheaf [τ𝒜]=-[𝒜], so that [𝒮_i,0]=0 and
the second eqn.(<ref>) reduces to (<ref>). Hence, as suggested by the physical arguments at the end of . <ref>, the non-perturbative flavor enhancement 
SO(2N_f)→Spin(2N_f) follows from the counting of the Fermi zero-modes implied by the axial anomaly.





  §.§.§ K_0(𝒞(Γ))_torsion: the Weil pairing


Let X∈𝒞(Γ)
The projection

    ⟨ S_i, F_Γ X⟩∈^n/B^n,

depends only on [X]. Rewrite the integral vector ⟨ S_i, F_Γ X⟩ in the -basis where B takes the normal form (<ref>)

    (⟨ S_1, F_Γ X⟩,⋯, ⟨ S_n, F_Γ X⟩) (w_1, w_2,⋯, w_f, u_1,1,u_2,1,⋯, u_1,s, u_2,s,⋯)
 and see its class  as an element of (ℚ^2/^2)^r

    (w_1, w_2,⋯, w_f, u_1,1,u_2,1,⋯, u_1,s, u_2,s,⋯)↦(u_1,1/d_1,u_2,1/d_1,⋯, u_1,s/d_s,u_2,s/d_s⋯) ∈ (ℚ^2/ℤ^2)^r.

The skew-symmetric matrix B then defines a skew-symmetric pairing

    2π i∑_s=1^r ϵ^ab u_a,s u^'_b,s/d_s∈ 2π i ℚ/ℤ.

The exponential of this expression is the canonical Weil pairing. Let us check one example.

[Pure SU(2)] The basis [P_1], [P_2] is canonical. Then the Weil pairing is

    (Z/2)^2× (Z/2)^2∋ (e,m) × (e^',m^')↦ (-1)^em^' -me^'.






 §.§ The cluster category as the UV line operators


We have seen that for a 𝒩=2 theory (with quiver property) the Grothendieck group K_0(𝒞(Γ)) is the extended 't Hooft group of additive and multiplicative conserved quantum numbers of the UV line operators and that this group is naturally endowed with all the structures required by physics, including the finer ones.

This amazing correspondence makes almost inevitable the identification of the the cluster category 𝒞(Γ) of the mutation-class of quivers with (super)potentials associated to a 4d 𝒩=2 model with the triangle category describing its UV BPS line operators. This identification has been pointed out by several authors working from different points of view <cit.>. In particular, the structure of the mutations of the Y–seeds in the cluster algebras lead to the Kontsevich-Soibelman wall crossing formula <cit.> (see <cit.> for  details).  This is just the action of the shift [1] on the cluster category which implements the quantum half monodromy 𝕂 (cfr. . <ref>).  

In section <ref> below we  check explicitly this identification by relating the geometrical description of the cluster category of a surface as given in the mathematical literature with the WKB analysis of line operators by GMN <cit.>.


For BPS line operators we also had a notion of `charge' which is useful to distinguish them, see . <ref>. We already mentioned there that both the definition and the properties of this `charge' have a precise correspondent in the mathematical notion of the index of a cluster object.
Now we may identify these two quantities. Note that, while the 't Hooft charge is invariant under quantum monodromy (i.e. under the shift [2]), the
index is not. This is the effect of non-trivial wall-crossing and, essentially, measures it <cit.>.

In . <ref> we saw that the index is fine enough to distinguish rigid objects of the cluster category. This is reminiscent  of our discussion in .<ref> about a (necessary) condition for UV completeness.  

In section . <ref>, building over refs.<cit.>,
we discuss how the interpretation of the cluster category 𝒞(Γ) as describing UV BPS line operators L_ind X(ζ) (labeled by the index of the corresponding cluster object X and the phase ζ of the preserved supersymmetry) leads to concrete expressions for their vacuum expectation values in the vacuum u

    ⟨ L_ind X(ζ)⟩_u.
 










 §.§ The perfect derived category 𝔓𝔢𝔯 Γ


To complete the understanding of the web of categories and functors 
describing the BPS physics of 
a 4d 𝒩=2 theory, it remains to discuss the physical meaning of the perfect category 
𝔓𝔢𝔯 Γ.
To the best of our knowledge,
an interpretation of the perfect category of a Ginzburg DG algebra has not appeared before in the physics literature.

We may extract some properties of the BPS objected described by the perfect category already from its Grothendieck group K_0(𝔓𝔢𝔯 Γ) and the basic sequence of functors

    0→ D^bΓ𝔓𝔢𝔯 Γ𝒞(Γ)→ 0.

The Grothendieck group K_0(𝔓𝔢𝔯 Γ) is isomorphic to the IR charge lattice Λ, so 𝔓𝔢𝔯 Γ is a category of IR BPS objects whose existence (i.e. “stability”) depends on the particular vacuum u. 𝔓𝔢𝔯 Γ yields the description of these physical objects from the viewpoint of the Seiberg-Witten low-energy effective Abelian theory. This is already clear from the fact that 𝔓𝔢𝔯 Γ contains the category describing the IR BPS particles i.e.
D^bΓ; BPS particles then form part of the physics described by 𝔓𝔢𝔯 Γ. A general object in 𝔓𝔢𝔯 Γ∖ D^bΓ differs from an object in the category D^bΓ in one crucial aspect: its total homology has infinite dimension, so (typically) infinite susy central charge and hence infinite energy. Then 𝔓𝔢𝔯 Γ is naturally interpreted as the category yielding the IR description of
 half-BPS branes of some kind. They may have infinite energy just because their volume may be  infinite.
 Although their central charge is not well defined, its phase is: it is just the angle θ corresponding to the subalgebra of supersymmetries under which the brane is invariant.
 
 On the other hand, the RG functor 𝗋 in (<ref>) associates to each IR object in 𝒪∈𝔓𝔢𝔯 Γ∖ D^bΓ a non-trivial UV line operator 𝗋(𝒪).
 This suggests a heuristic physical picture: let  𝒪∈𝔓𝔢𝔯 Γ∖ D^bΓ describe a BPS brane which is stable in the Coulomb vacuum u; this brane should be identified with the “state” obtained by acting with the UV line operator 𝗋(𝒪) on the vacuum u as seen in the low-energy Seiberg-Witten effective Abelian theory.  
 
 In order to make this proposal explicit, in the next section we shall consider a particular class of examples, namely the class 𝒮[A_1] theories <cit.>. In this case all three categories D^bΓ, 𝔓𝔢𝔯 Γ and 𝒞(Γ) are explicitly understood both from the Representation-Theoretical side (in terms of string/band modules <cit.>) as well as in terms of the geometry of curves on the Gaiotto surface C.
In this setting BPS objects are also well understood from the physical side since WKB is exact in the BPS sector.

Comparing the mathematical definition of the various triangle categories associated to a class 𝒮[A_1] model, and the physical description of the BPS objects, we shall check that the above interpretation of 𝔓𝔢𝔯 Γ is correct. 




  §.§.§ “Calibrations” of perfect categories
 To complete the story we need to introduce a notion of “calibration” on the objects of 𝔓𝔢𝔯 Γ which restricts in the full subcategory D^bΓ to the Bridgeland notion of stability.
The specification of a “calibration”
requires the datum of the Coulomb vacuum u and a phase θ=πϕ∈. Given an u (corresponding to specifying a central charge Z), the
ϕ-calibrated objects form
a full additive subcategory of 𝔓𝔢𝔯 Γ, 𝒦(ϕ), such that

    𝒫(ϕ)⊂𝒦(ϕ)⊂𝔓𝔢𝔯 Γ,    ∀ ϕ∈.
 
We use the term “calibration” instead of “stability” since it is quite a different notion with respect to Bridgeland stability (in a sense, it has “opposite” properties), and it does not correspond to the physical idea of stability.
These aspects are already clear from the fact that the central charge Z is not defined for general objects in  
𝔓𝔢𝔯 Γ. 

In the special case of the perfect categories arising from class 𝒮[A_1] QFTs, where everything is explicit and geometric, the calibration condition may be expressed in terms of flows of quadratic differentials, see . <ref>. 

We leave a more precise discussion of calibrations for perfect categories to future work. Here we limit ourselves to make some observations we learn from the class 𝒮[A_1] example.


 A phase πϕ∈ is called a BPS phase if the slice 𝒫(ϕ)⊂ D^bΓ contains non-zero objects. A phase πϕ is generic if it is not a BPS phase nor an accumulation point of BPS phases.


 In a class 𝒮[A_1] theory, assume there is no
BPS phase in the range [πϕ,πϕ^']. Then

    𝒦(ϕ)≅𝒦(ϕ^').

Moreover,
let πϕ be a generic phase. Then the ϕ-calibrated category  𝒦(ϕ)⊂𝔓𝔢𝔯 Γ has the form 

    𝒦(ϕ)≅𝖺𝖽𝖽 𝒯_ϕ

for an object 𝒯_ϕ∈𝔓𝔢𝔯 Γ such that

    𝗋(𝒯_ϕ)∈𝒞(Γ) is cluster-tilting.

In other words, the generic 𝒯_ϕ is a silting object of 𝔓𝔢𝔯 Γ.


We conjecture that something like the above Fact holds for general 4d 𝒩=2 theories.




§ CLUSTER AUTOMORPHISMS AND S-DUALITY





 §.§ Generalities


A duality between two supersymmetric theories  induces a (triangle) equivalence between
the triangle categories describing its 
BPS objects. The celebrate example is mirror symmetry between 
IIA and IIB string theories compactified on a pair of mirror Calabi-Yau 3-folds, ℳ, ℳ^∨.
At the level of the corresponding categories of BPS branes, mirror symmetry duality induces homological mirror symmetry, that is the equivalences of triangle categories <cit.>

    D^b(𝖢𝗈𝗁 ℳ)≅ D^b(𝖥𝗎𝗄 ℳ^∨),    
    D^b(𝖢𝗈𝗁 ℳ^∨)≅ D^b(𝖥𝗎𝗄 ℳ).


In fact, since to
a supersymmetric theory 𝒯 we associate a family of triangle categories, {𝔗_(a)}_a∈ I, depending on the class of BPS objects and the physical picture (e.g. IR versus UV), a dual pair of theories 𝒯, 𝒯^∨, yields
a family of equivalences of categories labeled by the index set I 

    𝔗_(a)𝔗_(a)^∨     a∈ I.

The several categories associated to the theory, {𝔗_(a)}_a∈ I, are related by
physical compatibility functors having the schematic form

    𝔗_(a)𝔗_(b)     a,b∈ I

(e.g. the `inverse RG flow' functor 𝗋
in eqn.(<ref>)). Physical consistency of the duality then require that we have commutative diagrams of functors of the form

    𝔗_(a)[r]^𝖽_(a)[d]_𝖼_(a,b)   𝔗_(a)^∨[d]^𝖼_(a,b)^∨
    𝔗_(b)[r]^𝖽_(b)   𝔗_(b)^∨

The philosophy of the present review is that the dualities are better understood in terms of such   diagrams of exact functors  between the relevant triangle categories. This idea may be applied to all kinds of dualities; here we are particularly interested in auto-dualities, that is, dualities of the theory with itself. The prime examples of auto-dualities is S-duality in 𝒩=2^* SYM and Gaiotto's 𝒩=2 generalized S-dualities <cit.>. 
One of the motivation of this paper is to use categorical methods to compute the group  𝕊 of S-dualities which generalize the PSL(2,) group for  𝒩=2^* as well as the results by Gaiotto.


An auto-duality induces a family of
exact functors 𝖽_(a)𝔗_(a)→𝔗_(a), one for each BPS category 𝔗_(a), such that:


  a) for all a∈ I, 𝖽_(a) is anautoequivalence of the triangle category 𝔗_(a);

  b) the {𝖽_(a)} satisfy physical consistency conditions in the form of commutative diagrams 

    𝔗_(a)[r]^𝖽_(a)[d]_𝖼_(a,b)   𝔗_(a)[d]^𝖼_(a,b)
    𝔗_(b)[r]^𝖽_(b)   𝔗_(b)




1) The group 𝔖 of generalized auto-dualities is the group of families 𝖽_(a) of autoequivalences satisfying eqn.(<ref>) modulo its  subgroup acting trivially on the physical observables. 
2) The group 𝕊 of (generalized) S-dualities is the quotient group
of 𝔖 which acts effectively on the (UV) microscopic local degrees of freedom of the theory.
   



With our definition of the S-duality group, the Weyl group of the flavor group is always part of the duality group 𝕊. It action on the free part of the cluster Grothendieck group is the natural one on the weight lattice.


 With this definition, the group 𝕊 for SU(2) SQCD with N_f=4 is <cit.>
 
    𝕊_SU(2), N_f=4=SL(2,)⋊Weyl(SO(8)).

  
 
  We shall see in Example <ref> that with this definition the S-duality group of a class 𝒮[A_1] theory is the tagged mapping class group of its Gaiotto surface, in agreement with the geometric picture in <cit.>, see also  <cit.>.
 
 

 §.§ Specializing to 𝒩=2 in 4d

 
 We specialize the discussion to the case of a 4d 𝒩=2 theory having the BPS quiver property. Such a theory is associated to a mutation-class of quivers with potential, hence to the three categories D^bΓ, 𝔓𝔢𝔯 Γ, 𝒞(Γ), discussed in the previous sections. They are related by the compatibility functors 𝗌, 𝗋 as in the exact sequence
 (<ref>). 
 


Applying Definition <ref> to the present set-up, we are lead to consider the  diagram of triangle functors 

    0 [r]     D^bΓ[d]_d_D[r]^𝗌   𝔓𝔢𝔯 Γ[d]_d_𝔓[r]^𝗋   𝒞(Γ)[d]_d_𝒞[r]    0
    
    0 [r]     D^bΓ[r]^𝗌   𝔓𝔢𝔯 Γ[r]^𝗋   𝒞(Γ)[r]    0
 
having exact rows and commuting squares,
where

    d_D∈Aut D^bΓ,   d_𝔓∈Aut 𝔓𝔢𝔯 Γ,  
    d_𝒞∈Aut 𝒞(Γ).

The group 𝔖 is the group of such triples (d_D,d_𝔓,d_𝒞) modulo the subgroup which acts trivially on the observables. The S-duality group 𝕊 is the image of 𝔖 under the homomorphism

    r𝔖→Aut 𝒞(Γ)/Aut 𝒞(Γ)_trivial,    
    (d_D,d_𝔓,d_𝒞)↦ d_𝒞.




  §.§.§ The trivial subgroup (Aut D^bΓ)_0


We start by characterizing the subgroup (Aut D^bΓ)_0⊂Aut D^bΓ of `trivial' auto-equivalences, i.e. the ones which leave the physical observables invariant. Since the Grothendieck group is identified with the IR charge lattice Λ, and charge is an observable, (Aut D^bΓ)_0 is a subgroup of the kernel Aut D^bΓ→Aut K_0(D^bΓ).
Next all ϱ∈(Aut D^bΓ)_0 should leave invariant the stability condition, that is the slicing 𝒫(ϕ), and hence the canonical heart 𝗆𝗈𝖽 J(Q,W) of D^bΓ. Since ϱ acts trivially on the Grothendieck group, it should fix all simples S_i.
Hence the projection Aut D^bΓ→Aut D^bΓ/(Aut D^bΓ)_0 factors through the quotient group

    Autph  D^bΓ :=Aut  D^bΓ/
    {autoequivalences preserving the simples S_i (element-wise)}.

