Mirror Construction for Nakajima Quiver Varieties
Jiawei Hu Siu-Cheong Lau Ju Tan



	In this paper, we construct the ADHM quiver representations and the corresponding sheaves as the mirror objects of formal deformations of the framed immersed Lagrangian sphere decorated with flat bundles.  More generally, framed double quivers of Nakajima are constructed as localized mirrors of framed Lagrangian immersions in dimension two.  This produces a localized mirror functor to the dg category of modules over the framed preprojective algebra.  
	
	For affine ADE quivers in specific multiplicities, the corresponding (unframed) Lagrangian immersions are homological tori, whose moduli of stable deformations are asymptotically locally Euclidean (ALE) spaces.  We show that framed stable Lagrangian branes are transformed into monadic complexes of framed torsion-free sheaves over the ALE spaces.
	
	A main ingredient is the notion of framed Lagrangian immersions. Moreover, it is important to note that the deformation space of a Lagrangian immersion with more than one component is stacky.   
	Using the formalism of quiver algebroid stacks, we find isomorphisms between the moduli of stable Lagrangian immersions and that of special Lagrangian fibers of an SYZ fibration in the affine A_n cases.





[
    [
    25 April 2024
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§ INTRODUCTION
 

The celebrated work of Atiyah-Drinfeld-Hitchin-Manin <cit.> showed that all anti-self dual solutions to Yang-Mills equations over ^4 for compact classical Lie groups can be encoded as linear data.  It reduces a highly non-trivial nonlinear partial differential equation to a much more manageable algebraic equation. Using the Penrose fibration, Donaldson <cit.> established a one-to-one correspondence between the isomorphism classes of the linear data and the isomorphism classes of framed holomorphic vector bundles over ^2 (where a framed bundle means the trivialization of the bundle is fixed at l_∞⊂^2). Subsequently, Nakajima formalized these linear data by introducing quiver representations <cit.> and use them <cit.> to construct framed torsion-free sheaves over P^2. These works explain the profound connections between Yang-Mills instantons, framed representations of the ADHM quvier Q^ADHM and framed sheaves of ^2. 



Our motivation comes from an observation of the relation between the ADHM quiver Q^ADHM and the framed immersed nodal sphere.  See Figure <ref>.  



According to homological mirror symmetry conjecture <cit.>, Lagrangian submanifolds in a symplectic manifold are dual to coherent sheaves on the mirror complex manifold.  The conjecture was proved for several important geometries such as <cit.>.  

Moreover, the Strominger-Yau-Zaslow program <cit.> asserted that mirror manifolds can be constructed as dual special Lagrangian torus fibrations.  
A crucial ingredient in the SYZ program is wall-crossing of open Gromov-Witten invariants bounded by Lagrangian torus fibers.  Wall-crossing was extensively studied, see for instance <cit.> from the algebraic aspects and <cit.> from the symplectic asnpects. Family Floer theory <cit.> applied to smooth SYZ fibers established a more direct relation of wall-crossing with the Fukaya category of Lagrangian submanifolds.

The nodal sphere  is a crucial object in the study of mirror symmetry since it provides an important class of singular SYZ fibers. Singular fibers of an SYZ fibration are the sources of walls of holomorphic discs of Maslov-index zero.  Among them, × T^n-2 for the nodal sphere  (where n is the dimension and T denotes a torus) is the most generic one.  

In <cit.>, Auroux carried out the SYZ mirror construction for the local model given by a conic fibration 

    π:_(a,b)^2-{ab=1}→,
 π(a,b)=ab using Lagrangian torus fibers, namely the Clifford and Chekanov tori.  The mirror manifold glued from the deformation spaces of tori is 
    {(u,v,z) ∈ (^2 - (0,0)) ×^×| uv = 1+z}.

The missing point (u,v)=(0,0) that needs to be filled in is dual to the singular fiber of π which is a nodal sphere.  

Floer theory for general compact Lagrangians and immersions were developed by <cit.>. <cit.> constructed a localized mirror functor using Maurer-Cartan deformations coming from immersed sectors. Following the formalism developed in <cit.>, gluing between the deformation space of the immersed sphere and that of the tori were deduced using Floer-theoretical isomorphisms in <cit.>.  The charts of immersed spheres were glued with those of tori to complete the SYZ mirror.

Let's get back to the ADHM quiver Q^ADHM.  The (unobstructed) deformation space of the immersed sphere was shown to be generated by two variables x,y, which correspond to the degree-one generators of the Floer theory at the immersed point.  It matches the two self-loops of the ADHM quiver.  However, the framing vertex of Q^ADHM is not present in the above description.

In the work <cit.>, vertices of the mirror quiver correspond to connected components of (the normalization of) a Lagrangian immersion. Framing prescribes the trivialization along infinity  l_∞⊂^2 on the sheaf side.  This indicates the corresponding Lagrangian should be non-compact in order to reflect the properties near infinity. Additionally, the framing vertex has no self loops, so the corresponding Lagrangian should be topologically trivial. Note that one transverse intersection point produces two generators in degree one.  Thus, the two arrows i,j in Q^ADHM should correspond to a transverse intersection point between the two Lagrangians.  In the SYZ setting, These indicate that the framing vertex should correspond to a (non-compact) Lagrangian section of the SYZ fibration. 

Wrapped or partially wrapped Floer theory <cit.> provides rich and powerful methods to study non-compact Lagrangians.  On the other hand, for the purpose of mirror symmetry for framed sheaves over a compactification, we shall treat the corresponding non-compact Lagrangians in a similar way to the compact ones without wrapping.  In particular, they have a finite-dimensional Floer cohomology.  For consistency with our Morse theoretical treatment for immersed Lagrangians, we will fix a Morse function on the non-compact Lagrangian component, and use pearl trajectory models <cit.> instead of actual Hamiltonian perturbations.

We show that the noncommutative deformation space of the framed Lagrangian immersion ^ shown in Figure <ref> equals to the ADHM quiver algebra. The main point is that the ADHM relations come from the obstruction of the framed Lagrangian immersion.

 
 Let ^:= xX+yY+iI+jJ and  _free^ be the completion of the path algebra of Q^ADHM. There exists a coordinate change on _free^ such that 
the formal deformation space of the framed Lagrangian immersion 𝕃^fr equals

    ^:=_free^ / (xy-yx+ij).



Taking the (unframed) immersed sphere as a reference Lagrangian , we have the mirror functor ^(,), where =xX+yY denotes the boundary deformations coming from the immersed generators. By Theorem <ref>, we have a stability condition on the formal deformation space of ^ induced from that of the quiver Q^ADHM <cit.>.


	^(,) transforms stable framed Lagrangian branes (^,^) to framed torsion-free sheaves over ^2 in the form of monads.




More generally, Kronheimer and Nakajima <cit.> extended the ADHM constructions to asymptotically locally Euclidean (ALE) spaces. Furthermore, Nakajima constructed hyperKähler quiver varieties of general quivers without self-loops in <cit.>, and utilized quiver varieties to construct moduli space of framed torsion-free sheaves over ALE spaces <cit.>. Among these works, a main ingredient is the following monadic complex, see Section <ref> and Theorem <ref> for notations: 
    0 → L^0((,),(,)) → E((,),(,)) ⊕ L^1((,), (F,)) → L^2((,),(,)) → 0.
 This complex possesses profound applications and appears at various points in the study of quiver varieties, such as the construction of instantons over an ALE space <cit.>, and Kashiwara crystal structure <cit.>, the definition of the Hecke correspondence <cit.>, and so on. We will focus on the construction and study of moduli spaces of framed torsion-free sheaves. 


Essentially, monad is a three-term complex that has cohomology concentrated at the middle term.  The cohomology of a monad yields a torsion-free sheaf, or a holomorphic vector bundle (with an induced connection from the trivial one) under a stronger fiberwise condition.  Moreover, quiver representations also naturally arise from monads. See for instance <cit.>.






It is natural to extend the mirror correspondence above for the ADHM quiver to general Nakajima quivers.  An important notion is a framed Lagrangian brane.  

We define a framing Lagrangian F of a convex symplectic manifold M to be a conical Lagrangian diffeomorphic to ℝ^n, which is equipped with a Morse function f_F with exactly one maximal point (and no other critical point). 
See Definition <ref>.
A framed symplectic manifold is a convex symplectic manifold together with a collection  of framing Lagrangians. 

A framed Lagrangian brane is defined to be the union of a compact Lagrangian immersion and  together with a formal deformation by a boundary cochain supported at their intersection. 


	In Seidel's Picard-Lefschetz theory <cit.>, a framed Lagrangian sphere is defined as a Lagrangian together with a fixed diffeomorphism class to a smooth sphere.  To distinguish from this, we call the object of our concern to be a framed Lagrangian immersion or a framed Lagrangian brane.  Our definition is more in the homological sense.  We have a subspace of the Floer cohomology of the compact Lagrangian component formed by the image of that of the framing.


In this paper, we focus on dimension two.  Namely, for each Nakajima's framed quiver, one has a corresponding framed symplectic surface constructed by plumbing.
We show that under the mirror functor,
framed Lagrangian branes are transformed to framed quiver representations in the form of monadic complexes.


   
   Let D:=(I,E) be a graph and  the compact Lagrangian immersion obtained from the plumbing construction. Then the associated quiver Q is the double quiver of the graph D. Moreover, up to a change of coordinates, the noncommutative deformation space  of  is the preprojective algebra of Q. Namely, 
    = _free/J,
 where J is the two-sided ideal generated by ∑_t(a)=vϵ(a)x_a̅x_a for all v ∈ I. 
   
   Furthermore, the noncommutative deformation space ^fr of the framed Lagrangian immersion ^fr is 
    ^fr= ^_free/ J^fr,
  where J^fr is a two-sided ideal generated by (∑_t(a)=vϵ(a) x_a̅ x_a)_v + (ij)_v for all v ∈ I.



	Assume the same setting as above. ^(,) transforms the framed Lagrangian brane (^fr,) to the following monadic complex:
	
    0 → L^0((,),(,)) → E((,),(,)) ⊕ L^1((,), (F,)) → L^2((,),(,)) → 0,
 where the differential is given by the deformed m_1^. This complex coincides with the monad constructed algebraically by Nakajima <cit.>.






In summary, Nakajima framed quiver varieties are identified with the moduli spaces of stable unobstructed deformations of framed Lagrangian branes under the mirror functor.  


	The relation between plumbing of Lagrangian two-spheres and preprojective algebras of quivers is a very interesting subject.  In <cit.>, Etgu-Lekili computed the wrapped Fukaya category of a plumbing of Riemann surfaces using Legendrian geometry, and obtained a multiplicative preprojective algebra.  It has a close relation with the singular string topology introduced by <cit.>.  Etgu-Lekili also studied Koszul duality among these algebras in <cit.>. This approach was further developed by <cit.>.
	
	Our study in this paper is closer to the SYZ program and family Floer theory, in which the mirror space is formulated as moduli of unobstructed deformations of Lagrangians <cit.>.  To obtain Nakajima quiver varieties, an important ingredient is framing on symplectic manifolds.  We localize over a formal neighborhood of immersed Lagrangians. Moreover, motivated from the study of ALE spaces, 
	we formulate a higher-rank version of stable deformations of Lagrangians branes, which is important to produce deformations of immersed Lagrangians to tori (for instance in the affine ADE cases). This naturally produces a mirror functor which transforms framed Lagrangians to monadic complexes.


Fundamentally, quiver theory is involved in this extended version of Lagrangian Floer theory because it encodes the stacky structure of deformation spaces of a Lagrangian immersion , especially when  has more than one components or when  is equipped with a higher-rank flat bundle.   In these situations, the deformation space is a quotient stack by a non-compact gauge group.  Quiver representation theory provides a tool to encode this for all dimensions simultaneously.

A local-to-global approach is provided by the notion of an algebroid stack <cit.> and its quiver version <cit.>.  In <cit.>,  is implicitly assumed to be equipped with a rank-one flat bundle.  In this paper, 
we formulate the notion of an affine stable nc chart in higher ranks (see Section <ref>), which allows the components of  to be equipped with higher-rank bundles.  

For general preprojective quiver algebras, the stable charts are noncommutative algebras that have highly non-trivial relations.  On the other hand, for affine ADE quivers, we expect that 
the preprojective relations can force their charts to be commutative.  This is motivated by the works of <cit.> who found the connection of the (multiplicative) preprojective algebra to the Kleinian singularities in the affine ADE case.  In particular, this implies that the loop algebras (which are singular affine varieties) are commutative.  We expect that commutativity also holds for the smooth affine charts under our formulation in suitable ranks.

We check this for affine A_n type and affine D_4 type quivers.


	Let _0 be the path algebra of the affine D_4 type quiver and δ⃗=(2,1,1,1,1). Then there exists some stable affine charts of the stable family _0 over  of rank δ⃗, where  is the noncommutative parameter space defined at the end of Section <ref>, which can be glued into the minimal resolution of D_4 singularity. In particular, these stable affine charts are commutative.



	Let  be the path algebra of the affine A_n type quiver and δ⃗=(1,⋯,1). Then there exists some stable affine charts of the stable family  over Λ of rank δ⃗, which can be glued into the minimal resolution of A_n singularity.
 	




From the perspective of Lagrangian Floer theory, affine ADE quivers in specific dimension vectors are special in the sense that the corresponding immersed Lagrangian branes have torus-type Lagrangian deformations (see Proposition <ref>).  In these cases, we can show that the moduli spaces are asymptotically locally Euclidean (ALE) gravitational instantons (Corollary <ref>).  

From this perspective, ALE spaces are mirror to the Weinstein neighborhoods of affine ADE-type immersed Lagrangians.  ADE surfaces form an important class of resolutions of singularities and are the main objects of study in McKay correspondence.  Its three-dimensional analogs provide a rich source of geometric surgeries, see for instance <cit.>.

A stability condition plays an important role in formulating an atlas of affine charts of a quiver algebra.  We take families of stable unobstructed deformations of a Lagrangian brane to construct mirror affine charts.  In Calabi-Yau cases, stable Lagrangians are expected to be represented by special Lagrangians, see for instance <cit.>.  We realize such a correspondence via Fukaya isomorphism over a quiver algebroid stack for the affine A_n case.


	Let D=(I,E) be the affine A_n Dynkin diagram. Let  be the Lagrangian immersion obtained by the plumbing construction, _i be the i-th immersed sphere and T_i be a family of Lagrangian tori that intersects the i-th spheres. There exist preisomorphism pairs between (,_0) and (_i,_i),i=1,⋯, n+1:
	
    α_i ∈ CF__0(U_0i)⊗_i((,_0),(_i,_i)),β_i ∈ CF__i ⊗_0(U_0i)((_i,_i),(,_0))

	and a quiver stack  corresponding to the minimal resolution of A_n singularity, whose charts are _0 and _i,i=1,⋯,n+1, that solves the isomorphism equations for (α_i,β_i) over the Novikov field Λ. Furthermore, there also exist preisomorphism pairs between (,_0) and (T_i,_i) over the quiver algebroid stack for i=1, ⋯, n.




In the above theorem, we use the stability condition introduced by <cit.> and the stability condition of a framed quiver representation introduced by Nakajima, as seen in <cit.>. Furthermore, we apply the technique of solving Fukaya isomorphism equations over the quiver algebroid stack, as introduced in <cit.>. This provides a precise identification between certain stable Lagrangians (defined in Section <ref>) and special Lagrangians.  See also <cit.> in the case of deformed conifolds and Atiyah flop.


	Additionally, we also analyze the Floer complex CF((^fr,),(^fr,)). In the affine A_n cases, we show that ^(^fr,)(^fr,) has cohomology exactly at the highest term which gives the mirror quiver representations.
	
	
		Let (^fr,) be the framed Lagrangian immersion in affine A_n. Then the Floer complex ^(^fr,)(^fr,) has only nontrivial cohomology at the top degree, which gives a framed quiver representation.
		
	
	The main challenge lies in demonstrating that the cohomology is trivial at the middle degree. This obstacle is overcome by observing that the Floer complex ^(^fr,)(^fr,) in affine A_n case is analogue to the Floer complex in ^2 case, smashed with /n.




 §.§ Notations
 
We will work over Novikov ring (field). We use the following notations: 

    Λ_+    ={∑_i=0^∞a_iT^A_i| A_i> 0 increases to  +∞, a_i∈ℂ};
    Λ_0    ={∑_i=0^∞a_iT^A_i| A_i≥ 0 increases to  +∞, a_i∈ℂ};
    Λ    ={∑_i=0^∞a_iT^A_i| A_i increases to  +∞, a_i∈ℂ}.


They admit a non-archimedean valuation map (resp. norm) 
    val(x)=min{λ_i| a_i≠ 0}   (resp.  |x|_𝔳=e^-val(x)).


We will extend  Floer group and Fukaya algebra over path algebras. Recall a quiver is a quadruple 
    Q = (I,E,h,t)
where I (and E) is the set of vertices (resp. arrows), and 
    h,t:E→ I
 are the maps assigning an arrow to its head (resp. tail). The functions h,t easily generalize to paths in Q. 

The path algebra kQ of a quiver Q is the k-algebra with its basis consisting of all paths in Q and multiplication defined by concatenation of paths extended by linearity. Paths are read from right to left as in composition of functions.  



 §.§ Acknowledgment
 

The first-named author thanks Xinyu Zhou for helpful discussions on non-Archimedean geometry.
The second-named author expresses his gratitude to Cheol-Hyun Cho, Hansol Hong and Yoosik Kim for numerous useful discussions and the collaborations on immersed Lagrangians and mirror functors, which form one of the main motivations for this paper.  The third-named author wants to express his gratitude to Kaoru Ono and Yan-Lung Leon Li for helpful discussions on  Kuranishi structures and related topics.  We are grateful to Naichung Conan Leung for  enlightening discussions on 3D mirror symmetry and moduli theory during our visits to The Chinese University of Hong Kong. 



§ REVIEW OF NC MIRROR FUNCTOR AND QUIVER STACKS




 §.§ Review of noncommutative mirror functor
 

Given a symplectic manifold M and a compact spin oriented Lagrangian immersion ⊂ M, we  associate a quiver Q to , whose path algebra encodes the boundary deformation of . Let's recall the construction in <cit.>.


	
  * Associate a quiver Q to ^1(). Namely, each component of (the domain of)  is associated with a vertex, and each generator in ^1() is associated with an arrow.
	
  * Let Λ_0 Q be the free path algebra of Q over Novikov ring. Each arrow a in Q is associated with a Novikov valuation such that val(a) >0. The valuation induces a filtration on Λ_0 Q. Take the completion of Λ_0 Q with respect to this filtration. Abuse the notation, we will denote the completion by Λ_0 Q. 
	
  * Extend the Fukaya algebra A of  over the path algebra Λ_0 Q and obtain a non-commutative A_∞-algebra 
    Ã^𝕃 = Λ_0 Q⊗_Λ_0^⊕(𝕃),
 
	whose unit is _𝕃 = ∑_L_i.  Λ_0^⊕⊂Λ_0 Q denotes ⊕_i Λ_0 · e_i where e_i are the trivial paths at vertices of Q.  The fibered tensor product means that an element a ⊗ X is non-zero only when tail of a corresponds to the source of X.  The A_∞-operations are defined by
	
    m_k (f_1 X_1,…,f_k X_k) := f_k … f_1   m_k (X_1,…,X_k)

	where X_l ∈()  and f_l ∈Λ_0 Q.
	
  * Extend the formalism of bounding cochains of <cit.> over Λ_0 Q, that is, we take 
	
    = ∑_l b_l B_l

	where B_l are the generators of ^1(), and b_l are the corresponding arrows in Q.  Then define the deformed A_∞-structure m_k^ as in <cit.> and via Equation (<ref>).
	
  * Quotient out the quiver algebra by the two-sided ideal R generated by coefficients of the obstruction term m_0^, so that m_0^ = W·_ over 
	
    := Λ_0 Q/R.
   is the space of noncommutative weakly unobstructed deformations of .
	We call (,W) to be the noncommutative local mirror of X probed by .
	
  * Extend the Fukaya category over , and enlarge the Fukaya category by including the noncommutative family of objects (,) where  in (<ref>) is now defined over .  This means for L_1,L_2 in the original Fukaya category, the morphism space is now extended as ⊗(L_1,L_2).  The morphism spaces between (,) and L are enlarged to be ((,),L) := ⊗_Λ_0^⊕(𝕃,L) (and similarly for (L,(,))).  We already have ((,),(,)) in Step 2 (except that Λ_0 Q is replaced by ).  The m_k operations are extended in a similar way as (<ref>).



	Consider the boundary deformation  of . There exists a well-defined A_∞-functor
	
    ^(,): (X) →(,W).




	m_0^=W·_ has degree 2.  Thus in the -graded situation, W=0, and the above (,W) reduces to the dg category of complexes of -modules.



	The formal boundary deformation is part of the data of the original Fukaya algebra, which is a formalism to extract the stacky deformation space from the original Fukaya algebra. Here, stacky means it's natural to consider the isomorphism classes of Maurer-Cartan elements, which involves quotient stack if we don't throw away unstable points. 




 §.§ Quiver Algebroid Stack
 

Philosophically, the noncommutative deformation space of  forms a local neighborhood of the mirror. It's natural to ask how to glue the local mirrors. It is worth noting that the noncommutative deformation space of distinct Lagrangian immersions could be path algebras with varying numbers of vertices. Hence, the gluing is complicated in general and can not be achieved by the usual algebra isomorphisms.  This issue can be addressed by the notion of quiver algebroid stack introduced by the second, third authors and their collaborator <cit.>. Quiver algebroid stack is essential for the gluing of the quiver algebras.

The algebroid stack and twisted complexes were initially introduced by Bressler-Gorokhovsky-Nest-Tsygan <cit.>. In the recent work <cit.>, the authors generalize these concepts for the purpose of mirror construction. More precisely, the authors introduce the localization of a path algebra and quiver algebroid stack, which provides a local-to-global approach to understand the path algebra of a quiver. Here we first recall the definitions of affine local chart and quiver algebroid stack. Following that, we will recall the mirror functors based on these constructions. Throughout this section, we will use Q to denote a connected quiver and I to denote the set of vertices. Let k be a field.

 
	Let S ⊂=k Q/R be a finite subset of paths such that each summand of an element in S has the same head and tail vertices. For each γ∈ S, we add one arrow, denoted by γ^-1, with s(γ^-1)=t(γ) and t(γ^-1)=s(γ), to the quiver Q.  Moreover, we take the ideal R̂ generated by R and γγ^-1 - e_t(γ), γ^-1γ - e_s(γ) to be the new ideal of relations.  The new quiver with relations k Q̂ / R̂ is called the localized algebra at S, and is denoted as (S^-1).





	Let B be a topological space.  A quiver algebroid stack consists of the following data.
	
		
  * An open cover {U_i: i ∈ I} of B.
		
  * A sheaf of algebras 𝒜_i over each U_i, coming from localizations of a quiver algebra _i(U_i) = k Q^(i)/ R^(i).
		
  * A sheaf of representations G_ij of Q^(j)_V over 𝒜_i(V) for every i,j and V open⊂ U_ij.
		
  * An invertible element c_ijk(v)∈(e_G_ij(G_jk(v))·𝒜_i(U_ijk)· e_G_ik(v))^× for every  i,j,k and v ∈ Q^(k)_0, that satisfies
		
    G_ij∘ G_jk(a)=c_ijk(h(a))· G_ik(a)· c_ijk^-1(t(a))

		such that for any i,j,k,l and v,
		
    c_ijk(G_kl(v)) c_ikl(v) = G_ij(c_jkl(v))c_ijl(v).

		In this paper, we always set G_ii=, c_jjk≡ 1 ≡ c_jkk.
	