An equivalence in the kernel of the projection Aut D^bΓ→Autph D^bΓ preserves (Q,W), the central charge Z, and the Grothendieck class λ. Hence it maps stable objects of charge λ into stable objects of charge λ. Comparing with eqn.(<ref>), we see that the net effect of an autoequivalence in the kernel is to produce an automorphism of projective varieties M_λ→ M_λ for each λ. Since the BPS states are the susy vacua of the 1d sigma-model with target space M_λ, this is just a change of variables in the SQM path integral, which leave invariant all physical observables[ 
The simplest example of such a negligible equivalence is the case of pure SU(2) whose quiver is the Kronecker quiver, 𝖪𝗋= ∙⇉∙.
The stable representations associated to the W boson are the simples in the homogeneous tube which form a ℙ^1 family (i.e. M_W boson≡ℙ^1) since the W boson belongs to a vector superfield. Then a negligible auto-equivalence is just a projective automorphism of ℙ^1.].
Since the auto-duality groups
are defined modulo transformations acting trivially on the observables,
Auteq  D^b is the proper auto-duality group 𝒮_IR at the level of the BPS category D^bΓ. 

The automorphisms of the quiver
extend to automorphisms of D^bΓ; let Aut(Q) be the group of quiver automorphisms modulo the ones which fix the nodes. Clearly,

    Autph D^bΓ= 
    Auteq D^bΓ⋊Aut(Q),

where

    Auteq  D^bΓ :=Aut  D^bΓ/
    {autoequivalences preserving the simples S_i (as a set)}.






  §.§.§ The duality groups 𝔖 and 𝕊



With the notation of section <ref>, Bridgeland in <cit.> and Goncharov in <cit.> showed that the following sequence

    0 →Sph D^bΓ→Auteq  D^bΓ→Aut_Q(CEG)→ 0

is exact. 
Here CEG stands for the cluster exchange graph (cfr. . <ref>): the clusters of the cluster algebra C_Γ are the vertices of the CEG and the edges are single mutations connecting two seeds; Aut_Q(CEG) is the graph automorphism group that sends the quiver to itself up to relabeling of the vertices. By construction this graph is connected.

 One has 

    Aut_Q(CEG)⊂Aut 𝒞(Γ),

i.e. the graph automorphisms (see <cit.>) are a subgroup of the autoequivalences of the cluster category.


Note that Auteq D^bΓ≡Auteq 𝔓𝔢𝔯 Γ,
the quotient group of Aut 𝔓𝔢𝔯 Γ by the subgroup fixing the Γ_i (as a set).
Indeed, all autoequivalences of 𝔓𝔢𝔯 Γ preserve
 the subcategory D^bΓ and hence restrict to autoequivalences of the bounded category; an autoequivalence ϱ∈Aut 𝔓𝔢𝔯 Γ which does not preserve the Γ_i's restricts to an element ϱ̅∈Aut D^bΓ which does not preserve the S_i's. Hence the restriction homomorphism
 
    Auteq 𝔓𝔢𝔯 Γ→Auteq D^bΓ,

 is injective. On the other hand, from eqn.(<ref>) we see that all autoequivalences in Auteq D^bΓ extend to autoequivalences in Auteq 𝔓𝔢𝔯 Γ:  indeed, the objects which are spherical in the subcategory D^bΓ remain spherical and 3-CY in the larger category 𝔓𝔢𝔯 Γ (cfr. eqn.(<ref>)), so the auto-equivalences is Sph D^bΓ extend to 𝔓𝔢𝔯 Γ; the autoequivalences in Aut_Q(CEG) are induced by quiver mutations, and hence induce auto-equivalences of 𝔓𝔢𝔯 Γ.  

Comparing with our discussion around eqn.(<ref>) we conclude:

 For a 4d 𝒩=2 theory with the BPS quiver property

    𝔖   ≅Auteq D^bΓ⋊Aut(Q),
    𝕊   ≅Aut_Q(CEG)⋊Aut(Q).





  §.§.§ Example: the group 𝕊 for SU(2) 𝒩=2^*



The mutation class of SU(2) 𝒩=2^* contains a single quiver, the Markoff one

   Q_Mar≡ ∙_1 @2->[r]     ∙_2 @2->[ld] 

              ∙_3 @2->[u]      

which is the quiver associated to the once punctured torus <cit.>. Clearly Aut(Q_Mar)≅_3, while all mutations leave Q_Mar invariant up to a permutation of the nodes. Consider the covering graph
CEG of CEG where we do not mod out the permutations of the nodes. Then CEG is the trivalent tree whose edges are decorated by {1,2,3}, the number attached to an edge corresponding to the nodes which gets mutated along that edge. One can check that there are no identifications between the nodes of this tree.

One may compare this ({1,2,3}-decorated) trivalent tree  with the ({1,2,3}-decorated) standard triangulation of the upper half-plane ℍ
given by the reflections of the  geodesic triangle of vertices 0,1,∞ (see ref.<cit.>). One labels
the nodes of a triangle of the standard triangulation by elements of {1,2,3}, and then extends (uniquely) the numeration to all other vertices so that the vertices of each triangle get different labels.
The sides of a triangle are numbered as their opposite vertex.
The dual of this decorated triangulation is our decorated trivalent graph CEG, see figure <ref>.
The arithmetic subgroup of the
hyperbolic isometry group,
PGL(2,)⊂ PGL(2,) preserves the standard triangulation of ℍ while permuting the decorations {1,2,3}. Since permutations are valid S-dualities, we get

    𝕊≅ PGL(2,)≅ PSL(2,)⋊_2

where the extra _2 may be identified with the Weyl group of the flavor SU(2).
Thus we recover as S-duality group in the usual sense (≡ the kernel of 
𝕊→Weyl(F))
the modular group PSL(2,) <cit.>. 
In the case of SU(2) 𝒩=2^* we have

    K_0(𝒞_Mar)≡coker B≅_2⊕_2⊕,

as expected for a quark in the adjoint representation (since π_1(G_eff)=_2), with the free part the weight lattice of SU(2)_flav. Hence the flavor Weyl group acts on K_0(𝒞_Mar) as -1, that is, as the cluster auto-equivalence [1]. Notice that the cluster category is 2-periodic, as expected for a UV SCFT with integral dimensions Δ.
 
  





 §.§ Relation to duality walls and 3d mirrors


The UV S-duality group 𝕊 has a clear interpretation: it is the usual S-duality group of the 𝒩=2 theory (twisted by the flavor Weyl group).
What about its IR counterpart 𝔖?

For Argyres-Douglas models we can put forward a precise physical interpretation based on the findings of <cit.>. Similar statements should hold in general. 


Given an element of the S-duality group, σ∈𝕊 we may construct a half-BPS duality wall in the 4d theory <cit.>: just take the theory for x_3<0 to be the image through σ of the theory for x_3>0 and adjust the field profiles along the hyperplane x_3=0 in such a way that the resulting Janus configuration is 12-BPS.  It is a domain wall interpolating between two dual 𝒩=2 theories in complementary half-spaces. On the wall live suitable 3d degrees of freedom interacting with the bulk 4d fields on both sides <cit.>. 
In this construction we may use a UV duality as well as an IR one <cit.>. Hence we expect to
get duality walls for all elements of 𝔖.
An element 𝔰∈𝔖 acts non-trivially
on the central charge Z so, in general, as we go from x_3=-∞ to x_3=+∞ we induce a non-trivial flow of the central charge Z in the space of stability functions. If lim_x_3→±∞Z is such that all the bulk degrees of freedom get an infinite mass and decouple, we remain with a pure 3d 𝒩=2 theory on the wall. Of course this may happen only for  special choices of 𝔰.
Thus we may use (suitable) 4d dualities to engineer 3d 𝒩=2 QFTs.  

The engineering of 3d 𝒩=2 theories as a domain wall in a 4d 𝒩=2 QFT, by central-charge flow in the normal direction, is precisely the set-up of
ref.<cit.>. 
In that paper one started from a 4d Argyres-Douglas of type 𝔤∈ ADE. The Z-flow along the x_3-axis was such that 
asymptotic behaviors as x_3→-∞ and x_3→+∞ were related in the UV by the action of the quantum half-monodromy 𝕂, that is, in the categorical language by the shift [1]∈𝕊. Two choices of IR duality elements, 𝔰, 𝔰^'∈𝔖, which produce the half-monodromy in the UV,  differ by an element of the spherical twist group (cfr. eqn.(<ref>))

    𝔰^'𝔰^-1∈Sph D^b.

The arguments at the end of . <ref> imply that for Argyres-Douglas of type 𝔤 the group Sph D^b is isomorphic to the Artin braid group of type 𝔤, ℬ_𝔤. 

  
As the title of ref.<cit.> implies, the explicit engineering of a 3d 𝒩=2 theory along those lines requires a specification of a braid, i.e. of an element of ℬ_𝔤. More precisely, in .5.3.2 of ref.<cit.> is given an explicit map (for 𝔤=A_r)

    (a braid in ℬ_𝔤)⟷(a 3d𝒩=2 Lagrangian).

So the Lagrangian description/Z-flow engineering of the 3d theories are in one-to-one correspondence with the 𝔰∈𝔖 such that r(𝔰)=[1]. It is natural to think of the 3d Lagrangian theory associated to 𝔰∈𝔖 as the duality wall associated to the IR duality 𝔰.
 Distinct 𝔰 lead to 3d theories which  superficially look quite different. However,  in this context, 3d mirror symmetry is precisely the statement that two theories defined by different IR dualities 𝔰,𝔰^'∈𝔖 which induce the same UV duality, r(𝔰^')=r(𝔰) produce equivalent 3d QFTs. From this viewpoint 3d mirror symmetry is a bit tautological, since the condition r(𝔰^')=r(𝔰) just says that the two 3d theories have the same description in terms of 4d
 microscopic degrees of freedom.   




 §.§ S-duality for Argyres-Douglas and SU(2) gauge theories


When (Q,W) is in the mutation-class of an ADE Dynkin graph (corresponding to an Argyres-Douglas model <cit.>) or of an ADE acyclic affine quiver (corresponding to SU(2) SYM coupled to matter such that the YM coupling is asymptotically-free <cit.>) to get 𝕊 we can equivalently study the automorphism of the transjective component of the AR quiver associated to the cluster category 𝒞(Γ): the inclusion above is due to the fact that we only consider the transjective component:

Let C be an acyclic cluster algebra and Γ_tr the transjective component of
the Auslander-Reiten quiver of the associated cluster category 𝒞(Γ). Then Aut^+C is the quotient
of the group Aut Γ_tr of the quiver automorphisms of Γ_tr, modulo the stabilizer Stab(Γ_tr)_0 of the points of this component. Moreover, if Γ_tr≅Δ, where Δ is a tree or of type Â then

    Aut  C = Aut^+C ⋊_2

 and this semidirect product is not direct.

In order to understand why this is the relevant component, we first recall that the Auslander-Reiten quiver of a cluster-tilted algebra always has a unique component containing local slices, which coincides with the whole
Auslander-Reiten quiver whenever the cluster-tilted algebra is representation-finite. This component
is called the transjective component and an indecomposable module lying in it is called a transjective
module. With this terminology, the main result is:

Let C be a cluster-tilted algebra and M, N be indecomposable transjective C-modules.
Then M is isomorphic to N if and only if M and N have the same dimension vector.

Therefore, since the dimension vector is the physical charge, we focus our attention to this class of autoequivalences. The classification results are summarized in table <ref>
where 

    H_p,q:=r,s|r^p=s^q, sr=rs
    
    G=τ,σ,ρ_1,ρ_n|ρ_1^2=ρ_n^2=1, τρ_1=ρ_1τ,τρ_n=ρ_nτ, τσ=στ, σ^2=τ^n-3,
    ρ_1σ=σρ_n, σρ_1 = ρ_nσ




[SU(2) with N_f≤3] SU(2) SQCD with N_f=0,1,2,3 correspond, respectively, to the following four affine 𝒩=2 theories <cit.>

    Â_1,1,  Â_2,1,  Â_2,2,  D̂_4.

A part for the flavor Weyl group Weyl(𝔰𝔭𝔦𝔫(2N_f)) (cfr. Example <ref>) we get a duality group  generated by the shift [1].
As discussed around eqn.(<ref>), this is equivalent to the shift of the Yang-Mills angle θ

    θ→θ-4π +N_fπ.

The case N_f=0 is special; physically one expects that the shift of θ by -2π should also be a valid S-duality. This shift should correspond to an auto-equivalence ξ of the N_f=0 cluster category with ξ^2=τ. Indeed, this is what one obtains from the automorphism of the transjective component see figure <ref>.
Alternatively, we may see the cluster category of pure SU(2) as the category of coherent sheaves on ℙ^1 endowed with extra odd
morphisms <cit.>. In this language τ acts as the tensor product with the canonical bundle τ↦⊗𝒦 (cfr. eqn.(<ref>)). Let ℒ be the unique spin structure on ℙ^1; we have the obvious auto-equivalence ξ↦⊗ℒ.
From ℒ^2=𝒦 we see that ξ^2=τ.






§ COMPUTER ALGORITHM TO DETERMINE THE S-DUALITY GROUP


The identification of the S-duality group with Aut_Q(CEG) yield a combinatoric characterization of S-dualities which leads to an algorithm to search S-dualities for an arbitrary 𝒩=2 model having a BPS quiver. This algorithm is similar in spirit to the mutation algorithm to find the BPS spectrum <cit.> but in a sense more efficient. The algorithm may be easily implemented on a computer; if the ranks of the gauge and flavor groups are not too big (say < 10), running the procedure on a laptop typically produces the generators of the duality group in a matter of minutes.





 §.§ The algorithm


The group Aut_Q(CEG) 
may be defined in terms of the transformations under quiver mutations of the d-vectors which specify the denominators of the generic cluster variables <cit.>. The actions of the elementary quiver mutation at the k–th node, μ_k, on the exchange matrix B and the d-vector d_i are

    μ_k(B)_ij   =
    - B_ij,     i=k  or j=k
    
    B_ij+max[-B_ik,0] B_kj+B_ik max[B_kj,0]—–   otherwise.
    μ_k(d)_l   ={[                                           d_l,                                           l≠ k; -d_k+max[∑_imax[B_ik,0]d_i,∑_imax[-B_ik,0]d_i]                                            l=k ].
 



A quiver mutation μ=μ_k_sμ_k_s-1⋯μ_k_1 is the composition  of a finite sequence of elementary quiver mutations μ_k_1, μ_k_2,⋯,μ_k_s. We write 𝖬𝗎𝗍 for the set of all quiver mutations.
Aut_Q(CEG)
is the group of quiver mutations which  
 leave invariant the quiver Q up to a permutation π of its nodes, 
modulo the ones which leave the d-vector invariant up to π: 

    Aut_Q(CEG)={μ∈𝖬𝗎𝗍 | ∃ π∈S_nμ(B)_i,j=B_π(i),π(j)}/{μ∈𝖬𝗎𝗍 | ∃ π∈S_nμ(B)_i,j=B_π(i),π(j) and μ(d)_i=d_π(i)},

while 𝕊=Aut_Q(CEG)⋊Aut(Q).
[A_2 cluster automorphisms]

Consider the quiver ∙_1→∙_2. The CEG is the pentagon in figure <ref>: every vertex is associated to a quiver of the form ∙_1→∙_2 or ∙_2→∙_1. 

Thus, in this case every sequence of mutations gives rise to a cluster automorphism. For example, consider μ_1: the quiver nodes get permuted under π=(1  2). We explicitly check – for example using Keller applet[See <https://webusers.imj-prg.fr/ bernhard.keller/quivermutation/>.] – that

    (μ_source)^5=μ_1μ_2μ_1μ_2μ_1=1

since (μ_source)^5 leaves the d-vectors invariant. From figure <ref> one sees that _5 is indeed the full automorphism group of the CEG of A_2. This result is coherent with
the analysis leading to table <ref>, as well as with the 
 tagged mapping class group of the associated Gaiotto surface, see Example <ref>.