	


In particular, we can define an affine chart of a quiver algebra using quiver algebroid stack:
 
	An affine chart of a quiver algebra  is 
    (=k Q'/R',G_01,G_10)

	where Q' is a quiver with a single vertex and R' is a two-sided ideal of relations; 
    G_01: →_loc and  G_10: _loc→
 
	are representations that satisfy
	
    G_10∘ G_01 =   Id;
    
    		G_01∘ G_10(a) =    c(h(a))  a  c(t(a))^-1

	for some function c: I → (_loc)^×
	that satisfies c(v) ∈ e_v_0·_loc· e_v, where v_0 denotes the image vertex of G_01.  Here, _loc is a localization of  at certain arrows and e_v denotes the trivial path at the vertex v.


The quiver algebroid stack arises naturally when we want to glue the noncommutative deformation space of a collection of immersed Lagrangians using (pre)isomorphism pairs.


	Let (L_1,_1), (L_2,_2) be two Lagrangian immersions. An isomorphism pair is a pair of degree 0 intersection points (α,β), where α∈ CF^0((L_1,_1),(L_2,_2)), β∈ CF^0((L_2,_2),(L_1,_1)), satisfying 
	

		m_1^_1,_2(α)=m_1^_2,_1(β)=0,   m_2^_1,_2,_1(α,β)=_L_1,   m_2^_2,_1,_2(β,α)=_L_2.
	
 


Let _1,⋯,_n be compact spin oriented immersed Lagrangians. We denote their nc unobstructed deformations by _i. Now we can transform an nc family of objects (_j,_j).  Let's define

    ℱ^(_i,_i)((_j,_j)) := (_i ⊗_(Λ^⊕)_i^∙(_i,_j) ⊗_(Λ^⊕)_j_j^, d = (-1)^|·|m_1^_i,_j(·)).

For an algebra , recall that ^ is the opposite algebra which is the same as  as a set (and the corresponding elements are denoted as a^), with multiplication a^ b^ := (ba)^. The concatenation is read from right to left with h(a^)=t(a).
ℱ^(_i,_i)((_j,_j)) is a (graded) (_i,_j)-bimodule, where the right _j-module structure on _j^ is by taking
a^· b := (ab)^ = b^ a^.  The tensor product over (Λ^⊕)_i and (Λ^⊕)_j means that an element a_i X a_j^ is non-zero only if t(X)=t(a_i) and h(X)=t(a_j^)=h(a_j).



Indeed, as a generalization of Step (6) in localized mirror construction, we shall extend the whole Fukaya category over

    T(_1,…,_n):=⊕_m≥ 0⊕_|I|=m (_i_0⊗⋯⊗_i_m)
 
that is understood as a product of the deformation spaces. ⊕_k≥ 0 means that we take the completion over Λ_0, meaning that we allow infinite series with valuation in Λ increasing to infinity. One can extend the Fukaya category over T(_1,…,_n) in an analogue way. Namely, the space of Floer chains and A_∞ operations have been extended over T(_1,…,_n).  For two Lagrangians L_0,L_1 that are not any of these _i's, the morphism space is T(_1,…,_n)⊗(L_0,L_1).  The morphism spaces involving (_i,_i) are extended as (_j⊗ T(_1,…,_n) ⊗_i)  ⊗_(Λ_0^⊕)_i ⊗ (Λ_0^⊕)_j^∙(_i,_j), T(_1,…,_n)⊗_i ⊗_(Λ_0^⊕)_i^∙(_i,L), and _i ⊗ T(_1,…,_n) ⊗_(Λ_0^⊕)_i^∙(L,_i).  All coefficients can be pulled to the left. More details and discussions about the coefficients can be found in the Chapter three of <cit.>.

Given a collection of Lagrangian immersions, quiver algebroid stack naturally emerges as a geometric mirror as shown in <cit.>. 

	Given symplectic manifold M and a collection of immersed Lagrangians ={_0, ⋯, _N }. Suppose there exist isomorphism pairs among these Lagrangians. Then
	there exists an A_∞ functor 
    ^:(M) →(),
 where  is a quiver stack constructed by gluing the noncommutative deformation spaces of , () is the dg category of twisted complexes over .


Furthermore, there exists a universal twisted complex 𝕌, which induces a functor ^:=⊗ -: dg  - mod(). In some interesting cases,  corresponds to the universal bundle over the moduli space of stable -module. And it enables us to compare these two A_∞ functors.





§ FRAMED FUKAYA CATEGORY




 §.§ Framed symplectic manifold

Let (M,ω) be a (non-compact) convex symplectic manifold of dimension 2n (see for instance <cit.> in the exact setting, and <cit.> in the monotone setup). In other words, M is assumed to be conical at infinity: there exists a compact subset M_c ⊂ M such that its complement can be identified with an infinite cone by a symplectomorphism 

    ϕ: (M∖ M_c^∘,ω)  ≅(∂ M_c× [1,∞), d(rα))

where M_c^∘ is the interior of M_c, (∂ M_c,α) is a contact manifold and r is the coordinate of [1,∞).  Moreover,

	
  * The Liouville vector field Z characterized by ω(Z,·) = θ points strictly outwards along ∂ M_c. 
	
	
  * ϕ is induced by the flow of Z for time log r.

	
  * Outside the compact set M_c, the symplectic form is exact: ω = dθ where θ = ϕ^*(rα).  



A conical almost complex structure J is taken, which is an ω-compatible almost complex structure on M satisfying J^*θ = dr for r large enough. 

We are going to define framed Lagrangian immersions, which consist of compact Lagrangian immersions, together with a  non-compact Lagrangian submanifold which we call the framing, with fixed gauge and formal perturbations.

  A framing Lagrangian of a convex symplectic manifold M  is a Lagrangian F⊂ M diffeomorphic to ℝ^n  equipped with a Morse function f_F with exactly one maximal point and no other critical point.

Moreover, we require the framing is conical which means F is of the form F = F_c∪_∂ F_c(∂ F_c× [1,∞)) where 

    
  * F_c=F∩ M_c intersects ∂ M_c transversally, 
    
  * θ|_F-F_c vanishes near the boundary ∂ F_c = ∂ M_c∩ F_c.  In particular, the complement F-F_c is an exact Lagrangian with a constant primitive function.
 







    A framed convex symplectic manifold is a tuple (M,) where M is a convex symplectic manifold and  is a collection of framing Lagrangians that are disjoint with each other.  We denote by || the cardinal number of the framing collection. 
    





	 

 

 
	Let's recall the  plumbing  construction of an exact symplectic manifold (see for example <cit.>), which is automatically a Liouville domain, whose completion is a convex symplectic manifold. 
	It is obtained from gluing by a symplectomorphism between two disk bundles D^*S_1 and D^*S_2 over two spheres that identifies the fiber and base in opposite directions.

Let M_1,M_2 be Riemannian manifolds, and we consider the total spaces of their disk cotangent bundles D^*M_1, D^*M_2.  Consider open balls B_i ⊂ D_i such that the disk bundles D^*B_i ⊂ D^*M_i for i=1,2 are symplectomorphic under  (x_1,y_1)  ↦ (-y_2,x_2), where x_i,y_i denote the base and fiber coordinates respectively.
The glued manifold 
can be completed to a Liouville manifold (after rounding off the corner).  Moreover, M_i are are embedded exact Lagrangian submanifolds. 

In the resulting Liouville manifold, we can take a cotangent fiber of M_i to be a framing Lagrangian. 


 
Consider the self-plumbing of ^2, which is obtained by symplectic gluing of the total space T^*^2 along the disc bundles D^*B_i for open balls around the north and south poles similar to the above example.  The glued manifold is completed to a Liouville manifold.  The zero-section becomes an immersed Lagrangian, and we can take a cotangent fiber of a point in ^2 away from the north and south poles to be a framing F for the space.
	
By the Weinstein neighborhood theorem, this can be identified as a neighborhood of the immersed sphere constructed by ^1-reduction as follows, which has an advantage of possessing a Lagrangian torus fibration.  
Let ϵ≠ 0 in  and consider the conic fibration π:X:={(a,b)∈^2: ab ≠ϵ}→∖{ϵ} defined by (a,b) ↦ ab. This fibration admits a Hamiltonian S^1-action on X given by 
    S^1 × X → X,    (θ, (a,b)) ↦ (e^iθa, e^-iθb),
 whose moment map is μ(a,b)=1/2(|a|^2-|b|^2).  We have the Lagrangian torus fibration (μ,log |ab-ϵ|): X →^2.  

The fiber at μ=0, log |ab-ϵ|=log |ϵ| is an immersed sphere.  Moreover, 
under such an identification, the framing F is given as a section of the Lagrangian fibration.  In particular, it intersects with the immersed sphere at a single point.












 §.§ Framed Fukaya category
 


	
	Let (M,) be a framed convex symplectic manifold.  Let L ⊂ M be a compact Lagrangian immersion that satisfies the following conditions. 
	
		
  * L ∩≠∅ and L intersects  transversely.
		
  * L ∩ does not contain the maximum point of the Morse function f_ on .
	
	Without loss of generality, we can just assume L ⊂ M_c.  The union L ∪ is called a framed Lagrangian immersion.


The above conditions on L can be easily achieved by a Hamiltonian perturbation.  Next, we follow the arguments of <cit.> to make sure that the moduli space of pseudo-holomorphic discs bounded by L∪ is compact by using the convexity assumption.  
Without loss of generality, L is assumed to be contained in M_c, and hence 
all the intersections L ∩ are contained in M_c. 




Let (M,) be a framed convex symplectic  manifold, and L a compact Lagrangian immersion.  All the pseudo-holomorphic curves u:(𝕊,∂𝕊)→ (M,L∪) satisfying u(∂𝕊)⊂ L∪ must be contained in M_c.



Assume there is such a non-constant pseudo-holomorphic curve u that intersects ∂ M_c×(1,∞). Choose a regular value 1+ϵ of r∘ u and let Σ be the preimage of [1+ϵ,∞) under r∘ u. 
The boundary ∂Σ is stratified into  ∂_FΣ and ∂_inΣ which lie in  and the interior of the Riemann surface  respectively. For convenience, we can additionally assume the primitives over F-F_c are 0 for all framings F.  Then we can rewrite the  energy of u|_Σ as

    ∫_Σu^*ω    = ∫_∂Σθ∘(du)
        = ∫_∂_inΣθ∘ (du) = ∫_∂_inΣ-d(r∘ u)∘ j

where the last equality follows from the contact-type condition of J (and j is the complex structure on the domain ). Now j(ξ) points inwards along the ∂_inΣ where ξ is tangent to ∂_inΣ and respecting its orientation. Take ϵ→ 0, the energy of u|_Σ is non-positive which is impossible. Therefore we conclude the image of u must be contained in M_c. 



We now discuss moduli spaces of stable maps from a bordered Riemann surface with immersed Lagrangians boundary conditions. Fix k∈ℤ_≥ 0 and β∈ H_2(X,L). We consider bordered Riemann surfaces  with k+1 boundary marked points representing β, and write z={z_0,z_1,⋯, z_k}. Let u:Σ→ M be a J-holomorphic map with u(∂Σ) ⊂ L. Such map is called stable if the automorphism group (Σ, u,z) is finite. need modifications Since our manifold is exact, there is no bubbling phenomenon and in this case, (Σ, u, z) is stable if and only if for each irreducible component of Σ, it is either non-constant or it has at least 3 boundary marked points when it's diffeomorphic to a disc. 

In the embedded case, one defines the moduli space ℳ_k+1(β, J) as compactifaction space of equivalence class of (Σ, u, z). But in our setting, since the Lagrangian is immersed we need to impose more assumptions. Let R be  the set of ordered pairs (p_-,p_+)∈ L× L such that p_-≠ p_+ and ι(p_-) = ι(p_+), and define an involution σ that sends (p_-,p_+) to (p_+,p_-). The corners corresponding to the immersed sectors are described by a map α: I → R where I is a subset of [k]. An element of ℳ_k+1(α,β, J) is a stable map u: (Σ, ∂Σ)→ (M,ι(L)) from a genus 0 bordered Riemann surface with k+1 marked points. For i∈ I, we require that 

    (lim_z↑ z_iu(z), lim_z↓ z_iu(z)) = α(i) ∈ R

where z↑ z_i (z↓ z_i) means z approaches z_i in counterclockwise (clockwise). In other words, there is no sheet jump at the immersed point. 

Define the evaluation maps ev_i: ℳ_k+1(α,β, J) → R∐ L by 

    ev_i([u,z])= 
    		(u(z_i),u(z_i)) ∈ L,     i∉I
    α(i) ∈ R,    i∈ I

for i∈ [k]. For i=0 (the output marked point), we also consider the evaluation map ev: 

    ev_i([u,z])= 
    		(u(z_0),u(z_0)) ∈ L,     i∉I
    σ∘α(0) ∈ R,    i∈ I

where σ is the involution map defined above. 


	The virtual dimension of ℳ_k+1(α,β, J) is  
	
    ℳ_k+1(α,β, J) = μ(α,β)+n +∑_i=0^k(1-codim(α(i))) -3




The above ensured the compactness of moduli spaces of pseudoholomorphic discs, and so we can define Fukaya category and Floer cohomologies.
There are different choices of models for chains, including the de Rham model, singular chain model and Morse model.  In this paper, we use the Morse model <cit.> for clean intersections, which counts pearl trajectories, namely certain unions (fiber products) of gradient flow trajectories and pseudo-holomorphic discs.  <cit.> constructed a Morse model via homotopy from their singular chain model that incorporated with their virtual perturbation techniques.  See also <cit.> for a review and the construction of a homotopy unit.

Consider (L_0,…,L_k), where L_i are either a component of  or a compact Lagrangian immersion satisfying the conditions in Definition <ref>.  When L_i is not intersecting transversely with L_i+1, we need to choose a perturbation of L_i+1.  This can occur when L_i,L_i+1 are both compact Lagrangian immersions or both are the same framing Lagrangian .  When the intersection is clean (in particular for the case that L_i=L_i+1), we can take a Morse perturbation: namely we choose a Morse function on the clean intersection and take its critical points as Floer generators.  For the case that L_i,L_i+1 are both compact Lagrangian immersions, we additionally require that the Morse function (on the clean intersection of L_i and L_i+1) does not have a critical point in L_i ∩ L_i+1∩ (which can be empty).  When the intersection is not clean, one can take a Hamiltonian perturbation on L_i+1.  We additionally require that the Hamiltonian perturbation preserves  and L_i ∩ϕ(L_i+1) ∩ = ∅.

For the case L_i=L_i+1 = F, we always use the prefixed Morse function f_F as the perturbation, which has a unique degree-zero critical point that represents the unit _F.  After fixing these perturbations between L_i and L_i+1 (starting with i=0 and proceeding to i=k-1), we can then consider the moduli spaces of pearl trajectories bounded by (L_0,…,L_k) and their virtual perturbations to define the m_k operations.  

As in <cit.> and <cit.>, the perturbations have to be carried out inductively on (L_0,…,L_k) and the disc classes.  Namely, the codimension-one boundaries of disc moduli are fiber products of moduli spaces of discs that have lower energies or fewer numbers of Lagrangian boundary labels.  The Hamiltonian and virtual perturbations on disc moduli need to be chosen inductively on (E,k) (where E stands for energy and k+1 is the number of boundary labels) to be compatible with those chosen on boundaries.  This gives the m_k operations for framed Lagrangian immersions that satisfy the A_∞ equations.

To extend the use of formal perturbations and Morse theory, let's first enlarge the category by equipping a function on every compact Lagrangian immersion, whose differential gives a formal Hamiltonian perturbation to the Lagrangian.  For a pair ((L,f),(L,g)) where f and g are functions equipped on a Lagrangian L and (g-f) is Morse, we take the critical points and gradient flow trajectories of (g-f) in the definition of Floer theory (for every tuple of Lagrangians that have ((F,f),(F,g)) as adjacent pairs).  If (g-f) is not Morse, we further take a slight perturbation g+ϵ h such that (g+ϵ h-f) is Morse.  This is regarded as a formal perturbation on (L,g).  Proceeding in this way, we have A_∞ operations on Lagrangians equipped with (possibly zero) functions.

On a framing Lagrangian F, we consider either (F,0) or (F,f_F) where f_F is the prefixed Morse function on F.  (F,f_F) will be used as a part of the reference Lagrangian later.  In the consideration of Floer theory, we have the pairs ((F,0),(F,0)), ((F,f_F),(F,f_F)), ((F,0),(F,f_F)), and ((F,f_F),(F,0)).  For the first two cases, we have clean intersections and we proceed like before by taking a formal perturbation by f_F on the second Lagrangian.  In the first three cases, we take f_F as the governing Morse function which has a unique maximum as the degree zero critical point.  On the other hand, for the last case, we take 0-f_F = -f_F as the governing Morse function which has a unique minimum as the degree n critical point.  Note that the minimum point can only appear as an output of a pearl trajectory.



 §.§ Formal families of framed Lagrangian branes


Let's fix a reference framed Lagrangian immersion ∪ (,-f_).  We use this to construct a mirror functor from the framed Fukaya category to the derived category of framed quiver representations.

We carry out the quiver construction of <cit.> for ∪ to give a framed quiver Q^.  In this setup, the reference framed Lagrangian has more than one component.  The moduli space of its deformations is a stacky quotient by a non-compact Lie group.  Quiver path algebra provides a canonical way to encode the stacky moduli.

For convenience of descriptions, we shall assume  and  are graded and relatively spin, so that the associated Floer cochains have integer degrees and the m_k operations have well-defined signs respectively.  Without these assumptions, Floer cochains have degrees in _2, and m_k are defined up to signs respectively.  We will also take canonical models, namely we take representatives of a basis of the Floer cohomology of ∪.

In other words, each framing Lagrangian in  (which is disjoint to each other) corresponds to a framing vertex, and each component of  corresponds to a vertex of the quiver.  Each degree one Floer generator between different components of ∪ corresponds to an arrow of Q^ between corresponding vertices.  On each component of , we fix a basis of its degree-one cohomology and associate self loops z,w with relation zw=1 at the corresponding vertex to each basic element.  (This means these are invertible self loops.)  These self loops encode the homolomies around corresponding loops for families of flat connections.

Let Λ Q^ denote the path algebra of Q^ over Λ.  We take 

    (∪,∪)_Λ Q^fr := Λ Q^fr⊗(∪,∪)
 
and extend the A_∞ operations m_k on it.  In particular, let = ∑_a a ⊗ X_a ∈(∪,∪)_Λ_0 Q^fr, where the sum is over the arrows a corresponding to jumps between corresponding Lagrangian branches, and consider m_0^,∇ := ∑_k ≥ 0 m_k^∇(,…,) for the family of flat connections ∇ parametrized by the self loops z introduced above, which is the obstruction of defining the Floer cohomology.

 
	Let ^ := Λ_0 Q^ / I where I is the two-sided ideal generated by the coefficients of m_0^,∇.  (∪,^) is called to be a universal family of framed Lagrangian branes supported on ∪.



	In above, we have assumed ∪ has a -grading and m_0^ has degree two.  In general, I is the ideal generated by the coefficients of m_0^,∇ mod out the span of the unit 1_∪ (which is the sum of the unit on each component of ∪). The coefficient W of 1_∪ in m_0^,∇ is called the disc potential.  (^,W) is called to be a Landau-Ginzburg model.


Below, we will define general families of framed Lagrangian branes which involve a choice of dimension vectors.  In such a general formulation, a Lagrangian brane may not have commutative unobstructed deformations, and we have to consider noncommutative families.

First, we fix dimension vectors w⃗ and v⃗ for the components of the framing Lagrangian  and 
 for the components of  respectively.  We equip the framing Lagrangian  with the trivial vector bundle in the rank specified by w⃗ on each component.  Moreover, the components of  are decorated with local systems ℰ of corresponding ranks specified by v⃗. 
We call ∪ equipped with these bundles to be  a Lagrangian brane.
Later, the local systems will be specified by the representing matrices of the self loops z introduced above.  v⃗ has positive entries, while w⃗ has non-negative entries.  When an entry of w⃗ is zero, we delete the corresponding component of  in the union ∪.  In particular, we can consider unframed Lagrangian branes by setting w⃗=0.

The morphisms between  (,v ,w)-framed Lagrangian brane L and (,v' ,w')-framed Lagrangian brane L' is required to satisfy the following condition: suppose p is an intersection point between F_i and F_i', which is represented by a maximal point of the Morse function,  then γ∈ Hom(ℱ_i|_p,ℱ_i'|_p) will not only preserve the basis but also preserve the order. This implies if γ is nontrivial, then w = w'. 





Conceptually, a family of (framed) Lagrangian branes over  (or ^) comes from the pull-back of the universal family by a map between  (or ^) and  ( or ^ resp.). Such a map is given by a rank v⃗ (resp. (v⃗,w⃗)) representation of the quiver algebra  (resp. ^) over another quiver algebra  (or ^).  




A representation G in rank v⃗ of a quiver algebra  by another quiver algebra  consists of an assignation f:I_→ I_, together with a family of maps g_h,t:E_h,t→ Mat_v_h× v_t() indexed by the ordered pairs (h,t)∈ V× V, where E_h,t is set of all arrows with head h and tail t. Moreover, for an arrow a∈ E_h,t, the entries of g_h,t(a) are required to be  arrows from f(t(a)) to f(h(a)).   The representation G is required to preserve relations of  and .
	
A framed representation G in rank (v⃗,w⃗) of a framed quiver algebra ^ by another framed quiver algebra ^ is a representation in the above sense, with the further condition that f: I_^→ I_^ maps a framing vertex to a framing vertex.




In other words, each vertex v_i of the quiver for  is mapped to a vertex of , and each arrow of  is represented by a matrix of arrows in . This representation is done in a manner that respects head and tail vertices, together with the relations for algebras mod idempotents at vertices; any expression in arrows of  that equals to zero is still zero in  upon substitution according to the representation. 



For simplicity, we will denote the representation defined above by G:→ Mat().  We also think of this representation as defining a family  over .





Now we define a noncommutative family of Lagrangian brane (,).


	Let  be an algebra (possibly noncommutative). A family of Lagrangian branes (,) over  is a representation G:→ Mat() in the above sense, where  is the universal family associated with (,) (see Definition <ref>).  This means there's a function that maps generators of the degree-one cohomology of the normalization of  to  elements in GL(), and maps other generators of CF^1() to elements in Mat() of suitable ranks. 




One can extend the Fukaya algebra of (, ) over  and consider the formal boundary deformations of (, ). 
	An A_∞-functor ^(,): (X) →dg given by the family (,) over  
(in the -graded case) can be obtained by employing constructions similar to those in <cit.> (by replacing  with its representation in Mat()). 










 




One would like to compare the formal deformation spaces of two (framed) Lagrangian immersions, which motivates the notion of a (framed) isomorphism.  First, we define isomorphisms between framed quiver algebras.  Next, we define isomorphisms between formal deformations of framed Lagrangian branes.





	Two framed quiver path algebras _1^ and _2^ are framed isomorphic if there exist framed representations G_21:_1^→_2^, G_12:_2^→_1^ and gerbe terms c_121, c_212, satisfying the following conditions: 
		
  * G_12 and G_21 are mutually inverse up to gerbe terms. More precisely, G_ij∘ G_ji(a)=c_iji(h(a))· a · c_iji^-1(t(a)) for (i,j)=(1,2) or (i,j)=(2,1). 
		
  * The gerbe terms at the framing vertices are trivial, i.e. c(v_f)=e_v_f for all gerbe terms c and framing vertices v_f.
	


A framed quiver algebroid stack is locally glued by framed quiver path algebras, which is motivated from taking compactification of the (unframed) quiver algebroid stack.