 The explicit expression (<ref>) of the S-duality group is the basis of a computer search for S-dualities.
Schematically: let the computer generate a finite sequence of nodes of Q, k_1,⋯, k_s, then construct the corresponding mutation μ_k_sμ_k_s-1⋯μ_k_1=μ, and check whether it leaves the exchange matrix B invariant up to a permutation π. If the answer is yes, let the machine check whether
 μ(d)_i≠ d_π(i). If the answer is again yes
the computer has discovered a non-trivial S-duality and prints it. Then the computer generates another  sequence and go cyclically through  the same steps again and again.
After running the procedure for some time t, we get a print-out with a list ℒ_t of non-trivial S-dualities of our 𝒩=2 theory.
A Mathematica Code performing this routine is presented in Appendix <ref>.


If the S-duality group is finite (and not too huge) ℒ_t will contain the full list of S-dualities. However, the most interesting S-duality groups are infinite, and the computer cannot find all its elements in finite time.
This is not a fundamental problem for the automatic computation of the S-duality group. Indeed, the S-duality groups, while often infinite,
are expected to be finitely generated, and in fact finitely presented. If this is the case, we need only that the finite list ℒ_t produced by the computer contains a complete set of generators of 𝕊.
Taking various products of these generators, and checking which products act trivially on the d-vectors, we may find the finitely many relations. The method works better if we have some physical hint on what the generators and relations may be.

Of course, the duality group obtained from the computer search is a priori only a subgroup of the actual 𝕊 because there is always the possibility of further generators of the group which are outside our range of search. However, pragmatically, running the procedure for enough time, the group one gets is the full one at a high confidence level. 



 §.§ Sample determinations of S-duality groups


We present a sample of the results obtained 
by running our Mathematica Code.  

[SU(2) 𝒩=2^* again] The CEG automorphism group for this model was already described in . <ref>.
Recall that PSL(2,ℤ) is the quotient of the braid group over three strands, ℬ_3 by its center Z(ℬ_3)

    PSL(2,ℤ)≅ℬ_3/Z(ℬ_3).
 
Running our algorithm for a short time returns a list of dualities which in particular contains the two standard generators of the braid group σ_1,σ_2∈ℬ_3, which correspond to  the following sequences of elementary quiver mutations:

    σ_1:=μ_1μ_2,    and    σ_2:=μ_1μ_3,  with permutation π=(1 3 2).

One easily checks the braid relation

    σ_1σ_2σ_1=σ_2σ_1σ_2  up to permutation,

as well as that the generator of the center Z(ℬ_3), (σ_2σ_1)^3,
acts trivially on the cluster category: indeed, it sends the initial dimension vector d⃗=-Id_3 × 3 to itself. From eqn.(<ref>) we conclude that the two S-dualities σ_1, σ_2 generate a PSL(2,) duality (sub)group. 
In facts, 𝕊/PSL(2,)≅_2 where the class of the non-trivial _2 element may be
represented (say) by μ_1. Indeed the map

    𝕊→_2≡Weyl(F_flav)
 
send the mutation μ to (-1)^ℓ(μ), where the length ℓ(μ) of
μ≡μ_k_sμ_k_s-1⋯μ_k_1 is s (length is well defined mod 2).

 

[SU(2) with N_f=4] We use the quiver in figure <ref> where for future reference we also draw the corresponding ideal triangulation of the sphere with 4 punctures <cit.>.

The following two even-length sequences of mutations leave the quiver invariant:

    S=μ_2μ_3μ_2μ_0μ_2μ_5μ_3μ_0,
    
    T= μ_5μ_2μ_0μ_3μ_5μ_3μ_4μ_2μ_4μ_1μ_4μ_2μ_4μ_5μ_1μ_2.

These sequences of mutations satisfy the following relations:

    S^4=1,   (ST)^6=1,   T  has infinite order.

Moreover, T and S commute with S^2 and (ST)^3. Write _2×_2 for the subgroup generated by S^2 and (ST)^3.
Then we have

    1 →_2×_2→⟨  S, T ⟩→ PSL(2,)→ 1.

Again 
this shows that
the duality sub-group ⟨ S,T ⟩ generated by S and T is equal to the
 the mapping class group of the sphere with four punctures (cfr. Proposition 2.7 of <cit.>). 
 In fact one has Aut_Q(CEG)/⟨ S,T ⟩≅_2; geometrically (see next section) the extra _2 arises because for class 𝒮[A_1] theories Aut_Q(CEG)
 is the tagged mapping class group of the corresponding Gaiotto surface (Bridgeland theorem<cit.>);
 the extra _2 is just the change
 in tagging. This extra _2 is also detected by the computer program which turns out dualities of order 12 and 8 which are not contained in ⟨ S,T ⟩ but in its _2 extension. 
Taking into account the S_4 automorphism of the quiver, we recover PSL(2,)⋉Weyl(𝔰𝔭𝔦𝔫(8))
with the proper triality action of the modular group on the flavor weights
<cit.>. For an alternative discussion of the S-duality group of this model as the automorphism group of the corresponding cluster category, see ref.<cit.>.


[E_6 Minahan-Nemeschanski] This SCFT is the T_3 theory, that is, the Gaiotto theory obtained by compactifying the 6d (2,0) SCFT of type A_2 on a sphere with three maximal punctures <cit.>. Since the three-punctured sphere is rigid, geometrically we expect a finite S-duality group. The homological methods of <cit.> confirm this expectation.
The computer search produced a list of group elements of order
2,3,4,5,6,8,9,10,12 and 18. Since, with our definition, the S-duality group should contain the 
Weyl group of E_6, we may compare this list with the list of orders of elements of Weyl(E_6),

    {2,3,4,5,6,8,9,10,12}.
 We see that the two lists coincide, except for 18.
Thus the S-duality group is slightly larger than the Weyl group, possibly just Weyl(E_6)⋊_2, where _2 is the automorphism of the Dynkin diagram.
Notice that this is the largest group which may act on the free part of the cluster Grothendieck group
(since it should act by isometries of the Tits form).



[Generic T_𝔤 theories]
By the same argument as in the 
previous Example, we expect the S-duality group to be finite for all T_𝔤 (𝔤∈ ADE) theories. 
We performed a few sample computer searches getting agreement with the expectation.




[E_7 Minahan-Nemeschanski]
The computer search for this example produced a list of group elements of order
2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18 and 30. Since our S-duality group contains the 
Weyl group of the flavor E_7, we  compare this list with the list of orders of elements of Weyl(E_7),
{2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 30}. We see that the two lists coincide. It is reasonable to believe that the full S-duality group coincides with Weyl(E_7). This is also the largest group preserving the flavor Tits form. 




 §.§ Asymptotic-free examples


As an appetizer, let us consider a 
𝒩=2 gauge theory 
with a gauge group of the form SU(2)^k coupled to (half-)hypermultiplets in some representation of the gauge group so that all Yang-Mills couplings g_i (i=1,…,k) have strictly negative β-functions. As discussed in . <ref>, the fact that the theory is asymptotically-free means that its cluster category 𝒞 is not periodic. However, its Coulomb branch is parametrized by k operators whose dimension in the UV limit g_i→0 becomes Δ=2. As in Example <ref>, this implies the existence of a 1-periodic sub-category ℱ(1)⊂𝒞. Iff all YM couplings g_i are strictly asymptotically-free, the category ℱ(1) consists of k copies of the 1-periodic sub-category of pure SU(2), Example <ref>.
In such an asymptotic-free theory the S-duality group is bound to be `small' since all auto-equivalence σ of the cluster category should preserve the 1-periodic sub-category ℱ(1); therefore, up to (possibly)  permutations of the various SU(2) gauge factors, σ should restrict to a subgroup of autoequivalences of the periodic category ℱ(1)_pure of pure SU(2) SYM.
As we saw in Example <ref>, the S-dualities corresponding to shifts of the Yang-Mills angle θ preserve[ Physically this is obvious. Mathematically, consider e.g. the shift shift θ→θ-4π+N_fπ in SU(2) with N_f flavors. It corresponds to the auto-equivalence ↦[1], which acts trivially on the 1-periodic subcategory.] the subcategory ℱ(1)_pure.
Thus
besides shifts of the various theta angles, permutations of identical subsectors, and flavor Weyl groups/Dynkin graph automorphism, we do not expect additional S-dualities in these models.
Let us check this expectaction against the computer search for dualities in a tricky example.   


[SU(2)^3 with 12(2,2,2)] 
A quiver for this model is given in figure <ref>. 

In this case the cluster Grothendieck group K_0(𝒞_pris)  is pure torsion, since a single half-hyper carries no flavor charge.
The three SU(2) gauge couplings g_i are asymptotically-free and the cluster category 𝒞_pris is not periodic but it contains the
1-periodic subcategory ℱ(1)⊂𝒞_pris described above[ Notice that there is no periodic sub-category associated to the quark sector; this is related to the absence of conserved flavor currents in this model.]. The S-duality group is then expected to consists of permutations of the three SU(2)'s and the three independent shifts of the Yang-Mills angles θ_i→θ_i-2π, that is,
𝕊=S_3⋉^3.

The computer algorithm produced the following three commuting generators of the cluster automorphism group of infinite

    θ_1=μ_23μ_22μ_11,    θ_2=μ_21μ_23μ_12,    θ_3=μ_22μ_21μ_13.





These three generators are identified with the three θ-shifts.


 Since the model is of class 𝒮[A_1] (with irregular poles), the S-duality group may also be computed geometrically (see section 7). The
computer result is of course consistent with geometry: each θ_i translation correspond to a twist around one of the three holes on the sphere: their order is clearly infinite and the three twists commute with one another.




 §.§ Q-systems as groups of S-duality


The above discussion may be generalized to all 𝒩=2 QFTs having a weakly coupled Lagrangian formulation. If the gauge group G is a product of k simple factors G_i, we expect the S-duality group to contain a universal subgroup ^k consisting of shifts θ_i→θ_i-b_iπ, with b_i the β-function coefficient of the i-th YM coupling. One may run the algorithm and find the universal subgroup; however, just because it is universal, its description in terms of quiver mutations also has a universal form which is easy to describe.

We begin with an example.


[Pure SYM: simply-laced gauge group] If the gauge group G is simply-laced, the exchange matrix of its quiver may be put in the form <cit.>[ In particular, coker B=Z(G)^∨⊕Z(G) is the correct 't Hooft group for pure SYM.]

    B=([  0 -C; -C  0 ])≡ C⊗ iσ_2,     C≡the Cartan matrix of G.

For instance, the quiver for SU(N) SYM is represented in figure <ref>.

These quivers are bipartite:
we may color the nodes black and white so that a node is linked only to nodes of the opposite color. Quiver mutations at nodes of the same color commute, so the product 
    ν=∏_i whiteμ_i
 is well-defined. Moreover interchanging (black) ↔ (white) yields the opposite quiver Q^opp which is isomorphic to Q via the node permutation π=1⊗σ_1. The effect of the canonical mutation ν on the quiver is to invert all arrows i.e. it gives back the same quiver up to the involution π. Thus ν corresponds to an universal duality of pure SYM. One checks that it has infinite order, i.e. generates a subgroup of S-dualities isomorphic to .

This sub-group  of S-dualities has different physical interpretations/applications in statistical physics <cit.> as well as in the context of the Thermodynamical Bethe Ansatz <cit.>.
Indeed, consider its index 2 subgroup generated by the square of ν

    ν^2=∏_j  blackμ_j ∏_i whiteμ_i.

The repeated application of the S-duality ν^2 generates a recursion relation for the cluster variables which is known as the Q-system of type G. It has deep relation with the theory of quantum groups; moreover it generates a linear recursion relation of finite length and has many other “magical” properties
<cit.>.

We claim that the duality ν^2
corresponds to a shift of θ.
Indeed, the cluster category in this case is the triangular hull of the orbit category of D^b(𝗆𝗈𝖽 Â_1,1⊗ G) and ν^2 corresponds to the auto-equivalence τ⊗Id
<cit.>. Comparing the action of τ⊗Id in the covering category with the Witten effect (along the lines of .<ref>) one gets the claim.


[SYM with non-simply laced gauge group]
The authors of ref.<cit.> defined Q-systems also for non-simply laced Lie groups. To a simple Lie group G one associates a quiver
and a mutation ν^2 which generates a group  which has all the required “magic” properties.
In ref.<cit.>  it was shown that the non-simply-laced Q-system does give the quiver description of the BPS sectors of the corresponding SYM theories.
The Q-system group is again the group of S-dualities corresponding to θ-shifts.  


[General 𝒩=2 SQCD models]
We may consider the general Lagrangian case in which the gauge group is a product of simple Lie groups, ∏_j G_j and we have hypermultiplets in some representation of the gauge group.
The quivers for such a theory may be found in refs.<cit.>.
For instance figure <ref> shows the quiver for SU(M)× SU(N) gauge theory coupled to 2 flavors of quark in the (N,1) and a quark bifundamental in the (N̅,M). It is easy to check that the two canonical mutations of the subquivers associated to the two simple factors of the
gauge group 

    ν_∘=∏_i=∘μ_i,    ν_□=∏_i=□μ_i,
  
leave the quiver invariant up to the permutation 
∘↔∙ and, respectively, □↔▪. The construction extends straightforwardly to any number of gauge factors G_j and all matter representations. The conclusion is that  we have a canonical  subgroup of the S-duality group per simple factor of the gauge group. It corresponds to  shifts of the corresponding θ-angle.
If the matter is such that the β-function vanishes, the full cluster category becomes periodic, and we typically get a larger S-duality group.



One can convince himself that the sequence of mutations μ does not change the quiver and that its order, in all the above cases, is infinite, as it is for the shift is the θ_j's. 




§ CLASS 𝒮 QFTS: SURFACES, TRIANGULATIONS, AND CATEGORIES



In this section we focus on a  special class of 𝒩=2 theories: the Gaiotto 𝒮[A_1] models <cit.>. We study them for two reasons: first of all they are interesting for their own sake, and second for these theories the three categories D^bΓ, 𝔓𝔢𝔯 Γ, 
and 𝒞(Γ) have geometric constructions, directly related to the WKB analysis of <cit.>. Comparing the categorical description with the results of refs.<cit.> we check the correctness of our physical interpretation of the various categories and functors. 


Class 𝒮[A_1] theories are obtained by the compactification of the 6d (2,0) SCFT of type A_1 over a complex curve C having regular and irregular punctures <cit.>. 
If there is at least one puncture, these theory have the quiver property <cit.>, and their quivers with superpotentials are constructed in terms of an ideal triangulation of C <cit.>. In the geometrical setting of Gaiotto curves, we can interpret the categories defined in section <ref> as categories of (real) curves on the spectral cover of the Gaiotto curve C. 

When only irregular punctures are present, the quiver with potential arising from these theories <cit.> has a Jacobian algebra which is gentle <cit.>, and hence all triangle categories associated to its BPS sector, eqn.(<ref>), have a simple explicit description.[ In facts, there is a systematic procedure, called gentling in ref.<cit.> which allow to reduce the general class 𝒮[A_1] model to one having a gente Jacobian algebra.] 
When only regular punctures are present, the 𝒩=2 theory has a Lagrangian formulation (which is weakly coupled in some corner of its moduli space)
and is UV superconformal. In particular the corresponding cluster category is 2-periodic, as the arguments of . <ref> imply.




 §.§ UV and IR descriptions

The main reference for this part is <cit.>. 