	Let B be a topological space.  A framed quiver algebroid stack consists of the following data:
		
			
  * An open cover {U_i: i ∈ I} of B.
			
  * A sheaf of framed quiver algebras 𝒜^_i over each U_i, coming from localizations of a quiver algebra ^_i(U_i) = k Q^(i)/ R^(i).
			
  * A sheaf of framed representations G_ij of 𝒜^_j(V) over 𝒜^_i(V) for every i,j and V open⊂ U_ij.
			
  * An invertible element c_ijk(v)∈(e_G_ij(G_jk(v))·𝒜^_i(U_ijk)· e_G_ik(v))^× for every  i,j,k and v ∈ Q^(k)_0, that satisfies
			
			
    G_ij∘ G_jk(a)=c_ijk(h(a))· G_ik(a)· c_ijk^-1(t(a))
 such that for any i,j,k,l and v,
			
    c_ijk(G_kl(v)) c_ikl(v) = G_ij(c_jkl(v))c_ijl(v).

			Furthermore, the gerbe terms at the framing vertices are trivial. In this paper, we always set G_ii=, c_jjk≡ 1 ≡ c_jkk.
			
		



	Given any framed quiver algebroid stack ^, there exists an induced unframed quiver stack , whose local charts and transition maps are the same as those in ^ after setting the arrows whose head or tail vertex is a framing vertex to be zero.



	Let (L_1^,^_1), (L_2^,^_2) be two framed Lagrangian immersions. A pair of degree 0 intersection points (α^,β^), where α^∈ CF^0((L_1^,^_1),(L^_2,^_2)), β^∈ CF^0((L_2^,^_2),(L_1^,^_1)), is called a framed isomorphism pair if there exists a framed quiver algebroid stack ^ over which (α^,β^) satisfies
	

		m_1^^_1,^_2(α^)=m_1^^_2,^_1(β^)=0,   m_2^^_1,^_2,^_1(α^,β^)=_L_1^,   m_2^^_2,^_1,^_2(β^,α^)=_L_2^.
	
 


In general, the gluing of nc deformation spaces of framed Lagrangian immersions can be restricted to the gluing of compact Lagrangian immersions.

 
	Let L_s^ be framed Lagrangian immersions with nontrivial topology and _s (resp. _s^) be the formal boundary deformation of L_s (resp. L_s^) for s=1,2. Suppose there exists a quiver algebroid stack ^ and an isomorphism pair (α^,β^) between (L_1^,_1^) and (L_2^,_2^) that solves the Fukaya isomorphism equations over ^. Then the restriction of (α^, β^) to the compact Lagrangian immersions solves the Fukaya isomorphism equations between (L_1,_1) and (L_2,_2) over the unframed quiver algebroid stack . Here,  is a quiver stack induced by ^. 
 


	Let :={F_1, ⋯, F_r} be the set of framing Lagrangians and I_k, J_k be the immersed sectors, where I_k ∈^1(F_k, (L_1^,_1^)) and J_k ∈^1((L_1^,_1^),F_k) for k=1, ⋯,r.  By definition, _1^=_1+ ∑_k i_k I_k + ∑_k j_k J_k, which is similar for _2^. Observe that the preisomorphism pair (α^,β^) between (L_1^,_1^) and (L_2^,_2^) is of the following form:
	
    α^=α+ ∑_l _F_l + ∑_m M_m+ ∑_n N_n,   β^= β + ∑_l _F_l+ ∑_p P_p+ ∑_q Q_q,

	where α∈^0((L_1,_1),(L_2,_2)), β∈^0((L_2,_1),(L_1,_2)),  _F_l is the fundamental class of the framing Lagrangian F_l, M_m ∈^0((L_1,_1),), N_n ∈^0(,(L_2,_2)), P_p∈^0((L_2,_2),) and Q_q ∈^0(,(L_1,_1)). We want to show (α,β) forms an isomorphism pair between (L_1,_1) and (L_2,_2) over ^. 
	
	First, we show m_1^_1,_2(α)=0. Since N_n ∈^0(,L_2), the outputs of m_1^_1^,_2^(N_n) at (L_1,L_2) are contributed by the polygons, which have J_k ∈(L_1,) as corners. Thus, the coefficients of the outputs are multiples of j_k in m_1^_1^,_2^(N_n). The situation is similar for the outputs of m_1^_1^,_2^(∑_l _F_l + ∑_m M_m) at (L_1,L_2). Therefore, the outputs of m_1^_1,_2(α) are the same as those of m_1^_1^,_2^(α^) after setting i_k=j_k=0 for all k. Hence, m_1^_1,_2(α)=0 over . Using the same idea, one can show m_1^_2,_1(β)=0. Furthermore, the outputs at other components have similar properties. One can also obtain m_1^_1,_2(M_m)=0 and m_1^_2,_1(Q_q)=0 over  for all m and q.
	
	Next, we consider m_2^_1,_2,_1(α,β). Since (α^,β^) is an isomorphism pair over ^, we have m_2^_1^,_2^,_1^(α^,β^)=∑_j _L_1j+ ∑_l _F_l. Similar to the above argument, it suffices to show m_2^_1^,_2^,_1^(M_m, Q_q) is not a multiple of _L_1j for all m,q,j.
	
	Suppose m_2^_1^,_2^,_1^(M_m, Q_q)= _L_1j. We have M_m ∈^0((L_1j,_1),F_k) and Q_q ∈^0(F_k,(L_1j,_1)) for some k. Otherwise if the intersections happen in the different components, the coefficient of the output will be a multiple of i and j. Notice that M_m and Q_q are closed. The existence of M_m and Q_q induces an injective homomorphism from HF(L_1j) into HF(F_q), but F_q is diffeomorphic to ^n. This contradicts to the fact that L_1j has nontrivial topology. Hence, m_2^_1,_2,_1(α,β)=_L_1. Similarly, we obtain  m_2^_2,_1,_2(β,α)=_L_2. Thus the prposition holds.



	Let L_0 be the union of two spheres, and L_i be the i-th immersed spheres in affine A_1 case for i=1,2. Let ={F_1, F_2} be the collection of framing Lagrangians and L_j^ be the corresponding framed Lagrangian immersions for j=0,1,2 as shown in the Figure <ref>.  Denote the deformation parameter of L_j^ by _j^ with formal deformation space _j^ for j=0,1,2.
	
	
	
	
	There exists a framed quiver algebroid stack ^ and framed isomorphism pairs (α_ij^, α_ji^) that solve Fukaya isomorphism equations over ^. More precisely, we have the following preisomorphism pairs: 
    α_10^=P_1+ (u_2^L)^-1Q_1+_F_1+_F_2,   α_01^=β_1^1+_F_1+_F_2

	
    α_20^=(v_1^L)^-1P_2+ Q_2+_F_1+_F_2,   α_02^=β_2^1+_F_1+_F_2

	
    α_12^= B_12^1+_F_1+_F_2,   α_21^=B_21^1+_F_1+_F_2.

	
	We construct the quiver stack ^ as follows:
	
		
  * The underlying topological space of ^ is the polyhedral set  P of K_^1. We endow P with a topology {∅,{U_i}_0≤ i ≤ 2}  where U_i is the complements of the i-th facet. 
		
  * We associate each open set U_i a sheaf which will be obtained by localizing the corresponding global sections _0 for i=0 or _i for 1≤ i≤ n. If there is no ambiguity we will use _0 or _i to denote the sheaves. Then _0(U_1) (resp. _0(U_2)) is defined by localizing _0 at {u_2^L} (resp. { v_1^L }); on the intersection, it's the algebra localized at the union of corresponding elements.  
		
  * The transition maps  G_i0:_0|_U_0i→_i|_U_0i and G_0i for i=1,2 are defined by 
    G_10:=  
    			u_1^L→  u_1 
    
    			v_1^L→  v_1 
     
    			u_2^L→ 1  
    
    			v_2^L→ v_1u_1+i_11j_11
    
    			i_k → i_1k    k=1,2  
    
    			j_k →  j_1k    k=1,2
    		,   G_01:=
    			u_1→ u_2^L u_1^L   
    
    			v_1→ v_1^L (u_2^L)^-1
    
    			i_11↦ i_1 
    
    			j_11↦ j_1 
    
    			i_12↦ u_2^L i_2 
    
    			j_12↦ j_2 (u_2^L)^-1.

		
		
    G_20:=  
    			u_2^L→  u_2 
    
    			v_2^L→  v_2 
     
    			u_1^L→  u_2v_2-i_21j_21
    
    			v_1^L→ 1  
    
    			i_k → i_2k    k=1,2  
    
    			j_k →  j_2k    k=1,2
    		,   G_02:=
    			u_2→ u_2^L (v_1^L)^-1
    
    			v_2→ v_1^L v_2^L 
    
    			i_21↦ i_1 
    
    			j_21↦ j_1 
    
    			i_22↦ v_1^L i_2 
    
    			j_22↦ j_2 (v_1^L)^-1.

		
  * The gerbe terms are defined by  
    c_010(v_1)= e_1      c_010(v_2)=u_2^L    c_020(v_1)= e_1      c_020(v_2)=v_1^L.

		
		
  * We define the transition maps between the first chart and second chart to be G_ik(x):=G_i0∘ G_0k(x), and the corresponding gerbe terms are trivial. More precisely, 
    G_21=G_20∘ G_01=
    				u_1→ u_2(u_2v_2-i_21j_21)    
    
    				v_1→ u_2^-1
    
    				i_11↦ i_21
    
    				j_11↦ j_21
    
    				i_12↦ u_2 i_22
    
    				j_12↦ j_22 u_2^-1,    G_12=G_10∘ G_02=
    				u_2→ v_1^-1   
    
    				v_2→ v_1(v_1u_1+i_11j_11) 
    
    				i_21↦ i_11
    
    				j_21↦ j_11
    
    				i_22↦ v_1 i_12
    
    				j_22↦ j_12 v_1^-1.

	
	
	
	
		
		
		
	
	
	
	The above data indeed defines a quiver algebroid stack ^. We will take G_01∘ G_10(u_1^L) as an example. The rest is similar: 
	
    G_01∘ G_10(u_1^L)= G_01(u_1)= u_2^Lu_1^L= c_010(v_2) · u_1^L · c_010(v_1)^-1.

	
	The framed quiver algebroid stack is constructed by analyzing the coefficients in the Fukaya isomorphism equations.  For example, by direct calculation, we obtain 
    m_1^^(P_1+ Q_1+_F_1+_F_2)= (j_1-j_11)J_11+ (u_1^L-u_1)X_1+ (v_1^L-v_1)Y_1+ (u_2^L-1)B_1^1+ (j_2-j_12)J_12+(i_1-i_11)I_11+(i_2-i_12)I_12.
 Therefore, we construct a framed quiver algebroid stack over which the coefficients vanish. This provides the transition maps G_10 and G_01 in ^. And m_1^^ (α_10^) vanishes over ^. Similarly, one can check that α_ij satisfies the Fukaya isomorphism equations over ^. Furthermore, we also have m_2^_i,_j,_k(α_ij,α_jk)=α_ik for i,j,k=0,1,2.
	
	By Proposition <ref>, the framed isomorphism restricts to an isomorphism pair between compact Lagrangian immersions. The resulted unframed quiver algebroid stack  is the quiver algebroid stack corresponding to minimal resolution of A_1 singularity. Readers can find more details about the quiver stack corresponds to A_n resolution in Theorem <ref>.
	









 §.§ Localization and charts in general ranks


Given a Lagrangian brane (,). The noncommutative deformation space is in general stacky. In order to analyze the local structure of the nc deformation space, we generalize the localized path algebra. We define the localization of a path algebra at a matrix of arrows. We denote a path algebra with relations by =k Q/R. In this case, even though a single path may not be invertible, a matrix of arrows can possess both left and right inverse. 

 
	Let S=(s_ij)_1 ≤ i ≤ m, 1 ≤ j ≤ n be a matrix of paths satisfying the following condition:
	
		
  * The paths in the same row (column) have the same heads (tails). 
		