In the deep UV a class 𝒮[A_1] 𝒩=2 theory is described by the Gaiotto curve C, namely a complex curve of genus g
with a number of punctures x_i∈ C. Punctures are of two kinds: regular punctures (called simply punctures) and irregular ones (called boundaries). The i–th boundary carries a positive integer k_i≥1 (the number of its cilia); sometimes it is convenient to regard regular punctures as  boundaries with k_i=0. Iff k_i≤ 2 for all i, the 𝒩=2 theory is a Lagrangian model with gauge group[ When g=0, the theory is defined only if b≥ 1 or b=0 and p≥ 3; in case p=0, b=1 we require k≥ 4; when g=1 we need p+b≥ 1. Except for the case p=0, b=1, corresponding to Argyres-Douglas of type A, m in eqn.(<ref>) is ≥0. m=0 only for Argyres-Douglas of type D <cit.>.]

    𝒢=SU(2)^m,     m=3g-3+p+2b  where 
    p=#{regular punctures}
    
    b=#{boundaries}.
 
If b=0 the theory is superconformal in the UV, and the space of exactly
marginal coupling coincides with the moduli space of genus g curves with p punctures, ℳ_g,p, whose complex dimension is m ≡ the rank of the gauge group 𝒢.
Instead, if b≥ 1 (and m≥ 2),
b out of the m SU(2) factors in the Yang-Mills group 𝒢 have asymptotically free couplings; these b YM couplings
go to zero in the extreme UV,
so that the UV marginal couplings are again equal in number to the complex deformations ℳ_g,p+b of C.

If some of the boundaries have k_i≥ 3, we have a gauge theory with the same gauge group SU(2)^m coupled to “matter” consisting, besides free quarks (in the fundamental, bi-fundamental, and three-fundamental of 𝒢), in an Argyres-Douglas SCFT of type D_k_i for each boundary[ Argyres-Douglas of type D_1 is the empty theory and the one of type D_2 a fundamental quark doublet.] <cit.>. The space of exactly marginal deformations is as before.

The IR description of the model is given by the Seiberg-Witten curve Σ which, for class 𝒮[A_1], is a double cover of C. More precisely, one considers in the total space of the ℙ^1-bundle

    ℙ(K_C⊕𝒪_C)→ C    where K_C   canonical line bundle on C
    𝒪_C   trivial line bundle on C

the curve 

    Σ≡{ y^2=ϕ_2(x) z^2 |  (y,z) homogeneous coordinates in the fiber}→ C,

where ϕ_2(x) is a quadratic differential on C with poles of degree at most k_i+2 at x_i.
The Seiberg-Witten differential is the tautological one

    λ= y/z dx

whose periods in Σ yield the 𝒩=2 central charges of the BPS states.

The dimension of the space of IR deformations is then[ This formula holds under the condition ℳ_g,p+b=3g-3+p+b≥0.] [ Here the asymptotically-free gauge couplings are counted as IR deformations.]

    s   = H^0(C, P K_C^2)≡ 3g-3+2p+2b+∑_i k_i
       where P=∑_i(k_i+2)[x_i],
  
so that the total space of parameters, UV+IR, has dimension

    n=m+s= 6g-6+3p+3b+∑_ik_i.

There are two kinds of IR deformations, normalizable and unnormalizable ones. The unnormalizable ones correspond to deformations of the Lagrangian, while the normalizable ones  to moduli space of vacua (that is, Coulomb branch parameters);
their dimensions are[ As written, these equations hold even if the condition in footnote <ref> is not satisfied. Notice that we count also the dimensions of the internal Coulomb branches of the matter Argyres-Douglas systems.]

    s_nor   ≡(Coulomb branch)=3g-3+p+b+∑_i (k_i-[k_i/2]),
     s_un-nor   =s-s_nor.






The double cover Σ→ C is branched over the zeros 
w_a∈ C of the quadratic differential ϕ_2(x). Their number is

    t= 4g-4+2p+2b+∑_i k_i,

but their positions are constrained by the condition that the divisor ∑_a[w_a] is linear equivalent to PK_C^2, so that their positions 
depend on only t-g parameters;
ϕ_2(x) depends on one more parameter

    s=t-g+1

since the positions of its zeros fix ϕ_2(x) only up to an overall scale (that is, up to the overall normalization of the Seiberg-Witten differential, which is the overall mass scale).

Therefore, up to the overall mass scale, giving the cover Σ→ C is equivalent to specifying the zeros w_a∈ C of the quadratic differential. Indeed, double covers are fixed, up to isomorphism, by their branching points. 
We shall refer to the points w_a∈ C as decorated points on the Gaiotto curve C. 

In summary: the UV description of a class 𝒮[A_1] amounts to giving the datum of the Gaiotto curve C, that is, a complex structure of a genus g Riemann surface and a number of marked points x_i∈ C together with a non-negative integer k_i at each marked point. To get the IR description we have to specify, in addition, the decorated points
w_a∈ C (whose divisor is constrained to be linear equivalent to PK_C^2). We may equivalently state this result in the form:


In theories of class 𝒮[A_1], to go from the IR to the UV description we simply delete (i.e. forget)
the decorated points of C.


We shall see in .<ref> below that this `forget the decoration' prescription is precisely the map denoted 𝗋 in the exact sequence of triangle categories of eqn.(<ref>).










 §.§ BPS states


In class 𝒮[A_1] 𝒩=2 theories, the natural BPS objects are described by (real) curves η on the Seiberg-Witten
curve Σ which are calibrated by the Seiberg-Witten differential <cit.>

    λ= y/z dx≡√(ϕ_2(x)) dx,

that is, they are required to satisfy the condition (we set ϕ≡ϕ_2 (dx)^2),




















    √(ϕ)|_η=e^iθ dt,    here t ∈,

for some real constant θ, and are maximal with respect to this condition. Being maximal, η may terminate only at marked or decorated points.[ We call
marked/decorated points in Σ the pre-images of marked/decorated points on C.]  BPS particles have finite mass, i.e. they correspond to calibrated arcs η with |Z(η)|<∞ where 
the central charge of the would be BPS state η,  is 

    Z(η)=∫_ηλ.

In this case, the parameter θ in eqn.(<ref>) is given by θ= Z(η).
Arcs η associated to BPS particles may end only at decorations.
All other maximal calibrated arcs have infinite mass and are interpreted as BPS branes <cit.>.

There are two possibilities for BPS particles:


  * they are closed arcs  connecting zeros of ϕ. These calibrated arcs are rigid and hence correspond to BPS hypermultiplets;

  * they are loops. Such calibrated arcs form ℙ^1-families and give rise to BPS vector multiplets.




  
Conserved charges. From eqn.(<ref>) we see that the central charge of an arc η
factors through its homology class η∈ H_1(Σ,). More precisely,
since the Seiberg-Witten differential λ is odd under the covering group _2≅𝖦𝖺𝗅(Σ/C), Z factors 
through the free Abelian group

    Λ≡ H_1(Σ,)_odd,
       rank Λ=2(g(Σ)-g(C))+#{k_i even}.

Applying the Riemann-Hurwitz formula[ Compare eqns.(6.26)-(6.28) in ref.<cit.>.] to the cover Σ→ C, we see that the rank of Λ is equal to the number n of UV+IR deformations, see eqn.(<ref>). In turn n is the number of conserved charges (electric, magnetic, flavor) of the IR theory. Hence the group homomorphism 

    ZΛ→,     [η]↦∫_ηλ,

is the map which associates to the IR charge γ∈Λ of a state of the 𝒩=2 theory
the corresponding central charge Z(γ). An arc with homology
[η]∈Λ then has `mass'

    ∫_η |λ|≥ |Z(η)|

with equality iff and only if it is calibrated, that is, BPS.


To get the corresponding  UV statements,
we apply to these results our
RG principle, that is, we forget the decorations. BPS particles then disappear (as they should form the UV perspective), while BPS branes project to arcs on the Gaiotto curve C. Several IR branes project to the same arc on C. The arcs on C have the interpretation of UV BPS line operators, and the branes which project to it are the objects they create in the given vacuum (specified by the cover Σ→ C) which may be dressed (screened) in various ways by BPS states, so that the IR-to-UV correspondence is
many-to-one in the line sector. 

The UV conserved charges is the projection of Λ; over ℚ all electric/magnetic charges are projected out by the oddness condition, and we remain with just the flavor lattice. However over  the story is more interesting and we get <cit.>

    Λ_UV≅^#{k_i even}⊕2-torsion.


Comparing with our discussion in the Introduction, we see that
Λ and Λ_UV should be identified with the Grothedieck group of the triangle categories D^bΓ and 𝒞(Γ), respectively. We shall check these identifications below.









 §.§ Quadratic differentials


We want to study the BPS equations to find the BPS spectrum of the theory. We start by analyzing the local behavior of the flow of the quadratic differential.
The quadratic differential near a zero can be locally analyzed in a coordinate patch where ϕ∼ w; thus we have to solve the following equation:

    √(w) dw/dt=e^i θ,

which gives w(t)=(3/2te^i θ+w_0^3/2)^2/3. We plot here the solution:

The three straight trajectories, which all start at the zero of ϕ, end at infinity in the poles of ϕ, i.e. the marked points on the boundaries of C. Since all zeros of ϕ are simple by hypothesis, it is possible to associate a triangulation to a quadratic differential by selecting the flow lines connecting the marked points as shown in the following figure: 

Moreover, by construction, it is clear that to each triangle we associate a zero of ϕ. These are the decorating points Δ of section <ref>.
If we make θ vary, we deform the triangulation up to a point in which the triangulation jumps: at that value of θ=θ_c, two zeros of ϕ are connected by a curve η: this curve is the stable BPS state. From what we have just stated, it will be clear that the closed arcs of section <ref> will correspond to BPS states.
Before and after the critical value of θ=θ_c, the triangulation undergoes a flip. Flips of the triangulation correspond to mutations at the level of BPS quiver (see section <ref> on how to associate a quiver to a triangulation). This topic is develop in full details in <cit.>.

Moreover, the second class of BPS objects, i.e. loops representing vector-multiplets BPS states, appears in one-parameter families and behave as in example <ref>; the map X of section <ref> will allow us to write the corresponding graded module X(a) ∈ D^b(Γ).


What about the curves connecting punctures or marked points (but not zeros of the quadratic differential)?[ These are the objects in the perfect derived category 𝔓𝔢𝔯 Γ] To answer this question, we have to take a detour into line operators.

We point out that the space of quadratic differential is isomorphic to the Coulomb branch of the theory. In a recent paper of Bridgeland <cit.> it is stated that the space of stability conditions of D^b Γ satisfy the following equation:

    Stab^0(D^b Γ)/Sph(D^bΓ)≅Quad(S,M).

Thus, the Coulomb branch is isomorphic to the space of stability conditions, up to a spherical twist. Indeed, a stability condition is a pair (Z,𝒫), where Z is the stability function (i.e. the central charge) and the category 𝒫 is the category of stable objects, i.e. stable BPS states (see section <ref> for more details).



 §.§ Geometric interpretation of defect operators

The main reference for this section is <cit.> and also <cit.>. There it is explained how to use M-theory to construct vertex operators, line operators and surface operators by intersecting an M5 brane with an M2 one. In particular,
 

  * a vertex operator corresponds to a point in the physical space: it means that the remaining two dimensions of the M2 brane are wrapped on the Gaiotto surface S, forming a co-dimension 0 object.

  * a line operator corresponds to a one-dimensional object in the physical space: the remaining one dimension of the M2 brane is wrapped around the surface S as a 1-cycle (i.e. non self-intersecting closed loop) or as an arc connecting two punctures or marked points.

  * a surface operator has two spacial dimensions and thus it is represented by a puncture over the surface C.

These operators are physically and geometrically related. A vertex operator, since it is a sub-variety of codimension 0 over the complex curve S, can be interpreted as connecting line operators (i.e the loops at the boundary of the sub-variety representing the vertex operator). Moreover, a line operator γ can be interpreted as acting on a surface operator x and transporting the point corresponding to the surface operator around the curve S; the action on its dual Liouville field (<cit.>) is the monodromy action along the line associated to the line operator γ. Finally, as explained in <cit.>, when we describe the curve S via a quadratic differential ϕ, we can interpret the poles of ϕ as surface operators and the arcs connecting marked points and the closed loops correspond to Verlinde line operators[ Traditionally, they are defined by
composing a sequence of elementary operations on conformal blocks, each corresponding
to a map between spaces of conformal blocks which may differ in the number or type
of insertions. Roughly speaking, one inserts an identity operator into the original
conformal block, splits into two conjugate chiral operators ϕ_a and ϕ̅_a, transports ϕ_a
along γ and then fuses the operators ϕ_a and ϕ̅_a back to the identity channel.]. If we consider two curves γ_1, γ_2 that intersect at some point, the line operators L(γ_1,ζ),L(γ_2,ζ) corresponding to these curves do not commute (see section <ref>). In the case of self-intersecting arcs, we get more general Verlinde operators: as line operators, they can be decomposed into a linear combination of non self-intersecting line operators by splitting the intersection in all possible pairs.


 §.§ Punctures and tagged arcs

The main reference, for this short section, is <cit.>. In there it is pointed out how the arcs over the Gaiotto curve C are tagged arcs: at each singularity (extrema of the arc) we have to choose the eigenvalue of the monodromy operator around that singular point. In particular, in section 8 of <cit.>, we discover that for irregular singularity, the tagging is not necessary, since it is equal to an overall rotation of the marked points around that boundary component. For punctures, on the other hand, it is not the case: we have to specify a _2-tagging. This boils down to a tagged triangulation, as explained in <cit.>. Therefore, when considering surfaces with punctures, we have to consider tagged arcs and not simple arcs. This observation will be important when we discover that the S-duality group is the tagged mapping class group of the Gaiotto surface C (see section <ref>).

































 §.§ Ideal triangulations
surfaces and quivers with potential


Let C be the Gaiotto curve of a class 𝒮[A_1] model. The  invariant of the family of QFTs obtained by continuous deformations of it is the topological type of C.
More precisely, we
define the underlying topological surface S of C by the following steps: i) forget the complex structure, and
ii) replace each irregular punture with a boundary component ∂ S_i with k_i≥ 1 marked points (ordinary punctures on C remain punctures on S). S is then the invariant datum which describes the continuous family of 𝒮[A_1] theories.

 An ideal triangulation of S is a maximal set of pairwise non-isotopic arcs ending in punctures and marked points which do not intersect (except at the end points) and are not homotopic to a boundary arc.[ A boundary arc is the part of a boundary component between two adjacent marked points.] All ideal triangulations have the same number of arcs <cit.>

    n=6g-6+3p+3b+∑_ik_i.

Note that is the same number as the number of UV+IR deformations, eqn.(<ref>), as well as the number of IR conserved charges rank Λ, eqn.(<ref>).

To an ideal triangulation T of the surface of S we associate a quiver with superpotential (Q,W). 
The association S↔ (Q,W) is intrinsic in the following sense:

 Let (Q,W) be the quiver with potential associated to an ideal triangulation T of the surface S. A quiver with potential (Q^',W^') is mutation equivalent to (Q,W) if and only if it[ This is slightly imprecise since, in presence of regular punctures W contains free parameters <cit.>. The statement in the text refers to the full family of allowed W's.] arises from an ideal triangulation T^' of the same surface S. 


In view of Corollary <ref>, an important result is:

 The quiver with superpotential of an ideal triangulation is always Jacobi-finite.


Thus an ideal triangulation T defines a Jacobi-finite Ginzburg DG algebra Γ≡Γ(Q,W) and therefore also the three triangle categories D^bΓ,
𝔓𝔢𝔯 Γ and 𝒞(Γ) described in . 2.5, 2.6. Then

 Up to isomorphism, the three triangle categories D^bΓ,
𝔓𝔢𝔯 Γ and 𝒞(Γ) are independent of the chosen triangulation T, and hence are intrinsic properties of the topological surface S.




It remains to describe the quiver with potential (Q,W) associated to the ideal triangulation T. The nodes of Q are in one-to-one correspondence with the arcs γ_i of T (their number being equal to the number of IR charges, as required for the BPS quiver of any 𝒩=2 theory). Giving the quiver, is equivalent to specifying its exchange matrix:


 For any triangle D in T={γ_i}_i=1^n which is not self-folded, we define a matrix B^D = (b^D)_ij ,1≤ i ≤ n,1≤ j≤ n as follows.


  * b^D_ij = 1 and b^D_ji = -1 in each of the following cases:


  * γ_i and γ_j are sides of D with γ_j following γ_i
in the clockwise order;

  * γ_j is a radius in a self-folded triangle enclosed by a loop γ_l, and γ_i and γ_l are sides of
D with γ_l following γ_i in the clockwise order;

  * γ_i is a radius in a self-folded triangle enclosed by a loop γ_i, and γ_l and γ_j are sides of
D with γ_j following γ_l in the clockwise order;


  * b^D_ij = 0 otherwise.

Then define the matrix B^T := (b_ij ),1≤ i ≤ n,1≤ j≤ n by b_ij =∑_D b^D_ij , where the sum is taken over all triangles in T that are not self-folded. The matrix B^T is a skew-symmetric matrix whose incidence graph is the quiver Q associated to the triangulation.




  
The superpotential. The superpotential W is the sum of two parts. The first one is a sum over all internal triangles of T (that is, triangles having no side on a boundary component). The full subquiver over the three nodes of Q associated with an internal triangle of T has the form

    ∙[dl]_α
    ∙[rr]_β      ∙[ul]_γ

Such a triangle contributes a term
γβα to W. The second part of W is a sum over the regular punctures. Let γ_1,γ_2⋯, γ_n  be the set of arcs ending at the puncture p taken in the clockwise order. The full subquiver of Q over the nodes corresponding to this set of arcs: it is an oriented n-cycle. The contribution to W from the puncture p is λ_p times the associated n-cycle, where λ_p≠0 is a complex coefficient. 




  
No regular puncture: gentle algebras. Suppose S has no regular puncture. Since an arc of T belongs to two triangles (which may be internal or not), 

    at a node of Q end (start) at most 2 arrows.
 The superpotential is a sum over the internal triangles
∑_iγ_iβ_iα_i
and the Jacobi relations are of the form

    the arrows α, β arise from the same internal triangle ⟹ αβ=0.

A finite-dimensional algebra whose quiver satisfies (<ref>) and whose relations have the form (<ref>) is called a gentle algebra <cit.>, a special case of a string algebra. Thus, in absence of regular punctures, the Jacobian algebra J(Q,W) is gentle. Indecomposable modules of a gentle algebra may be explicitly constructed in terms of string and band modules <cit.> (for a review in the physics literature see <cit.>). A gentle algebra is then automatically tame; in particular, the BPS particles are either hypermultiplets or vector multiplets, higher spin BPS particles being excluded. 

How to reduce the general case of a class 𝒮[A_1] theory to this gentle situation is explained in ref. <cit.>.
 



We give here an example of the quiver associated to an ideal triangulation T, whose incidence matrix is B^T:





 §.§ Geometric representation of categories


The main reference is <cit.>. There is a precise dictionary between curves over a decorated marked surface and the objects in the category DΓ. 

Then, let Δ be the set of decorated points: to each triangle of T, choose a point in the interior of that triangle. With respect to a quadratic differential ϕ of section <ref>, the decorated points correspond to the simple zeros of ϕ. The marked points, on the other hand, correspond to poles of ϕ of order m_i+2. Let S_Δ be the surface S with the decorated points. The basic correspondence between geometry and categories is, on the one hand, between objects in D(Γ) and curves over S_Δ and on the other hand between morphisms in D(Γ) and intersections between curves. Here follow the complete dictionary:


  * Recall that an object S∈ D^b (Γ) is spherical iff 
    Hom_D^bΓ(S,S[j])≅ k (δ_j,0+δ_j,3).
 A spherical object in the category D^b (Γ) corresponds to a simple[ A simple arc does not have self intersections.] closed[ A closed arc starts and ends in Δ.] arc (CA) between 2 points in Δ. In particular, simple objects are elements of the dual triangulation and are all spherical. These are some of the BPS hypermultiplets. We can describe these curves as elements of the relative homology with -coefficients H_1(S,Δ,ℤ) over the curve S and the set of points Δ: indeed, the operation of “sum” is well defined and it corresponds to the relation that defines the Grothendieck group K_0(D^bΓ). We now describe the map that associates a graded module (or equivalently a complex) to a closed arc. Consider a closed arc γ that is in minimal position with respect to the triangulation: every time γ intersects the triangulation, we add to the complex a simple shifted module S_i[j] and we connect it to the complex with a graded arrow: the grading of the map depends on how the curve and the decorated points are related. In particular, the grading – corresponding to the Ginzburg algebra grading – is defined in the following figure[ The figure is taken from <cit.>.]:

We call X both the map X: CA(S_Δ) → D^b (Γ) and X:OA(S_Δ) →𝔓𝔢𝔯 Γ, where CA(S_Δ) are the arcs of the decorated surface S_Δ connecting at most two points in Δ and OA(S_Δ) are the arcs connecting marked points. Notice that for the open arc (OA) case, we also have to take into account the tagging at the punctures. In particular, the situation is the following:


  * For an open curve ending on a puncture inside a monogon: if the curve is not tagged, then it intersects only the monogon boundary; if the curve is tagged, then the curve intersect the ray inside the monogon.

  * For an open curve ending on a puncture that is not inside a monogon, then the untagged curve does intersect the curve it would intersect as if it were untagged; if the curve is tagged, then it is as if the curve made a little loop around the puncture and so changes the intersection.



Let us consider the quiver A_2 again. Then, consider the curves γ_1, γ_2 and γ_3 as in the picture. 

By applying the rules above we get:
  
    X(γ_1):    S_1 a=1→S_2  ≅ k1→k

  
    X(γ_2):    S_1[-1] a^*=1←S_2  ≅ k[-1]1←k

  
    X(γ_3):    S_2 t_1=1→S_2[-2]  ≅ 0 0→k⊕ k[-2] 1



This map also works for the graded modules corresponding to closed loops: both those starting at the zeros of the quadratic differential and those not. As we will see in the example <ref> of the Kronecker quiver, only certain loops are in the category D^b Γ. All other possible loops belong to 𝔓𝔢𝔯 Γ. Moreover, even in the case with punctures, the algorithm to get (graded) modules form the curves is the same (thanks to proposition 4.4 of <cit.>).

  * Since every simple object is a spherical objects in D^b Γ – in particular, since D^b Γ is 3-CY, the simple objects are 3-spherical – we can define the Thomas-Seidel twist T_S associated to such a spherical object S by the following triangle

    Hom_D^b Γ^∙(X,S)⊗ S → X → T_S(X) →.

These twists are autoequivalences of DΓ. Geometrically, these spherical twists correspond to braid twists associated to the simple closed arc γ_S: let BT(S_Δ) denote the full group generated by all the braid twists. The action of the braid twist is like in figure

Moreover, being T^* the dual triangulation of the triangulation T, 
    CA(S_Δ)=BT(S_Δ)· T^*
 as shown in <cit.>.

  * In general, braid twists of S_Δ correspond to spherical twists of D^b Γ: BT(T_S_Δ)=Sph(D^b (Γ)). Moreover, the quiver representing the braid relations is exactly the quiver Q: to each vertex we associate a twist T_S_i and to each single arrow i → j a braid relation T_S_iT_S_jT_S_i=T_S_jT_S_iT_S_j. If there is no arrow between i and j, then [T_S_i,T_S_j]=0.


  * Rigid and reachable objects in 𝔓𝔢𝔯 Γ correspond to the simple open arcs, i.e. simple curves connecting marked points. The other objects in 𝔓𝔢𝔯 Γ correspond to generic arcs: both those arcs connecting two different punctures or marked points and closed loops encircling decorations or boundaries or punctures. We can describe these curves as elements of the relative homology H_1(S_Δ,M,ℤ) over the curve S_Δ (where the points in Δ are topological points in S_Δ) and the set of marked points M: indeed, the operation of “sum” is well defined and it corresponds to the relation that defines the Grothendieck group K_0(𝔓𝔢𝔯 Γ). 

  * Let T be the triangulation of the surface. The arcs of the triangulations are associated to the Γ e_i objects in 𝔓𝔢𝔯 Γ and the elements of the dual triangulations are the simple objects in D^bΓ. This is the geometrical version of the simple-projective duality:

    Hom^j(Γ e_i,S_k[l])=δ_lj δ_ki.

The choice of a heart in D^bΓ corresponds to the choice of the simple objects; thus, via the simple-projective duality it also corresponds to the choice of a triangulation T. The relation between the Grothendieck groups of D^bΓ and 𝔓𝔢𝔯 Γ is via the Euler form (which corresponds to the intersection form, as pointed out in item <ref>) and it corresponds to Poincaré duality of the relative homology groups.


























  * Flips of the triangulation (forward and backward) correspond to right and left mutations μ_i^±. Two different flips of the same arc are connected by a braid twist assocaited to that simple closed arc: T_S_i=μ^+_i(μ_i^-)^-1. 


  * Hom spaces correspond to intersection numbers:[ CA=closed arc, OA=open arc.] the full proof of the following facts can be found in <cit.>:

    Hom(X(CA),X(CA))=2 Int(CA,CA)


    (X(OA),X(CA))= Int(OA,CA)

The intersection numbers between arcs in S_Δ are defined as follows:
 

  * For an open arc γ and any arc η, their intersection number is the geometric
intersection number in S_Δ -M:

    Int(γ, η) = min{|γ^'∩η^'∩  (S_Δ - M)||γ^'≅γ, η^'≅η}.


  * For two closed arcs α, β in CA(S_Δ), their intersection number is an half integer
in 1/2ℤ and defined as follows:

    Int(α,β) = 1/2 Int_Δ (α,β) + Int_S-Δ(α,β),

where

    Int_S_Δ-Δ(α,β) = min{|α^'∩β^'∩  S_Δ -Δ| | α≅α^', β≅β^'}

and

    Int_Δ(α,β) = ∑_Z∈Δ|{t | α(t) = Z}| ·|{r  | β(r) = Z}|.

Let T_0 be a triangulation and η any arc; it is straightforward to see Int(T_0,η) ≥ 2 for a loop, and the equality holds if and
only if η is contained within two triangles of T_0 (in this case, η encircles exactly one decorating
point).


[Sphere with three holes and one marked point per each boundary component.]
In this particular case (the following results do not hold in general), the matrix corresponding to the bilinear form Int(-,-) can be obtained by taking the Cartan matrix of the quiver with potential (Q,W), i.e. the matrix whose columns are the dimensions of the projective modules, inverting and transposing it. 
In particular, the incidence matrix for a sphere with three boundary components with m_i=1, ∀ i ∈{1,2,3} – over the basis of simples (corresponding to the edges of the dual triangulation) – is 



0.35!
!<0cm,0cm>;<0.5cm,0cm>:<0cm,0.5cm>::
!(0,0) *+∙="b1"
!(5,0) *+∙="b3"
!(2.5,-2.5) *+∙="b2"
!(-1,0) *+*<15pt>="b4"
!(5.8,0.7) *+*<15pt>="b5"
!(2.5,-3.6) *+*<15pt>="b6"
!(2.5,-1.5) *+*<105pt>="center"
"b1" -@[blue] "b2"^3 "b2" -@[blue] "b3"^2 "b3" -@[blue] "b1"_1
"b1" -@`(10,7.5),(9,-6)@[blue] "b3"^4
"b3" -@`(2.5,-12),(-2,-3)@[blue] "b2"^5
"b2"-@`(-11,0),(2,4)@[blue]"b1" ^6
   
Int=(
[  1/2 -1/2  1/2    0    0    0;  1/2  1/2 -1/2    0    0    0; -1/2  1/2  1/2    0    0    0;   -1    0    0  1/2  1/2 -1/2;    0   -1    0 -1/2  1/2  1/2;    0    0   -1  1/2 -1/2  1/2;      ])



The Euler characteristic of D^b (Γ), as a bilinear form defined in <ref>, on the other hand, is an antisymmetric integral matrix that is the antisymmetric part of Int(-,-):

    χ(-,-)=(
    [  0  1 -1 -1  0  0; -1  0  1  0 -1  0;  1 -1  0  0  0 -1;  1  0  0  0 -1  1;  0  1  0  1  0 -1;  0  0  1 -1  1  0;    ]).


Notice that the skew-symmetric matrix we have just found corresponds to the matrix B^T associated to the ideal triangulation of figure <ref>. 

  * Relations amongst the exchange graphs (EG) of the surface S (to each vertex of this graph we associate a triangulation and to each edge of the graph a flip of the triangulation) and the cluster exchange graph (CEG) of the cluster algebra (to each vertex of the graph we associate a cluster and to each edge a mutation):

    EG(S) = CEG(Γ)


    EG(D^b (Γ))/Sph(D^b (Γ))=CEG(Γ)


  * A distinguished triangle in D^b (Γ) corresponds to a contractible triangle in S_Δ whose edges are 3 closed arcs (α,β,η) as in the figure, such that the categorical triangle is X(α) → X(η) → X(β) →.

Exploiting the group structure of the homology group H_1(S,Δ,ℤ), we see that we have the following relation:

    [α]-[η]+[β]=0;

this is the defining relation of the Grothendieck group K_0(D^bΓ).

  * The triangulated structure of 𝔓𝔢𝔯 Γ is less easy to represent geometrically. Before proceeding with an example, we define the left and right mutations in 𝔓𝔢𝔯 Γ starting from the silting set. 

A silting set ℙ in a category D is an Ext^>0-configuration, i.e. a maximal collection of non-isomorphic indecomposables
such that Ext^i(P, T) = 0 for any P, T ∈ℙ and integer i > 0. The forward mutation μ^-_P at an element P ∈ℙ is another silting set ℙ^-_P, obtained from ℙ by replacing P with

    P^- = Cone(P →⊕_T ∈ℙ
    {P}DHom_irr(P, T)⊗ T ),

where Hom_irr(X, Y ) is the space of irreducible maps X → Y , in the additive subcategory
𝖺𝖽𝖽 ⊕_T ∈ℙ T of D. The backward mutation μ^+_P at an element P ∈ℙ is another silting
set ℙ^+_P, obtained from ℙ by replacing P with

    P^+ = Cone(
    ⊕_T ∈ℙ
    {P}Hom_irr(T, P) ⊗ T → P)[-1]



Notice that equations (<ref>) and (<ref>) are exactly the same as (<ref>) and (<ref>): the notation of the latter is more straightforward for the next computations. 

We give now an example to show how one can get the triangulated structure of per Γ and we relate it to the group structure of H_1(S_Δ,M,).
[A_2 example]
The silting set for the quiver A_2: ∙_1 →∙_2 is ℙ={Γ e_1,Γ e_2} which we denote as {Γ_1,Γ_2}. We apply the left mutation corresponding to a forward flip: it gives the following triangle

    Γ_2 →Γ_1 →Γ_2^-→ .