		
  * Let v be a column of paths such that the head of i-th entry is the  tail of i-th column of S, then Sv= 0 implies v=0. The similar condition holds for rows of paths w and wS=0. 
	
	Then we define the localized algebra at S in the following way:
	
	For each entry s_ij, we add one path, denoted by γ_ji, with h(γ_ji)=t(s_ij) and t(γ_ji)=h(s_ij), to the quiver Q. This forms a matrix of paths γ. Moreover, we take the ideal R̂ generated by R and (γ S - diag(e_t(s_11), ⋯, e_t(s_1n)), S γ - diag(e_h(s_11), ⋯, e_h(s_m1))) to be the new ideal of relations.  The new quiver algebra with relations k Q̂ / R̂ is called the localized algebra at S, and is denoted as S^-1.

 

	Definition <ref> is a special case of Definition <ref> by taking S to be a diagonal matrix of arrows.





Similarly to definition <ref>, we can define an affine chart of rank w⃗ for a path algebra.

 
	Let  be a quiver algebra with a reference vertex v_1 and w⃗ be a dimension vector which equals 1 at v_1.  An affine chart of rank w⃗ centered at v_1 is a triple 
    (_i=k Q_i/R_i,G_0i,G_i0)

	where Q_i is a quiver with a single vertex and R_i is a two-sided ideal of relations; 
    G_0i: _i → S^-1 and  G_i0: S^-1→ Mat(_i)
 
	are representations with v_1 being the image vertex of G_0i that satisfy 
	
    G_i0∘ G_0i =   Id,
    
    		G_0i∘ G_i0(a) =    c(h_a)  a  c(t_a)^-1.

	The map c assigns each vertex v an invertible matrix of paths (which we call Gerbe terms) in S^-1  with the properties:
	
		
  * each entry is a path from v to v_1,
		
  * the number of rows equals the rank of v.
	
	Here, S^-1 is the localization of  at some matrix of paths S as described in Definition <ref>.



Let's give an example of affine chart of rank w⃗ for a quiver algebra .

 

Let Q be the quiver of the  doubling of affine D_4 as shown in the Figure <ref> (a). We consider the path algebra _0 of Q restricted to the complex moment map level zero, in other words, the elements in _0 satisfy ∑_i=1^4 a_i b^i=0 and b^ia_i=0 for i=1,⋯, 4. In the following, we will write down an affine local chart for _0 explicitly, whose dimension vector w⃗ is (2, 1,1,1,1) where the order of components agree with subscripts of vertices.



	
		[row sep=huge,column sep=large]
			    v_1 [d, "a_1"', bend right]                        

	v_4 [r,  bend right]      v_0 [u, "b_1"', bend right] [r, bend right] [d, bend right] [l, bend right]     v_2 [l, bend right] 
    v_3 [u, bend right]                                         
	    
	[row sep=huge]
		∙["X"', loop, distance=4em, in=30, out=330] ["Y"', loop, distance=4em, in=150, out=90] ["Z"', loop, distance=4em, in=270, out=210]
		 

	 a)     b)

	 
	figureDouble quiver of affine D_4 and the underlying quiver of _2
	
	

	


	We localize _0 at the matrix of arrows [ a_1 a_2 ], that is, we insert the reverse arrows α^1 and α^2 so that 
    [ a_1 a_2 ]·[ α^1; α^2 ] =e_0,

	
    [ α^1; α^2 ]·[ a_1 a_2 ]= [ e_1   0;   0 e_2;     ].

	We then localize it at diag(b^2a_1,b^3a_1,b^4a_1), the above construction is equivalent to localize _0 at a block matrix S and we will denote it by S^-1_0. 
	
	Let _2 be the quiver algebra with relations ⟨ X,Y,Z ⟩/([X,Y], [X,Z],[Y,Z], Z(X+1)-XY) whose  underlying quiver is depicted in  Figure <ref> (b). Thus, _2 ≅[X,Y,Z]/(XY-Z(X+1)). We define the following representations: 
	
    G_02: _2 → S^-1_0,     G_02:=
    	
    		X ↦ (b^2a_1)^-1(α^2a_3)(b^3a_1) 
    
    		Y ↦ (b^3a_1)^-1(b^3a_2)(b^2a_1) 
    
    		Z ↦ (b^4a_1)^-1(b^4a_2)(b^2a_1) 
    	.

    This is a well-defined representation. In other words, it represents the relation by zero. Let's show G_02(XY)=G_02(YX):
    
    G_02(X)G_02(Y)   = (b^2a_1)^-1(α^2a_3)(b^3a_2)(b^2a_1)
       =- (b^2a_1)^-1α^2(a_1b^1+ a_2b^2+ a_4b^4)a_2(b^2a_1)
       =- (b^2a_1)^-1(α^2a_4b^4a_2b^2a_1) 
       =  (b^2a_1)^-1(α^2a_4b^4(a_1b^1+ a_3b^3+ a_4b^4)a_1)= (b^2a_1)^-1(α^2a_4b^4a_3b^3a_1)
       =-(b^2a_1)^-1(α^2 a_2b^2a_3b^3a_1)
       = -(b^2a_1)^-1(b^2a_3)(b^3a_1).

    In the second line, we use ∑_i=1^4 a_ib^i=0. In the third line, we use b^2a_2=0 and the Equation (<ref>): α^2 a_1=0. Similarly, we get the fourth and fifth line. Finally, we apply  α^2 a_2 = e_2 to obtain the last line.
    
    G_02(Y)G_02(X)   = (b^3a_1)^-1(b^3a_2)(α^2a_3)(b^3a_1) 
       = (b^3a_1)^-1b^3(e_5-a_1 α^1)a_3(b^3a_1) 
       = - (b^3a_1)^-1(b^3a_1α^1 a_3)(b^3a_1)= -(α^1 a_3)(b^3a_1).

    In the second line, we use the Equation (<ref>): a_1 α^1+ a_2 α^2 = e_5. The last line comes from b^3 a_3=0. Notice that using the same relation 
    -(α^1 a_3)(b^3a_1)=- (b^2a_1)^-1(b^2a_1)(α^1 a_3)(b^3a_1) 
    
        	=    -(b^2a_1)^-1b^2(e_5- a_2 α^2)a_3b^3a_1 = -(b^2a_1)^-1b^2a_3b^3a_1.
 This shows G_02(X)G_02(Y)= G_02(Y)G_02(X). One can check G_02 respects other relations by similar calculations. Hence, G_02 is a well-defined representation.

    Conversely, we define G_20: S^-1_0→ Mat(_2) by first assigning G_20(a_1):=[ 1; 0 ] and G_20(a_2):= [ 0; 1 ]. Using the complex moment map equation, we can solve the other terms:  
    G_20: S^-1_0→ Mat(_2),    G_20:=
        
        	a_1 ↦[ 1; 0 ]
    
        	a_2 ↦[ 0; 1 ]
    
        	a_3 ↦[ -YX;   X ]
    
        	a_4 ↦[  -YX; -X-1 ]
    
        	b^1 ↦[       0 YXY-ZXY ]
    
        	b^2 ↦[ 1 0 ]
    
        	b^3 ↦[ 1 Y ]
    
        	b^4 ↦[ 1 Z ].

    
    Additionally, we have the following unavoidable gerbe terms: c(v_1)=e_1, c(v_2)=(b^2a_1)^-1, c(v_3)= (b^3a_1)^-1, c(v_4)= (b^4a_1)^-1, c(v_0)=[            α^1; (b^2a_1)^-1α^2 ]. 
    
    One can check that the representations G_20 and G_02 are mutually inverse up to gerbe terms. We will take a_3 as an example. The other terms can be verified similarly. 
    
    G_02∘ G_20(a_3)=G_02([ -YX;   X ])= [ -(b^3a_1)^-1(b^3a_2α_2 a_3b^3a_1);         (b^2a_1)^-1(α_2a_3b^3a_1) ]= [              α_1a_3b^3a_1; (b^2a_1)^-1(α_2a_3b^3a_1) ],
 where we use α_2a_2=e_5- α_1 a_1 and b^3a_3=0.
    Meanwhile, 
    c(v_0) · a_3 · (c(v_3))^-1= [             α_1; (b^2 a_1)^-1α_2 ]· a_3 · b^3a_1.

    
    
    
    
    	 
    
    




Therefore, _2 is a commutative affine chart of rank w⃗. In fact, _2 is the affine 2-space ^2_, since _2= [X,Y,Z]/(XY-(X+1)Z) ≅[X,T], where T:=Y-Z. 
	
	In the Proposition <ref>, we will explain how to construct the affine chart in rank w⃗ of _0 using stability condition. 
	







 §.§ ζ_-stable Lagrangian
 
In this subsection, we first recall the stability condition of quiver representations, and then define ζ_-stable Lagrangians. This definition is motivated by the relations between (formal) deformation spaces of a Lagrangian immersion and quiver representations.
 
Let's fix notations for quiver representations. Recall that we denote a quiver by Q=(I,E) where I (resp. E) denotes the set of vertices (resp. arrows). Let V=∑_i∈ I V_i be an I-graded vector space, we define its dimension vector by V:= ( V_i)_i ∈ I∈^I_≥ 0. 
Fix a dimension vector V and a weight ζ_  with respect to V, quiver representations possess explicit stability conditions (Definition <ref>), see for instance <cit.> and <cit.>.

 We start this by recalling the stability condition of quiver representations.

 

 If V^1, V^2 are I-graded vector spaces, we introduce vector spaces 
    L(V^1, V^2):=⊕_i ∈ I(V^1_i, V^2_i)
 which is the space of linear maps between the vector spaces over the same vertices. We also introduce 
    E(V^1, V^2):=⊕_a ∈ E(V^1_t(a),V^2_h(a)).


Let V, W be I-graded vector spaces. We define 
    M(V,W):=E(V,V) ⊕ L(V,W) ⊕ L(W,V),
 
which is the representation space of the framed quiver Q^fr.

We now define the stability conditions. For ζ_= (ζ_,i)_i ∈ I∈ R^I, let ζ_· V:= ∑_i ζ_,i V_i.

	
	A point (B,a,b) ∈ M is ζ_-semistable if the following two conditions are satisfied:
	
		
  * If an I-graded subspace S of V is contained in Ker  b and B-invariant, then ζ_· S ≤ 0.
		
  * If an I-graded subspace T of V contains in Im  a  is B-invariant, then ζ_· T ≤ζ_· V.
	
	We say (B,a,b) is ζ_-stable if the strict inequalities hold in 1,2 unless S=0, T=V respectively.


When W=0 i.e. B∈ E(V,V), the stability conditions are slightly different.


	Suppose that ζ_· V=0. A point B ∈ E(V,V) is ζ_-semistable if it satisfies the following condition: if an I-graded subspace S of V  is B-invariant, then ζ_· S ≤ 0. 
	
	A point B is ζ_-stable if the strict equality holds unless S=0 or S=V.


This definition coincides with the case W≠ 0 for ζ_-semistability, but not for ζ_-stability.


Let (L^,_̃1̃) and (L^,_̃2̃) be two families of Lagrangian branes over a -algebra . Denote the compact components of L^ by _j and framing components by F_j, for j=1, ⋯, n. We also denote 
    L:=∪_j    F:=∪ F_j   _i:= _̃ĩ|_L    w_i:= _̃ĩ|_F,
 for i=1,2. The Floer complex CF^∙((L^,_̃1̃),(L^,_̃2̃)) has the following interesting components. In particular, these provide a nice decomposition of the Floer complex of the framed Lagrangian branes obtained by  the plumbing.

First, we introduce the following notion to represent certain degree i-th Floer complex of each corresponding compact component:


    L^i((L,_1),(L,_2)):= ⊕_j CF^i((_j,_1), (_j,_2))=⊕_j ⊕_p ∈𝒳^i(_j,_j)⊗_(_1|_p, _2|_p),
where 𝒳^i(_j,_j) denotes the degree i generators of the Floer complex. Since we use the Morse model for clean intersections, the generators are critical points of the Morse functions.