The element Γ_2^- is an infinite complex of the form S_1 t_1→Γ_1[-2]. At a geometrical level it corresponds to the red curve in this figure:

As one can see, these three open arcs do not seem to be geometrically easily related. The next step, which is fundamental for consistency of the geometric representation, is to consider the following triangle:

    Γ_1[-2]→Γ_2^- → S_1 →

This triangle is exact and moreover S_1 ∈ D^bΓ. This implies that in the cluster category, both Γ_2^- and Γ_1[-2] map to the same curve. We can exploit the geometric effect of the shift (see item <ref>) to compute Γ_1[-2]: it is the green curve in the next figure

When we map to the cluster category via the forgetful functor (see item <ref>), we have that both Γ_1[-2] and Γ_2^- are sent to the same object: S_2. Indeed, the corresponding cluster category is made of 5 indecomposables which form the following periodic AR diagram

         Γ_1 [dr]          P_1[dr]      

            Γ_2  [ur]          S_2 [ur]         S_1 

where we can read the corresponding triangles in the cluster category. The geometric picture is


We can now generalize the above results by stating that the triangulated structure is consistent with the group structure of the relative homology group H_1(S_Δ,M,ℤ) paired with the relative homology of H_1(S,Δ,ℤ): indeed, we see that in H_1(S_Δ,M,ℤ) – after choosing opposite direction for the red and green path of figure <ref>– we have

    [Γ_1[-2]]-[Γ_2^-]+[S_1]=0.

The relations defining the Grothendieck K_0(𝔓𝔢𝔯 Γ), such as

    [Γ_2]-[Γ_1]+[Γ_2^-]=0,

are less obvious from a homological viewpoint: there is no other way but compute them explicitly when needed.

  *  The Amiot quotient 𝔓𝔢𝔯 Γ/ D^b (Γ) <cit.> – through which the cluster category is defined – corresponds to the forgetful map F:S_Δ→ S. For a short reminder of triangulated quotients, see section <ref>. In particular, we recover easily the results of <cit.>: the indecomposable objects of the cluster categories are string modules or band modules. Geometrically a string is an open arc and the procedure to associated a module to it is the same as the one described in item <ref> for closed arcs. So indeed, since open curves are indecomposable objects of 𝔓𝔢𝔯 Γ, as pointed out in item <ref>, the following diagram – at least for string modules – commutes:

     
  𝒪A(S_Δ) [r]^F [d]^X     OA(S) [d]^X 

𝔓𝔢𝔯 Γ[r]^π    𝒞(Γ)
              

We expect no difference in the case of band modules (which correspond to families of loops).
 
  *  In the cluster category 𝒞(Γ) and in the perfect derived category 𝔓𝔢𝔯 Γ, the shift [1] corresponds to a global anticlockwise rotation of all the marked points on each boundary component. For punctures, the action of [1] corresponds to a change in the tagging. In 𝒞(Γ), it is equivalent to the operation τ (the AR translation), as defined in <cit.>. In particular we see that in presence of only regular punctures, [2] flips twice the tagging getting back to the original situation, that is, in this case the cluster category is 2-periodic, as expected on physical grounds.






























Let us consider here a simple example that will allow us to clarify some aspects.
[A_2 again]

Let us consider the following curves over a disk with 5 marked points on the boundary and no punctures. The triangulation is made of the black lines 1 and 2 and the boundary arcs:

The curves a and d correspond to graded modules in D^b Γ as shown in the example <ref>; indeed, the curve d corresponds to the graded module S_1 ← S_2[-1] in D^b Γ.
The red curves b and c', on the other hand, cannot be associated to any closed curve, since they cannot intersect any closed curve in minimal position: they are elements only of 𝔓𝔢𝔯 Γ and not of D^b Γ. When we take the Amiot quotient, the green dots disappear and so do the curves a and d. Moreover, the curve c is homotopic to a boundary arc and b is homotopic to 2. Thus the quotient does what we expect: only the curves in 𝔓𝔢𝔯 Γ that are not in D^b Γ do not vanish. 
The intersection form in this example is 

    Int=
    (
    [  1 -1;  0  1;    ])

and thus the Euler characteristic is 

    χ=
    (
    [  0  1; -1  0;    ])

We can thus construct the Thomas-Seidel twists associated to the simple modules:

    T_S_1=Id-|S_1⟩⟨S_1|·χ=(
    [  1 -1;  0  1;    ])


    T_S_2=Id-|S_2⟩⟨S_2|·χ=(
    [ 1 0; 1 1;   ])

And we can explicitly check the braid relation 

    T_S_1· T_S_2· T_S_1=T_S_2· 
    T_S_1· T_S_2
 
both on K_0(D^b Γ)
 and geometrically. Moreover it is clear from the matrix representation that T_S_1 and T_S_2 generate SL_2(ℤ). We can also act with these twists on the graded modules:

    T_S_1· X(S_2)=X((-1,1))

The graded module whose dimension vector is (-1,1) is k[-1] a^*=1←k. This is consistent with the geometric picture, as one can verify:



[Kronecker quiver]

The surface corresponding to the Kronecker quiver is an annulus with one marked point on each boundary component. This theory corresponds to a pure SU(2) SYM. Thus, we shall find closed curves corresponding to BPS vector bosons. They must also appear a one-parameter family. In the following figure, the green loops around the two points represent a band module, the black dots the marked points; the black lines are the flow lines associated to the quadratic differential ϕ described in section <ref>.

The module corresponding to (one of) the green curve can be computed via the map X: a ↦ X(a) ∈ D^b Γ, starting at a generic point along the curve. We get

    X(a):   S_1 a=1,b=λ⟹ S_2 ≅ k a=1,b=λ⟹ k .

This module is stable (see section <ref>) since it is equivalent to the regular module in the homogeneous tube of -k 𝖪𝗋.














 §.§ Summary

We give here a sketchy summary of what we have written so far; recall that, given a quadratic differential ϕ(z) dz⊗ dz there are three kind of markings: the zeros of ϕ (decorations), the simple poles of √(ϕ) (punctures) and the irregular singularities of ϕ, which generate the marked points on the boundary segments.


  * BPS vector multiplets correspond to loops, not crossing the separating arcs of the flow of ϕ, for θ= Z(BPS)=θ_c. They belong to the category D^b Γ via the maps X.

  * BPS hypermultiplets correspond to arcs connecting two zeros of ϕ. They belong to the category D^b Γ via the map X and can be identified with elements in the relative homology H_1(S,Δ,), where Δ are the zeros of ϕ.

  * Surface operators corresponds to punctures and marked points.

  * The objects in the perfect derived category 𝔓𝔢𝔯 Γ are the screening states created by line operators acting on the vacuum. They correspond to arcs connecting marked points and punctures and thus belong to H_1(S_Δ,M,).

  * UV line operators correspond to arcs over S connecting marked points and punctures (but not zeros of ϕ).  They belong to the cluster category 𝒞(Γ) and can also be identified with elements of the relative homology H_1(S,M,ℤ), where M is the set of markings of the surface S. 

With this dictionary in mind, we can now exploit the categorical language to compute physical quantities (such as vacuum expectation values of UV line operators). 

A generalization of these concepts, in particular towards ideal webs and dimer models can be found in <cit.>. There it is argued that the 3-CY category D^bΓ is the physical BPS states category, in accordance with our analysis. Moreover, in the case in which we have a geometrical interpretation via bipartite graphs, the mapping class group of the punctured surface, is a subgroup of the full S-duality group.
Here we give a simple example.
[Pure SU(3)]
The pure SU(3) theory can be described as a S[A_2] Gaiotto theory over a cylinder with one full punctures per each boundary <cit.>. The mapping class group is generated by a single Dehn twist around the cylinder. It acts on the bipartite graph associated to the SU(3) theory and it must be a subgroup of the full S-duality group. It is isomorphic to ℤ. Since we also have that pure SU(3) theory can be described by the quiver

    𝖪𝗋⊠ A_2,

We explicitly find – using he techniques of <cit.> – that a cluster automorphism is given by τ_𝖪𝗋⊗τ_A_2, which generates a free group, thus isomorphic to ℤ.

























































§ CLUSTER CHARACTERS AND LINE OPERATORS




 §.§ Quick review of line operators


The main reference for this part is <cit.>. We are going to study, in the following section, the algebra of line operators: we shall discover that this algebra is closely related to the cluster algebra of Fomin and Zelevinski <cit.>. Recall that an IR line operator (also called framed BPS state <cit.>) is characterize by a central charge ζ and a charge γ. Similarly for a UV line operator. The starting point is to consider the RG flow:

    RG(·,α,ζ):{UV  line  operators}→{IR  line  operators}
    
    L(α,ζ) ↦∑_γ∈ΓΩ̅(α,ζ,γ,u,y) L(γ,ζ),

where L(α,ζ) is a supersymmetric UV line operator of UV charge α (see <ref>). We can think of it as a supersymmetric Wilson line operator:

    L(α,ζ):=exp(i α∫_time A+1/2(ζ^-1ϕ +ζϕ̅)).

where A is the gauge connection and ϕ and ϕ̅ are the supersymmetric partners. The idea is that cluster characters provide the coefficients Ω̅(α,ζ,γ,u,y); moreover, the OPE's of line operators can be identified with the cluster exchange relations. The physical definition of Ω̅(α,ζ,γ,u,y) as supersymmetric index is the following. Define the Hilbert space of our system with a line operator in it polarizing the vacuum: H_L,ζ,u=⊕_γ∈Γ_uH_L,ζ,u,γ, where Γ_u is the charge lattice and u a point in the Coulomb branch. When we restrict only to BPS states we have H^BPS_L,ζ,u. We now define the following index (i.e. a number that counts the line operators):

    Ω̅(α,ζ,γ,u,y)=Tr_H_L,ζ,u,γ(y^2J_3(-y)^2I_3),

where the I and J operators are the Cartan generators associated to the unbroken Lorentz symmetry SO(3) and unbroken R-symmetry SU(2)_R by the presence of the line operator which moves along a straight line in the time direction. In particular, if y=1:

    Ω̅(α,ζ,γ,u,1)=Tr_H_L,ζ,u,γ(1^2J_3(-1)^2I_3)=∑_m1^2m(-1)^0= H_L,ζ,u,γ^BPS.

This index corresponds to the Poincaré polynomial of the quiver Grassmannian Gr_γ(α), where we interpret the UV line operator L of charge α as the quiver representation of which we compute the cluster character (see <ref> for more details).


  §.§.§ Algebra of UV line operators

Let the OPE's of UV line operators be defined as follows

    L(α, ζ)L(α',ζ)=∑_β c(α,α',β)L(β,ζ).

From now on, let us fix the generic point of the Coulomb branch u. We also define the generating functions for the indexes Ω̅(L, γ)= Ω̅(L,γ,u=fixed,y=1):

    F(L)=∑_γΩ̅(L,γ)X_γ,

where the formal variable X_γ is such that X_γ X_γ'=X_γ+γ'. One can check that F(LL')=F(L)F(L'). This equality gives a recursive formula to compute F(LL'). Furthermore, we can study the wall-crossing of UV line operators via the formula of KS <cit.>:

    F^+(L_γ_c)=F^-(L_γ_c)∏_γ∏_m=-M_γ^M_γ(1+(-1)^mX_γ)^|γ,γ_c|a_m,γ

where γ_c is the charge of the wall we are crossing, and M_γ is the maximal value of the operator 𝒥_3=J_2+I_3 and the a_m,γ are the coefficients of the index:

    Ω̅(α, ζ,γ,u,y)=∑_m=-M_γ^M_γ a_m,γy^m.

We can transfer this formula on the newly defined variables to implement the wall-crossing more efficiently: 

    X_γ'=X_γ∏_γ∏_m=-M_γ^M_γ(1+(-1)^mX_γ)^γ,γ_ca_m,γ.

Moreover, the transformation of the indices Ω̅(L, γ) is an
automorphism of the OPE algebra. Thus, the algebra obeyed by the generating functionals
is in fact an invariant of the UV sector theory. The properties of the KS wall-crossing formula of <ref> and the fact that these generating functionals are invariants of the UV theory (i.e. of the cluster category C(Γ)), tell us that F(L) are exactly the cluster character.
The non-commutative generalization is done via the star product <cit.>:

    L(α, ζ)*_yL(α',ζ)=∑_β c(α,α',β,y)L(β,ζ)

and X_γ*_yX_γ'=y^γ,γ'_DX_γ+γ'. We define again the generating functions F(L)=∑_γΩ̅(L,γ,y)X_γ, and then verify that 

    F(L*_yL')=F(L)*_yF(L').

Indeed we discover that the non-commutative version of F behaves exactly like a quantum cluster character (usual cluster characters are obtained by setting y=1). We can thus analyze these objects from a purely algebraic point of view and study cluster algebras and cluster characters: we shall do this is the next section.


 §.§ Cluster characters

The main references for this section are <cit.>.
We begin by recalling some basic definitions and properties of cluster characters. Let 𝒞:=C(Γ) be a cluster category.
 A cluster character on 𝒞 with values in a commutative ring A is
a map

    X : obj(𝒞) → A

such that


  * for all isomorphic objects L and M, we have X(L) = X(M),

  * for all objects L and M of 𝒞, we have X(L ⊕ M) = X(L)X(M),

  * for all objects L and M of 𝒞 such that Ext^1_𝒞(L, M) = 1, we have

    X(L)X(M) = X(B) + X(B'),

where B and B' are the middle terms of the non-split triangles

    L →  B → M →   and    M → B^'→  L →

with end terms L and M.[ If B^' does not exist, then X(B^')=1.]


Let T=⊕_i T_i be a cluster tilting object, and let B=Ent_𝒞 T. The functor 
    F_T:𝒞→-B,     X ↦Hom(T,X)

 is the projection functor that induces an equivalence between 𝒞/𝖺𝖽𝖽 T[1] →𝗆𝗈𝖽-B. Then the Caldero-Chapoton map<cit.>, 

    X^T_?: ind 𝒞→ℚ(x_1, . . . , x_n)

is given by

    X^T_M ={[                                x_i   if  M ≅Σ T_i; ∑_e χ(Gr_e F_TM)∏^n_i=1 x_i^S_i,e_D-S_i,FM  else, ].

where the summation is over the isoclasses of submodules[ Recall that a module N is a submodule of M iff there exists an injective map N → M.] of M and S_i are the simple B-modules. Moreover, the Euler form and Dirac form in the formula above, are those of 𝗆𝗈𝖽-B.
We now recall some properties of quiver Grassmannians[ Gr_e(FM):={N ⊂ FM  |  N =e}, i.e. it is the space of subrepresentations of M with fixed dimension e.] and in particular their Euler Poincaré characteristic χ (with respect to the étale cohomology).
 Let Λ be a finite dimensional basic ℂ-algebra. For a Λ-module M we define the F-polynomial to be the generating function for the
Euler characteristic of all possible quiver grassmanians, i.e.

    F_M := ∑_eχ(Gr_e(M))y^e ∈ℤ[y_1, . . . , y_n]

where the sum runs over all possible dimension vectors of submodules of M.

Moreover, we assume that S_1, . . . , S_n is a complete system of representatives
of the simple Λ-modules, and we identify the classes [S_i] ∈ K_0(Λ) with the
natural basis of ℤ^n.
Let Λ be a finite dimensional basic ℂ-algebra. Then the
following holds:


  * If 
    0 → L i→ M π→ N → 0
 is an Auslander-Reiten sequence in
Λ-mod, then

    F_L  F_N = F_M + y^ N .


  * For the indecomposable projective Λ-module P_i with top S_i we have

    F_P_i = F_rad  P_i + y^ P_i

for i = 1, . . . , n.

  * For the indecomposable injective Λ-module I_j module with socle S_j
we have

    F_I_j = y_j  F_I_j/S_j + 1

for j = 1, 2, . . . , n.


The recursive relations of cluster characters and F-polynomials are the key tools to find a computational recipe: the next section is devoted to pointing out this algorithm. All aspects will be clarified in Example <ref>.