Besides, the framing Lagrangian F intersects transversally with L. These transversally intersection points are also of interest. Similarly, we denote 

    L^i((L,_1),(F,w_2))= ⊕_j ∈ I CF^i((_j,_1), (F_j,w_2))


    L^i((F,w_1),(L,_2))= ⊕_j ∈ I CF^i( (F_j,w_1),(_j,_2)).




Degree 1 intersection points are special, which are used for localized mirror constructions. We introduce

    E((L^,_1),(L^,_2)):= ⊕_p ∈ E⊗_(_1|_p, _2|_p),
 where E is the set of arrows in the associated quiver Q.



In particular, E((L^,),(L^,)) is the representation space of the associated quiver Q, which is of great interest since it encodes the Maurer-Cartan space of (L^,). In the plumbing construction, we show that CF^1((L^,_1),(L^,_2))=E((L^,),(L^,))= E((L,_1),(L,_2)) ⊕ L^1((L,_1),(F,w_2)) ⊕ L^1((F,w_1),(L,_2)) in Section <ref>.


Now we can define stability conditions. Given an unframed Lagrangian immersion  with a rank v trivial vector bundle . Denote its associated quiver by Q. For ζ_=(ζ_,i)_i ∈ I∈^|I|, let ζ_· rank  := ∑_i ∈ Iζ_,i v_i.


	Given a Lagrangian brane (,) of rank v. Suppose that ζ_·v=0. A deformation B ∈ MC((,)) ⊂ E((,),(,)) is ζ_-semistable if the following is satisfied:
	
		
  * If an I-graded trivial subbundle S of  is B-invariant, then ζ_· rank   S ≤ 0. 
	
	A deformation B is ζ_-stable if the strict inequality holds unless S=0 or S=.


The stability condition for the framed Lagrangian immersions is slightly different. Let (L^,) be a framed Lagrangian brane of rank (v⃗,w⃗).


	Given ζ_= (ζ_,i)_i ∈ I∈^I, where I is the index of compact components. A deformation (B,a,b) ∈ MC((L^,)) ⊂ E((L^,),(L^,)) is ζ_-semistable if the following two conditions are satisfied:
	
		
  * If an I-graded trivial subbundle S of  is contained in Ker(b) and B-invariant, then ζ_· rank   S ≤ 0.
		
  * If an I-graded trivial subbundle T of   contains Im(a) and is B-invariant, then ζ_· rank   T ≤ζ_· rank  .
	
	We say (B,a,b) is ζ_-stable deformation if the strict inequalities hold in (1), (2) unless S=0, T=V respectively.



	In both cases, a Lagrangian  with a ζ_-stable deformation is called ζ_-stable Lagrangian.



	
		
  * In most of the cases we are interested in, we take  to be the complex number , Novikov ring Λ_0 or Novikov field Λ.
		
  * We will omit the trivial vector bundles if the trivial bundles are clear in the context.
	


When performing localized mirror construction, the following stability condition is of great interest, which can be achieved by taking certain character ζ_.

By definition of noncommutative family, if the quiver of   has a single vertex, an -family over   rank w⃗ can be identified with an (, )-bimodule ^⊕ (∑ w_i), where the module structure is given by the representation G: → Mat(). If there is a vertex whose rank is one (WLOG, we assume it's v_1), then e_1(^⊕ (∑ w_i))≅. Under the above assumptions, we now introduce a special class of representations called stable families, which can be seen as a noncommutative analogue of the stable locus. 

 
	Let 𝔸 be a family over  of rank w where the vertex v_1 of  is of rank one and the quiver of  has a single vertex. We say a family  over  of rank w⃗ is stable if there's no proper -submodule M of ^⊕ (∑ w_i) such that e_1M =. 


[Example of a stable family]
	Let's show the family S^-1_0 over _2 of rank w⃗ constructed in example <ref> is indeed stable. 
	
	Let M be a S^-1_0-submodule of _2^⊕ 6, which satisfies e_1 M = _2. Because b^2a_1 is invertible in S^-1_0, e_2 M ≅_2. Similarly, vertices v_3 and v_4 also admit an invertible path from the vertex v_1, which implies e_3 M ≅ e_4 M ≅_2. Furthermore, since M is a S^-1_0-module, it's closed under a_1 and a_2 action. Thus, e_0M ≅_2^⊕ 2. Hence, M ≅_2^⊕ 6, which is stable.
	



	Let (,) be a Lagrangian brane of rank v⃗ where   |__v_0 has rank 1. A nc family of Lagrangian brane (,) over  is called stable if the deformation space M is a stable family over . 


Given a dimension vector v⃗, we can define a canonical nc family of -representations of rank v⃗ as follows.  Let k be a unital algebra. First, we define ' to be the free k-algebra generated by the variables corresponding to the matrix entries of _k(k^ v_t(a), k^ v_h(a)) for all arrows a.  The number of variables is ∑_a ∈ E v_t(a)v_h(a). ' can be understood as a quiver algebra with a single vertex.  If we take quotient of ' by commuting relations, we would obtain the coordinate ring of the representation space Rep(Q,v⃗). 

For a quiver algebra := kQ/I= k Q/ ⟨ r_1, ⋯, r_s ⟩, 

we define the associative k-algebra :='/R, where R is the ideal generated by the induced relations on the matrix entries according to r_1,…,r_s.

 will be called the noncommutative parameter space corresponding to the subvariety M(Q,v⃗) ⊂Rep(Q,v⃗) of 
-representations. 



Given a stable family of -representations over  in rank v⃗, we need to suitably localize  according to the stability condition (Definition <ref>) to get stable subfamilies.  We will take the collection of all localizations _loc of  (in the sense of Definition <ref>) such that the sub-families of -representations over _loc are stable.  A sub-collection of them will form stable affine charts by setting the invertible matrices to be identity.

We will carry out this procedure for the affine D_4 quiver in Section <ref>.














 §.§ Framed Lagrangian branes
 

For our purpose, it's necessary to enlarge the Fukaya category.
The objects of the enlarge Fukaya category of M consists of two types of Lagrangian submanifolds: a topologically closed, connected, relatively spin (possibly immersed) Lagrangian and a fixed collection of framing Lagrangians. The Lagrangians L are equipped with local systems, i.e. trivial vector bundles  with unitary holonomy over its normalization. We will write the object as a pair (L,ℰ). Note by our convexity assumption, even though we include noncompact Lagrangians, the m_k operation is still well-defined.  

The morphism space between transversely intersecting Lagrangians (L_1,_1), (L_2,_2) is defined as 

    ^∙((L_1,_1), (L_2,_2)):=⊕_p ∈ L_1 ∩ L_2Λ_0 ⊗_ℂ(_1|p,_2|p).


Let L be a topologically closed, connected, relatively spin immersed Lagrangian in M. We denote by ι: L̃↪ M an immersion with image L and by ℐ⊂ L the self-intersection. As in Akaho-Joyce <cit.>, the preimage of ℐ consists of two disjoint branches ℐ^- ⨿ℐ^+ ⊂L̃, each of which is diffeomorphic to I. Then the fiber product 
    L̃×_L L̃= ⨿_i=-1,0,1 R_i
 contains the diagonal R_0 and two immersed sectors 
    R_-1:={(p_-,p_+)∈L̃×L̃|    p_- ∈ℐ^-, p_+ ∈ℐ^+, ι(p_-)=ι(p_+) }


    R_1:={(p_+,p_-)∈L̃×L̃|    p_- ∈ℐ^-, p_+ ∈ℐ^+, ι(p_-)=ι(p_+) }


For an immersed Lagrangian (L,), the Floer complex is defined as 
    CF^∙((L,);Λ_0):=C^∙((L,);Λ_0) ⊕_p ∈ℐΛ_0 ⊗ Hom_(|_p_-, |_p_+) ⊕Λ_0 ⊗ Hom_(|_p_+, |_p_-),
 where p denotes doubly self-intersection points. C^∙((L,);Λ_0) can be defined using various models, see <cit.>. ?? <cit.> also analyzes the connection between Morse model and singular chain model. An A_∞-structure can be produced on the Floer complex CF^∙((L,);Λ_0) as in the work of Akaho-Joyce <cit.>, generalizing the work of Fukaya-Oh-Ohta-Ono <cit.> in the embedded case. In this paper, we shall use Morse model, which means we fix a Morse function f: L̃→. The cochain complex C^∙((L,);Λ_0):= C^∙((f,);Λ_0) is generated by the critical points of f. Namely, 
    C^∙((f,);Λ_0):= ⊕_p ∈ Crit(f)Λ_0 ⊗ Hom_(|_p, |_p).
  

For our purpose, we also consider an immersed Lagrangian with clean self-intersection. In this case, each connected component of the clean intersections is equipped with a Morse function f_i. As before, let ℐ⊂ L be the clean self-intersection. The fiber product L̃×_L L̃ also contains three components, which are diagonal and two branch jumps. The Floer complex of (L,) is defined as 
    CF^∙((L,);Λ_0):= ⊕_i ⊕_p ∈ Crit(f_i)Λ_0 ⊗ Hom_(|_p_-, |_p_+) ⊕Λ_0 ⊗ Hom_(|_p_+, |_p_-) ,
 
where f_i is the Morse function on the i-th component of the clean intersection. The Floer theory is defined by counting holomorphic pearl trajectories <cit.>. More details about this case can be found in <cit.>.

Using homotopy method <cit.> <cit.>, the algebra can be made to be unital as in <cit.>. The unit is denoted by 1_𝕃.  It is homotopic to the formal sum of the maximum points of the Morse functions on all components (representing the fundamental class), denoted by 1^▾_𝕃.  Namely, 1_𝕃-1_𝕃^▾=m_1(1_𝕃^h)
(assuming 𝕃 bounds no non-constant disc of Maslov index zero).  




For a pair ((L_1,_1),(L_2,_2)) of closed, connected, relatively spin, and embedded Lagrangian submanifolds intersecting cleanly, we define the Lagrangian Floer complex by considering the union L:=L_1 ∪ L_2. The union L_1 ∪ L_2 is an immersed Lagrangian with clean self-intersections. Denote the normalization by L̃:= L_1 ∐ L_2, which is equipped with a Morse function f. One has an A_∞-bimodule CF^∙((L_1,_1),(L_2,_2);Λ_0) for the Lagrangian Floer theory as before: 
    CF^∙((L_1,_1),(L_2,_2);Λ_0):= ⊕_p ∈ Crit(f|_R_1)Λ_0 ⊗ Hom_(_1|p, _2|p) ⟨ p ⟩.
 The space of Floer cochains CF^∙((L_1,_1),(L_2,_2);Λ_0) for two cleanly-intersect immersed Lagrangians is similar and we shall not repeat.







§ ADHM QUIVER AS A MIRROR TO THE FRAMED IMMERSED TWO-SPHERE
 

In this section, we aim to interpret the ADHM construction as a mirror symmetry phenomenon. Specifically, we will accomplish this by constructing the ADHM quiver representations and corresponding framed sheaves using the framed mirror construction, which is applied to the framed immersed sphere in ℂ^2 ∖{ab=ϵ}.







As described in Example <ref>, the zero section gives rise to an immersed sphere 𝕃 with two immersed generators X and Y;  we can take a cotangent fiber of  a point away from the north and south poles to be the framing F. 

An alternative description is to consider the S^1-reduced space of ℂ^2 ∖{ab=ϵ} which is a conic fibration over ℂ; 𝕃 is the fiber of  (μ,log |ab-ϵ|): M →^2 at (0, log |ϵ|), and the framing F is given as a section of this fibration. Throughout this section we will primarily use this perspective, only resorting to the plumbing construction for topological considerations.  

We fix a reference framed Lagrangian immersion ∪ (F,-f_F) and denote it by 𝕃^fr. It's evident that the framing F  intersects 𝕃 only once.  Any degree-one generators in the reference framed Lagrangian along with their corresponding arrows in the quiver will be labeled with a subscript 0. See Figure <ref>. 