  §.§.§ Computing cluster characters


The best way to compute cluster characters, is to exploit the results in <cit.>. The idea is to associate a Laurent polynomial to a path in the quiver. If the algebra is gentle, to a path we can associate a string module: computing the cluster character associated to string modules (up to an overall monomial factor) becomes a simple combinatorics problem. For any locally finite quiver Q, we define a family of
matrices with coefficients in ℤ[x_Q] = ℤ[x_i|i ∈ Q_0] as follows.
For any arrow β∈ Q_1, we set

    A(β) :=[[ x_t(β)      0;      1 x_s(β) ]]
       and   A(β^-1) :=[[ x_t(β)      1;      0 x_s(β) ]].

Let c = c_1 ... c_n be a walk of length n ≥ 1 in Q. For any i ∈{0, . . . , n} we set

    v_i+1 = t(c_i)

(still with the notation c_0 = e_s(c)) and

    V_c(i): =
    [[ ∏_α∈ Q_1(v_i,-), α≠ c_i^± 1,c_i-1^± 1x_t(α)                                           0;                                           0 ∏_α∈ Q_1(-,v_i), α≠ c_i^± 1,c_i-1^± 1x_s(α) ]].

We then set

    L_c =1/x_v_1... x_v_n+1[1, 1]V_c(1) ∏_i=1^nA(c_i)V_c(i + 1) [[ 1; 1 ]]
    ∈ℒ (x_Q).

If c = e_i is a walk of length 0 at a point i, we similarly set

    V_e_i(1): =
    [[ ∏_α∈ Q_1(v_i,-)x_t(α)                     0;                     0 ∏_α∈ Q_1(-,v_i)x_s(α) ]].

and

    L_e_i =1/x_i[1, 1]V_e_i[[ 1; 1 ]] ∈ℒ (x_Q).

In other words, if c is any walk, either of length zero, or of the form c = c_1 ... c_n,
we have

    L_c =1/∏^n_i=0 x_t(c_i)[1, 1]∏^n_i=0A(c_i)V_c(i + 1)[[ 1; 1 ]] ∈ℒ (x_Q).

with the convention that A(c_0) is the identity matrix. In general, we have the following result:

    X_M=1/x^n_ML_c,

where M is the string module associated to the path c and the monomial x^n_M is the normalization coefficient.


Let us consider the cluster category of A_4: its AR quiver is the following

We have also made the choice of tilting objects. The algebra End  T is given by the following quiver:[ The vertices are the T_i and the arrows j → i correspond to Hom_𝒞(T_i,T_j).]














with relations βα = γβ = αγ = 0. The Dirac form is the following:

    -,-_D=(
    [  0  1  0  0; -1  0 -1  1;  0  1  0 -1;  0 -1  1  0;    ]),

whereas the Euler form is a,b:=Hom_𝒞(a,b)-Hom_𝒞(a,b[1]).
Consider the B-module F_TM=(1,1,0,0). Its submodules are 0, S_1 and F_TM itself. The corresponding path is just the arrow c:2 → 1. By applying the formulas above we get:

    L_c   =   1/x_1x_2[1,1] A(c_0)· V_c(1)· A(c)· V_c(2)[[ 1; 1 ]]
       =   1/x_1x_2[1,1] Id[[ x_3   0;   0 x_4 ]] [[ x_1   1;   0 x_2 ]] [[ 1 0; 0 1 ]]  [[ 1; 1 ]]
       =   x_1x_3+x_4+x_2x_4/x_1x_2.

Notice that the denominator is exactly x^ FM: this is a general feature for the decategorification process <cit.>.




Moreover, we know that the Euler characteristic of a point is 1 and thus χ(Gr_0F_TM)=1=χ(Gr_F_TMF_TM). We then exploit the AR sequence 
    0 → S_1 → F_TM → S_2 → 0
 and get the recursive relation

    F_S_1F_S_2=F_F_TM+y_2,

which is equivalent to the following polynomial equation:

    1+y_1 χ_S_1+y_2 χ_S_2+y_1y_2 χ_S_1χ_S_2=1+χ_S_1y_1+χ_FMy_1y_2+y_2,

which implies that χ_S_1=χ_S_2=1 and this is consistent with the previous result. One can check this and many other computations using appendix <ref>.


One final remark is needed: we could have computed the cluster characters by a sequence of mutations of the standard seed of the cluster algebra associated to the quiver of B=End T. For the non-commutative case, i.e. when 
    x^α x^β=q^1/2α,β_D x^α+β,

this procedure is the only one we know to compute quantum cluster characters. From the physics point of view, this is the important quantity: since cluster variables behave like UV line operators, they must satisfy the same non-commutative algebra.



























 §.§ Cluster characters and vev's of UV line operators

Let us start by recalling how the vacuum expectation values of line operators are computed in <cit.>. The idea is associate to a loop over a punctured Gaiotto surface a product of matrices. In the case of irregular singularities, since these singularities can be understood as coming from a collision of punctures, loops can get pinched and become laminations. Thus to both string modules (associated to laminations) and band modules (associated to loops), we can associate a rational function in some shear variables Y_i. Their expression turns out to be equal to cluster characters: for string modules we can use section <ref>, whereas for band modules we can use the bangle basis of <cit.> and the multiplication formula or the Galois covering technique of <cit.>. We shall now give some detailed examples in which we apply what we just described.
[A_2 quiver]
The computations of <cit.> of the vev's of the UV line operators can be found in section 10.1. They have been made using the “traffic rule”. The idea is to follow the lamination path and create a sequence of matrices according to the traffic rule. In the end, one takes the trace of the product of matrices (loop case) or contract the product of matrices with special vectors (open arcs case). For example,

The matrix product is the following:

    L_1=(B_R· R· M_Y_2· L· E_R) (B_R· R· M_Y_2· L· E_R),

where the matrices are

    L=([ 1 1; 0 1 ])
    ,     R=([ 1 0; 1 1 ])
    ,     M_X=([   √(X)      0;      0 1/√(X) ]),

and the vectors are

    B_R=(1  0),  B_L=(0 1),  E_L=(1  0)^t,   E_R=(0  1).

Then we get:

    R.M_Y_2.L=
    (
    [          √(Y_2)          √(Y_2);          √(Y_2) √(Y_2)+1/√(Y_2);                 ]) ,

and finally

    L_1=√(Y_2)√(Y_2)=Y_2.

The other four line operators, corresponding to the four remaining indecomposable objects of the cluster category of 𝒞(Γ_A_2) (or equivalently the remaining four cluster variables) are:

    L_2= Y_1 + Y_2Y_1,  L_3=1/Y_2+Y_1/Y_2+ Y_1, L_4 = 1/Y_2+1/Y_2Y_1,  L_5 =1/Y_1.

On the other hand, the cluster characters of A_2 are:

    x_1, x_2, 1/x_2+x_1/x_2, 1/x_1+x_2/x_1,1/x_1x_2+1/x_1+1/x_2.

The following map (Y_1,Y_2) ↦ (1/x_2,x_1) transforms one set to the other. This map is the tropicalization map of Fock and Goncharov <cit.>:

    Y_i=∏_jx_j^B_ij.

Notice that this result was expected from the general algebraic properties of the line operators algebra and the cluster algebras.

We now proceed to a more interesting example: the pure SU(2) theory. The computations of section <ref> has to be modified a bit: as we will see, it is convenient to exploit the Galois covering techniques of <cit.>.
[Kronecker quiver]
Let us focus our attention to the non rigid modules, i.e. those belonging to the homogeneous tubes of the AR quiver of the cluster category C(Γ_Kr). There is a ℙ^1 family of these modules and, amongst them, two of them are string modules (those of the form 11λ=0⇒ 2). By the theorems of <cit.>, the cluster characters do not depend of the value of the parameter λ and we are thus free to choose the simplest one to compute the character. Geometrically, this family of modules corresponds to loops around the cylinder (see figure <ref>). With the traffic rule techniques – with the slight modification of taking the trace instead of using the B and E vectors – we compute the vev of the line operator whose e.m. charge is (1,1):

    L_(1,1)=√(Y_1Y_2)+√(Y_1/Y_2)+1/√(Y_1Y_2).

We can reproduce this result using cluster characters. The only observation is that we cannot simply use section <ref> to compute the character associated to the module dim M=(1,1): we have a path ambiguity. We thus have to construct a _2 Galois cover <cit.>, compute the character on the cover, and then project it down to the Kr quiver. The reason is that the Kronecker quiver has a double arrow and we have to be able to specify the path we follow uniquely. On the _2 cover the ambiguity is lifted and the character can be computed. The _2 cover is:

           ∙_3              ∙_2'  

            ∙_1  [ur]_c [dr]          ∙_2 [ul] [dl]     π→       

     ∙_4              ∙_1' @=>[uu]

where the covering map π sends 1,2 ↦ 1^' and 3,4 ↦ 2'. The character corresponding to the string c with respect to the covering quiver is

    1+x_1x_2+x_3x_4/x_1x_3

Therefore, if we identify the cluster variables following the covering map π, we get

    1+x_1'^2+x_2'^2/x_1'x_2'.

This result is consistent with what we find in literature (e.g. <cit.>).
Also in this case, we find that the tropicalization map [ Id est Y_i=∏_jx_j^B_ij.] sends the rational function <ref> to <ref>:

    (Y_1,Y_2) ↦ (x_2^-2,x_1^2).


In this final example we show how to compute the cluster character associated to a band module in a more complicated quiver.
[SU(2) with N_f=4]
We are interested in the module M corresponding to the purple loop in the following figure

Using the traffic rule, it is rather straightforward to compute the VEV of the line operator associated to the module M. The result – which is similar to the ones computed in <cit.> – is:

    L_M=Tr(L· M_Y_1· R· M_Y_2· R.M_Y_4· L· M_Y_5)=1 + Y_4 + Y_2 Y_4 + Y_4 Y_5 + Y_2 Y_4 Y_5 + Y_1 Y_2 Y_4 Y_5/√(Y_1Y_2Y_4Y_5)

The cluster character computation is more involved than the simple application of section <ref>: we exploit the techniques of <cit.>. The idea is similar to the traffic rule: we find a path over the hexagonal graph of <cit.> that is homotopic to the path considered. Then, to each edge of the hexagonal graph we associate a matrix with the following rule:

The matrices D and F are:

    D(x)=(
    [    0    x; -1/x    0;      ]),   F(x_α/x_β x_γ)=(
    [           1           0; x_α/x_β x_γ           1;             ]).

In our example we find:

    Tr(D(x_1)F(x_3/x_1x_2)F(x_3/x_4x_2)D(x_4)F^-1(x_6/x_4x_5)F^-1(x_6/x_1x_5))


    =x_2x_5x_1^2+x_2x_5x_4^2+x_3x_6x_1^2+x_3x_6x_4^2+2 x_1x_4x_3x_6/x_1x_2x_4x_5.

Again we can check that the result (<ref>) is the tropicalization of (<ref>).




§ ACKNOWLEGMENTS

We have benefit from discussions with Michele Del Zotto, Dirk Kussin and Pierre-Guy Plamondon. SC thanks the Simons Center for Geometry and Physics, where this work was completed, for hospitality.
















































































































































§ CODE FOR CLUSTER CHARACTERS


This is a short Mathematica code that computes the L_c polynomials of section <ref>. Up to an overall normalization factor, the L_c polynomials are the cluster characters. The algorithm follows precisely the procedure described in section <ref>.






































§ CODE FOR CLUSTER AUTOMORPHISMS


This short Mathematica script is useful to find generators and relations for the automorphisms of the cluster exchange graph. The formulas used to implement the mutations for the exchange matrix B_ij and the dimension vectors d_l (where l is an index that runs over the nodes) are the following:

    μ_k(B)_ij   =
    - B_ij,     i=k  or j=k
    
    B_ij+max[-B_ik,0] B_kj+B_ik max[B_kj,0]—–   otherwise.
    μ_k(d)_l   ={[                                           d_l,                                           l≠ k; -d_k+max[∑_imax[B_ik,0]d_i,∑_imax[-B_ik,0]d_i]                                            l=k ].
 
The procedure of this script is explained in section <ref>.




§ WEYL GROUP OF E_6


With this short Mathematica script, we explicitly construct the Weyl group of E_6 over the basis of simple roots.

We directly checked that the longest elements has length 36, that the order of the Weyl group is

    51840=2^7 3^45

and the order of each element belongs to this set:

    {1,3,2,5,4,6,12,8,10,9}.




10

alday2012wilson
L. F. Alday.
Wilson loops in supersymmetric gauge theories.
Lecture Notes, CERN Winter School on Supergravity, Strings, and
  Gauge Theory, 2012.

alday2010loop
Luis F. Alday, Davide Gaiotto, Sergei Gukov, Yuji Tachikawa,
  and Herman Verlinde.
Loop and surface operators in gauge theory and Liouville modular
  geometry.
Journal of High Energy Physics, 2010(1):1–50, 2010.

alim2013bps
Murad Alim, Sergio Cecotti, Clay Córdova, Sam Espahbodi,
  Ashwin Rastogi, and Cumrun Vafa.
BPS Quivers and Spectra of Complete N=2 Quantum Field
  Theories.
Communications in Mathematical Physics,
  323(3):1185–1227, 2013.

alim2014mathcal
Murad Alim, Sergio Cecotti, Clay Cordova, Sam Espahbodi,
  Ashwin Rastogi, and Cumrun Vafa.
N=2 quantum field theories and their BPS quivers.
Advances in Theoretical and Mathematical Physics,
  18(1):27–127, 2014.

amiot2009cluster
Claire Amiot.
Cluster categories for algebras of global dimension 2 and quivers
  with potential.
In Annales de l'Institut Fourier, volume 59, page 2525,
  2009.

amiot2011generalized
Claire Amiot.
On generalized cluster categories.
Representations of algebras and related topics, pages 1–53,
  2011.

aspinwall2009dirichlet
Paul S. Aspinwall et al.
Dirichlet branes and mirror symmetry.
 American Mathematical Soc. Vol. 4, 2009. 

aspinwall2006superpotentials
Paul S. Aspinwall and Lukasz M. Fidkowski.
Superpotentials for quiver gauge theories.
Journal of High Energy Physics, 2006(10):047, 2006.

assem2010gentle
Ibrahim Assem, Thomas Brüstle, Gabrielle Charbonneau Jodoin,
  and Pierre-Guy Plamondon.
Gentle algebras arising from surface triangulations.
Algebra & Number Theory, 4(2):201–229, 2010.

assem2013modules
Ibrahim Assem and Grégoire Dupont.
Modules over cluster-tilted algebras determined by their dimension
  vectors.
Communications in Algebra, 41(12):4711–4721, 2013.

assem2012friezes
Ibrahim Assem, Grégoire Dupont, Ralf Schiffler, and David
  Smith.
Friezes, strings and cluster variables.
Glasgow Mathematical Journal, 54(01):27–60, 2012.

assem2012cluster
Ibrahim Assem, Ralf Schiffler, and Vasilisa Shramchenko.
Cluster automorphisms.
Proceedings of the London Mathematical Society,
  104(6):1271–1302, 2012.

assem2006elements
Ibrahim Assem, Andrzej Skowronski, and Daniel Simson.
Elements of the Representation Theory of Associative
  Algebras: Volume 1: Techniques of Representation Theory,
  volume 65.
Cambridge University Press, 2006.

barot2008grothendieck
Michael Barot, Dirk Kussin, and Helmut Lenzing.
The Grothendieck group of a cluster category.
Journal of Pure and Applied Algebra, 212(1):33–46,
  2008.
  
  barot
  M. Barot, D. Kussin, and H. Lenzing,
 The cluster category of a canonical algebra,
   arXiv:0801.4540.


bridgeland2007stability
Tom Bridgeland.
Stability conditions on triangulated categories.
Annals of Mathematics, pages 317–345, 2007.

bridgeland2015quadratic
Tom Bridgeland and Ivan Smith.
Quadratic differentials as stability conditions.
Publications mathématiques de l'IHÉS,
  121(1):155–278, 2015.

brustle2011cluster
Thomas Brüstle and Jie Zhang.
On the cluster category of a marked surface without punctures.
Algebra & Number Theory, 5(4):529–566, 2011.

burban2008cluster
Igor Burban, Osamu Iyama, Bernhard Keller, and Idun Reiten.
Cluster tilting for one-dimensional hypersurface singularities.
Advances in Mathematics, 217(6):2443–2484, 2008.



caorsi2016homological
Matteo Caorsi and Sergio Cecotti.
Homological S-Duality in 4d N=2 QFTs.
arXiv preprint arXiv:1612.08065, 2016.

Cecotti:2010qn 
  S. Cecotti and C. Vafa.
 2d Wall-Crossing, R-Twisting, and a Supersymmetric Index.
 arXiv:1002.3638 [hep-th].

cecotti2010r
  S. Cecotti, Andrew Neitzke, and Cumrun Vafa.
R-twisting and 4d/2d correspondences.
arXiv preprint arXiv:1006.3435, 2010.

cecotti2011classification
  S. Cecotti and C. Vafa,
 Classification of complete N=2 supersymmetric theories in 4 dimensions,”
  Surveys in differential geometry 18 , 2013,
  [arXiv:1103.5832 [hep-th]].
  
  
  braidmirror
       S. Cecotti, C. Cordova and C. Vafa.
 Braids, Walls, and Mirrors,
 arXiv:1110.2115 [hep-th].
  
  Cecotti:2012gh 
  S. Cecotti and M. Del Zotto.
 4d N=2 Gauge Theories and Quivers: the Non-Simply Laced Case,
  JHEP 1210, 190 , 2012,
  [arXiv:1207.7205 [hep-th]].


cecotti2012quiver
  S. Cecotti.
 The quiver approach to the BPS spectrum of a 4d N=2 gauge theory,
 Proc. Symp. Pure Math.  90, 3 , 2015,
  [arXiv:1212.3431 [hep-th]].

cecotti2013categorical
 S. Cecotti.
 Categorical Tinkertoys for N=2 Gauge Theories.
 Int. J. Mod. Phys. A 28, 1330006 , 2013,
  [arXiv:1203.6734 [hep-th]].
  

  
  cecotti2013more
  S. Cecotti, M. Del Zotto and S. Giacomelli.
 More on the N=2 superconformal systems of type D_p(G),
 JHEP 1304, 153 , 2013,
  [arXiv:1303.3149 [hep-th]].


cecotti2014systems
  S. Cecotti and M. Del Zotto,
 Y-systems, Q-systems, and 4D 𝒩=2 supersymmetric QFT,
 J. Phys. A 47, no. 47, 474001, 2014,
  [arXiv:1403.7613 [hep-th]].

cecotti2015galois
S. Cecotti and M. Del Zotto.
 Galois covers of 𝒩=2 BPS spectra and quantum monodromy,
 Adv. Theor. Math. Phys.  20, 1227 , 2016,
  [arXiv:1503.07485 [hep-th]].


cecotti2015higher
  S. Cecotti and M. Del Zotto,
 Higher S-dualities and Shephard-Todd groups,
 JHEP 1509, 035, 2015,
  [arXiv:1507.01799 [hep-th]].
  
  toappear
S. Cecotti and M. Del Zotto,
 to appear.


chen2014tilting
Jianmin Chen, Yanan Lin, and Shiquan Ruan.
Tilting objects in the stable category of vector bundles on a
  weighted projective line of type (2, 2, 2, 2; λ).
Journal of Algebra, 397:570–588, 2014.

cordova2013line
Clay Córdova and Andrew Neitzke.
Line defects, tropicalization, and multi-centered quiver quantum
  mechanics.
arXiv preprint arXiv:1308.6829, 2013.

dehy2008combinatorics
Raika Dehy and Bernhard Keller.
On the Combinatorics of Rigid Objects in 2–Calabi–Yau
  Categories.
International Mathematics Research Notices, 2008, 2008.

del2013four
Michele Del Zotto.
Four-dimensional N=2 superconformal quantum field theories
  and BPS-quivers.
PhD thesis, Scuola Internazionale Superiore di Studi
  Avanzati, 2013.

del2014absence
Michele Del Zotto and Ashoke Sen.
About the absence of exotics and the Coulomb branch formula.
arXiv preprint arXiv:1409.5442, 2014.

Denef:2002ru 
  F. Denef,
 Quantum quivers and Hall / hole halos,
 JHEP 0210, 023 , 2002, 
  [hep-th/0206072].

derksen2008quivers
Harm Derksen, Jerzy Weyman, and Andrei Zelevinsky.
Quivers with potentials and their representations I: Mutations.
Selecta Mathematica, New Series, 14(1):59–119, 2008.

kedem2
P. Di Francesco and R. Kedem.
Q-systems, heaps, paths and cluster positivity.
Comm. Math. Phys. 293 (2009) 727802, arXiv:0811.3027 [math.CO]

DiFrancesco:2008mc 
  P. Di Francesco and R. Kedem,
Q-systems as cluster algebras II: Cartan matrix of finite type and the polynomial property,
Lett. Math. Phys.  89, 183 (2009)
  doi:10.1007/s11005-009-0354-z
  [arXiv:0803.0362 [math.RT]].

dimo1
    T. Dimofte,
Duality Domain Walls in Class 𝒮[A_1],
Proc. Symp. Pure Math.  88, 271, 2014.


dimo2
    T. Dimofte, D. Gaiotto and R. van der Veen.
RG Domain Walls and Hybrid Triangulations.
Adv. Theor. Math. Phys.  19, 137 (2015)
  [arXiv:1304.6721 [hep-th]].

dominguez2014caldero
Salomón Dominguez and Christof Geiss.
A Caldero–Chapoton formula for generalized cluster categories.
Journal of Algebra, 399:887–893, 2014.

drinfeld2004dg
Vladimir Drinfeld.
DG quotients of DG categories.
Journal of Algebra, 272(2):643–691, 2004.

drukker2009loop
Nadav Drukker, David R Morrison, and Takuya Okuda.
Loop operators and S-duality from curves on Riemann surfaces.
Journal of High Energy Physics, 2009(09):031, 2009.

dupont2011generic
Grégoire Dupont.
Generic variables in acyclic cluster algebras.
Journal of Pure and Applied Algebra, 215(4):628–641,
  2011.
  
  bookmap
  B. Farb and D. Margalit.
A Primer on Mapping Class Groups,
Princeton University Press, 2012.

fock2009cluster
V. Fock and A. Goncharov.
Cluster ensembles, quantization and the dilogarithm.
arXiv preprint math.AG/0311245.

fock2005dual
V. Fock and A. Goncharov.
Dual Teichmuller and lamination spaces.
arXiv preprint math/0510312, 2005.

fomin2008cluster
Sergey Fomin, Michael Shapiro, and Dylan Thurston.
Cluster algebras and triangulated surfaces. Part I: Cluster
  complexes.
Acta Mathematica, 201(1):83–146, 2008.

fomin2002cluster
Sergey Fomin and Andrei Zelevinsky.
Cluster algebras I: foundations.
Journal of the American Mathematical Society, 15(2):497–529,
  2002.

gaiotto2012n
Davide Gaiotto.
N=2 dualities.
Journal of High Energy Physics, 2012(8):1–58, 2012.


gaiotto2013wall
Davide Gaiotto, Gregory W Moore, and Andrew Neitzke.
Wall-crossing, Hitchin systems, and the WKB approximation.
Advances in Mathematics, 234:239–403, 2013.


gaiotto2013framed
Davide Gaiotto, Gregory W Moore, and Andrew Neitzke.
Framed BPS states.
Advances in Theoretical and Mathematical Physics,
  17(2):241–397, 2013.
  
gaiotto2014open
Davide Gaiotto.
Open Verlinde line operators.
arXiv preprint arXiv:1404.0332, 2014.


geigle
W. Geigle and H. Lenzing.
A class of weighted projective lines arising in representation theory of finite dimensional algebras,
Springer Lecture Notes in Mathematics 1273 (1987) 265-297.


ginzburg2006calabi
Victor Ginzburg.
Calabi-Yau algebras.
arXiv preprint math/0612139, 2006.

goncharov2016ideal
Alexander B. Goncharov.
Ideal webs, moduli spaces of local systems, and 3d Calabi-Yau
  categories.
arXiv preprint arXiv:1607.05228, 2016.

GH
P. Griffiths and J. Harris.
Principles of Algebraic Geometry,
Wiley (1994).

hooft1978phase
G. 't Hooft.
On the phase transition towards permanent quark confinement.
Nuclear Physics: B, 138(1):1–25, 1978.

hooft1979property
G. 't Hooft.
A property of electric and magnetic flux in non-Abelian gauge
  theories.
Nuclear physics: B, 153:141–160, 1979.

hooft1980confinement
G. 't Hooft.
Confinement and topology in non-abelian gauge theories.
Acta physica Austriaca: Supplementum, 22:531–586, 1980.

hooft1980topological
G. 't Hooft.
Which topological features of a gauge theory can be responsible for
  permanent confinement?
Recent developments in gauge theories. Proceedings of the NATO
  Advanced Study Institute on recent developments in gauge theories, held in
  Cargèse, Corsica, August 26-September 8, 1979, pages 117–133, 1980.

iyama2008mutation
Osamu Iyama and Yuji Yoshino.
Mutation in triangulated categories and rigid Cohen–Macaulay
  modules.
Inventiones mathematicae, 172(1):117–168, 2008.

kapustin2008homological
Anton Kapustin, Maximilian Kreuzer, and Karl-Georg Schlesinger.
Homological mirror symmetry: new developments and
  perspectives, volume 757.
Springer, 2008.

keller2005triangulated
Bernhard Keller.
On triangulated orbit categories.
Doc. Math, 10(551-581):21–56, 2005.

 kedem1
 R. Kedem, 
Q-systems as cluster algebras. 
J. Phys. A: Math. Theor. 41 (2008) 194011 (14 pages). arXiv:0712.2695 [math.RT]
  


keller2006differential
Bernhard Keller.
On differential graded categories.
arXiv preprint math/0601185, 2006.

keller2007derived
Bernhard Keller.
Derived categories and tilting.
London Mathematical Society Lecture Note Series,
  332:49, 2007.

keller2011cluster
Bernhard Keller.
On cluster theory and quantum dilogarithm identities.
In Representations of Algebras and Related Topics,
  Editors A. Skowronski and K. Yamagata, EMS Series of Congress
  Reports, European Mathematical Society, pages 85–11, 2011.

keller2007cluster
Bernhard Keller and Idun Reiten.
Cluster-tilted algebras are Gorenstein and stably
  Calabi–Yau.
Advances in Mathematics, 211(1):123–151, 2007.

keller2011deformed
Bernhard Keller and Michel Van den Bergh.
Deformed Calabi–Yau completions.
Journal für die reine und angewandte Mathematik
  (Crelles Journal), 2011(654):125–180, 2011.

keller2011derived
Bernhard Keller and Dong Yang.
Derived equivalences from mutations of quivers with potential.
Advances in Mathematics, 226(3):2118–2168, 2011.

kontsevich1994homological
Maxim Kontsevich.
Homological algebra of mirror symmetry.
arXiv preprint alg-geom/9411018, 1994.

Kontsevich:2008fj 
  M. Kontsevich and Y. Soibelman.
 Stability structures, motivic Donaldson-Thomas invariants and cluster transformations,
  arXiv:0811.2435 [math.AG].

kontsevich2014wall
Maxim Kontsevich and Yan Soibelman.
Wall-crossing structures in Donaldson–Thomas invariants,
  integrable systems and mirror symmetry.
In Homological mirror symmetry and tropical geometry, pages
  197–308. Springer, 2014.

kussin2013triangle
Dirk Kussin, Helmut Lenzing, and Hagen Meltzer.
Triangle singularities, ADE-chains, and weighted projective
  lines.
Advances in Mathematics, 237:194–251, 2013.

labardini2008quivers
Daniel Labardini-Fragoso.
Quivers with potentials associated to triangulated surfaces.
Proceedings of the London Mathematical Society, 98(3):797–839,
  2008.
 
 lenzing1 
  H. Lenzing, 
Hereditary categories, in Handbook of Tilting Theory, eds. L. Angeleri Hügel, D. Happel and H. Krause, London Mathematical Society Lecture Note Series 332 CUP (2007), pages 105-146.
  
  miyaki
  J.-I. Miyachi and A. Yekutieli, “Derived Picard groups of finite dimensional hereditary algebras”, arXiv:math/9904006.

musiker2013bases
Gregg Musiker, Ralf Schiffler, and Lauren Williams.
Bases for cluster algebras from surfaces.
Compositio Mathematica, 149(02):217–263, 2013.

neeman2014triangulated
Amnon Neeman. 
Triangulated Categories. (AM-148), volume 148.
Princeton University Press, 2014.

intematrix
M. Newman.
Integral Matrices.
 Academic Press, New York, 1972



palu2008cluster
Yann Palu.
Cluster characters for 2-Calabi–Yau triangulated categories.
In Annales de l'institut Fourier, volume 58, pages
  2221–2248, 2008.

palu2009grothendieck
Yann Palu.
Grothendieck group and generalized mutation rule for
  2–Calabi–Yau triangulated categories.
Journal of Pure and Applied Algebra, 213(7):1438–1449, 2009.

plamondon2011cluster
Pierre-Guy Plamondon.
Cluster algebras via cluster categories with infinite-dimensional
  morphism spaces.
Compositio Mathematica, 147(06):1921–1954, 2011.

qiu2016decorated
Yu Qiu.
Decorated marked surfaces: spherical twists versus braid twists.
Mathematische Annalen, 365(1-2):595–633, 2016.

qiu2013cluster
Yu Qiu and Yu Zhou.
Cluster categories for marked surfaces: punctured case.
arXiv preprint arXiv:1311.0010, 2013.

qiu2014decorated
Yu Qiu and Yu Zhou.
Decorated marked surfaces II: Intersection numbers and
  dimensions of Homs.
arXiv preprint arXiv:1411.4003, 2014.

reiten2010tilting
Idun Reiten.
Tilting theory and cluster algebras.
arXiv preprint arXiv:1012.6014, 2010.

segal2016all
Ed Segal.
All autoequivalences are spherical twists.
arXiv preprint arXiv:1603.06717, 2016.

seiberg1995electric
Nathan Seiberg.
Electric-magnetic duality in supersymmetric non-Abelian gauge
  theories.
Nuclear Physics B, 435(1-2):129–146, 1995.

seiberg1994monopoles
Nathan Seiberg and Edward Witten.
Monopoles, duality and chiral symmetry breaking in N=2
  supersymmetric QCD.
Nuclear Physics B, 431(3):484–550, 1994.

seidel2001braid
Paul Seidel and Richard Thomas.
Braid group actions on derived categories of coherent sheaves.
Duke Mathematical Journal, 108(1):37–108, 2001.

shapere1999bps
Alfred D. Shapere and Cumrun Vafa.
BPS structure of Argyres-Douglas superconformal theories.
arXiv preprint hep-th/9910182, 1999.

terash
   Y. Terashima and M. Yamazaki,
SL(2,R) Chern-Simons, Liouville, and Gauge Theory on Duality Walls,
JHEP 1108, 135,2011, 
  [arXiv:1103.5748 [hep-th]].
  
Witten:1979ey 
  E. Witten,
Dyons of Charge e θ/2 π,
Phys. Lett.  86B, 283 , 1979.

xie2012network
Dan Xie.
Network, Cluster coordinates and N=2 theory I.
arXiv preprint arXiv:1203.4573, 2012.