 The associated quiver Q^ADHM(resp. Q) of 𝕃^fr (resp. 𝕃) is shown in Figure <ref>.



	The construction of the framed Lagrangian immersion ^ can be intuitively justified as follows.  First, by Kontsevich's Homological mirror symmetry conjecture <cit.>, coherent sheaves are mirror to Lagrangian submanifolds.   In this case, let's consider rank-one framed torsion-free sheaves over ^2, which correspond to 
ideal sheaves of points on ^2.






On the other hand, by the conjecture of Strominger-Yau-Zaslow  <cit.>, the mirror space admits a dual torus fibration. A Lagrangian torus should correspond to a skyscraper sheaf, and a Lagrangian section should correspond to a line bundle over the mirror. 
Combining the pictures, the connected sum of a Lagrangian torus and a Lagrangian section should be mirror to an ideal sheaf of points up to degree. 
Hence, stable deformations of ^ ought to correspond to framed torsion-free sheaves.

More generally, we observe that the quiver associated with ^ coincides precisely with the ADHM quiver.



 §.§ Completion of path algebras

Because of the existence of constant polygons, in general we have series like 1+a_1xy +a_2(xy)^2+⋯, in order to make sense of this, we shall consider the completion of the path algebra.  Recall the valuation of the Novikov ring is defined by: 
    val:∑_i=1^∞a_iT^A_i↦ A_1 and val(0)=+∞.
 The valuation induces a non-Archimedean norm on Λ_0 given by 
    |c|_𝔳:= e^-val(c).
  

Let Λ_0 Q  be the free path algebra generated by arrows x_a.  Let's fix a function 𝔳 from the arrow set to [0,1], x_a↦ |x_a|_𝔳∈ [0,1].  The Gauss norm on Λ_0 Q is defined as 
    ‖∑_Ia_Iζ^I‖_𝔳=max_I{|a_Iζ^I|_𝔳}
 where ζ^I is an ordered monomial of x_a and |ζ^I| is the corresponding product of |x_a|_𝔳. 
One can associate a corresponding valuation on Λ_0 Q by setting val(z)=-log‖ z ‖. 

The valuation induces a filtration 
    F^pΛ_0 Q={z ∈Λ_0 Q: |z|_𝔳≤ e^-p}.

In the following, we will always take the completion of Λ_0 Q with respect to this filtration and denote it by Λ_0 Q_𝔳. The above construction makes Λ_0 Q_𝔳 a complete topological space with respect to the Gauss norm. 

When we take the tensor product of two complete path algebras, it's always assumed to be the complete tensor product. 



 §.§ Unobstructedness and deformation space

We will utilize linear combinations of immersed generators X,Y to deform the immersed sphere  (respectively, X_0, Y_0, I_0, J_0 for 𝕃^fr).

Let Λ_0Q_(a,b) be the completed path algebra of Λ_0 Q, where |x|_𝔳 =a, |y|_𝔳=b for a,b ∈ [0,1]. We take 
    _free := ⋂_0≤ a,b ≤ 1, 0≤ ab<1Λ_0Q_(a,b)

which contains the series that converge with respect to all the norms satisfying |x|_𝔳,|y|_𝔳≤ 1 and |xy|_𝔳<1.

We rephrase Lemma 3.3 of <cit.> as the following proposition. 
We find a coordinate transformation that renders m_0^ into the commutator. 


	Let = xX+ yY.  The obstruction term of  equals
	
    m_0^=((1+ ∑_i=1^∞  a_i(xy)^i)xy-(1+ ∑_1^∞  a_i(yx)^i) y x)P

	for some a_i ∈, where P is the minimum point of the Morse function. There exists a change of coordinates on (x,y) such that the unobstructed deformation space of  can be written as
	
    := _free/(xy-yx)≅Λ_0 [x,y][[xy]].
 
 

	
	The above expression of  m_0^ follows from Lemma 3.3 of <cit.>.  The coefficients of xy and yx are equal due to the anti-symmetric involution on .
	The coefficients a_i depend on the choice of Kuranishi structures on the moduli spaces of constant polygons.  This can be rewritten as 
    m_0^=((1+ ∑_i=1^∞  a_i(xy)^i)x y-y(1+ ∑_1^∞  a_i(xy)^i) x)P.
 We take (1+ ∑_i=1^∞ a_i(xy)^i)x as a new coordinate, which is still denoted by x by abuse of notation.  y is kept to be the same.  Since (1+ ∑_i=1^∞ a_i(xy)^i) is invertible and has norm 1 in _free, this is a norm-preserving isomorphism on _free.
	Then the unobstructed relation becomes (xy-yx), and hence the unobstructed deformation space is := _free/(xy-yx).  This is isomorphic to Λ_0 [x,y][[xy]] since |x|_𝔳,|y|_𝔳≤ 1 and |xy|_𝔳<1.
	
	

	
	
	
	
	
	
	




The immersed 2-sphere (, b=xX+yY, ) is unobstructed and its deformation space is commutative. 




	After changing coordinates, the noncommutative deformation space of the immersed Lagrangian L is 
	
    ≅Λ_0 [x,y][[xy]],

   for |x|_𝔳,|y|_𝔳≤ 1 and |xy|_𝔳 < 1 where a norm is taken. 


	This is a subring of Λ[[x,y]] consisting of elements of the form ∑_I a_Iζ^I such that |a_Iζ^I|_𝔳→ 0 as |I|→∞.








Now we consider the framed immersed sphere ^.  Below, we show that the unobstructed relation coincides with the ADHM relation up to a coordinate change.

We will take a similar completion _free^ of Λ_0 Q^ADHM by taking the intersection over all the valuations 𝔳 that have
|x_0|_𝔳,|y_0|_𝔳, |i_0|_𝔳, |j_0|_𝔳≤ 1 and |x_0y_0|_𝔳, |i_0j_0|_𝔳< 1.








	 
	 Let ^:= x_0X_0+y_0Y_0+i_0I_0+j_0J_0.  There exists a coordinate change on _free^ such that 
	 the formal deformation space of the framed Lagrangian immersion 𝕃^fr equals
	
    ^:=_free^/(x_0y_0-y_0x_0+i_0j_0).



	
	
	The unobstructed relations are coefficients of m_0^^ which have degree two. In this case,  the only degree two generator is the minimal point P of the immersed sphere component. Thus, we only have a single relation, which is contributed from pearl trajectories to P.  Such a trajectory is a union of a Morse flow line to P and either constant polygons (x_0y_0)^k+1, (y_0x_0)^k+1 or (i_0j_0)^k+1.  Thus the coefficient of  m_0^^ equals 
	
    (1+ ∑_k=1^∞  a_k(x_0y_0)^k)x_0 y_0-(1+ ∑_k=1^∞  a_k(y_0x_0)^k) y_0 x_0+ (1+∑_k=1^∞  b_k(i_0j_0)^k )i_0 j_0,
 
	for some constants a_k and b_k depending on the perturbation of the Kuranishi structure. This is the same as
    (1+ ∑_k=1^∞  a_k(x_0y_0)^k)x_0 y_0 - y_0(1+ ∑_k=1^∞  a_k(x_0y_0)^k)x_0 + (1+∑_k=1^∞  b_k(i_0j_0)^k )i_0 j_0.
     
We take (1+ ∑_i=1^∞ a_k(x_0y_0)^i)x_0 (resp. (1+∑_k=1^∞  b_k(i_0j_0)^k )i_0) as a new coordinate, which is still denoted by x_0 (resp. i_0) by abuse of notation.  y_0 and j_0 are kept to be the same.
	
	

Then the unobstructed equation becomes the ADHM equation: 
    x_0y_0- y_0x_0+ i_0j_0=0.
  
Hence, the noncommutative deformation space of 𝕃^ is ^:=_free^/(x_0y_0-y_0x_0+i_0j_0).



	The framed immersed 2-sphere (^, ^=x_0X_0+y_0Y_0+i_0I_0+j_0J_0, ^) is unobstructed. 



 
	
	Let _1 be a trivial vector bundle of rank k over the immersed sphere, and _2 a trivial vector bundle of rank r over the framing. We restrict to the subcategory generated by ^.  The Maurer-Cartan space of the framed Lagrangian brane (^,) is 
	
    ℳ(V,W):={(B_1,B_2, i, j)∈ Rep(Q^ADHM,V,W) ⊗Λ_0 |  [B_1,B_2]+ij=0}/GL(k),

	where Q^ADHM is the ADHM quiver, rank   V= k and rank   W=r.


	
	By definition, an element :=B_1X_0+B_2Y_0+iI_0+jJ_0 ∈ CF^1((^,),(^,)) is Maurer-Cartan if 
    m_0^=∑ m_k(, ⋯,)=0,
 which counts the same pearl trajectories considered in Theorem <ref>. Hence, up to coordinate change, the Maurer-Cartan space of ^ is the space of isomorphic representations of Q^ADHM, satisfying  [B_1,B_2]+ij=0. Restricting to the subcategory generated by ^,  can take Λ_0-valued deformation, since there  is no convergence issue. Thus, the result holds. 






 §.§ Framed torsion-free sheaves
  



As is known, ℂ^2 ∖{ab=ϵ} admits an explicit special Lagrangian torus fibration, see for instance Proposition 5.2 in <cit.>. More precisely, let γ(r) be the circle around ϵ with radius r. The Lagrangian tori T_γ(r),λ are special Lagrangian with respect to the holomorphic volume form Ω=da∧ db/ab-ϵ. Therefore, the ambient space ^2 ∖{ab=ϵ} admits a special Lagrangian fibration ^2 ∖{ab=ϵ}→^2 defined by ( 1/2(|a|^2-|b|^2),log |ab-ϵ|).
 




Notice that this fibration has a singular fiber over (ϵ,0), which is an immersed sphere T_γ(ϵ),0. Let's denote it by . In the following, we will perform the localized mirror construction using . 







By Lemma <ref>, the deformation space of  forms a formal neighborhood of the zero locus of xy in the affine space ^2, which is the localized mirror. Performing the localized mirror construction as in Section <ref>, we obtain an A_∞-functor ^(,): (X) →dg . We will restrict ^(,) to the Fukaya subcategory generated by the framed Lagrangian immersion ^. As a consequence, the mirror sheaves of the Lagrangian submanifolds can be extended to the affine space ^2_, as [x,y] ⊂ and there is no convergence issue after changing variables.  



In the following, we use the functor ^(,) to transform the framed Lagrangian brane itself into a monad.  We denote the family of objects to be transformed by (L^,) to distinguish from the reference Lagrangian . Here, L is equipped with a trivial rank n vector bundle _1 and the framing F is equipped with a Morse function f_F and a trivial rank r vector bundle _2.

As described in Section <ref>,  the Floer group is defined by counting pearl trajectories. An illustration of perturbed strip is depicted in Figure <ref>, which we will primarily consider in the rest of this section for a better demonstration of strip counting. 





 
	^(,) transforms the framed Lagrangian brane (L^,) into the monad of the torsion-free sheaves over ℂ^2.  


	
	The labeling is as shown in Figure <ref> and Figure <ref>. The associated quiver of L^ is exactly the ADHM quiver Q^ADHM. The unobstructed equation of L^ is [x_1,y_1]+i_1j_1=0, as computed in Corollary <ref>. Using the Morse model, α_1 (reps. α_2) can be identified with the maximal (reps. minimal) point of the Morse function. Consequently, L^ is transformed by  to the following complex:
	
    (^(,)((L^,)), m_1^): ⟨α_1 ⟩[rrr]^-[ x_1 · - · x y_1 · - · y       j_1 · ]         ⟨X̃, Ỹ, J̃⟩[rr]^-[ -(y_1 · - · y);    x_1 · - · x;          i_1 · ]      ⟨α_2 ⟩,

	where the condition that the differential square equals zero can be verified directly using the unobstructedness of the Lagrangians. 
	
	Since L is equipped with the rank n trivial bundle _1 and F with a rank r trivial bundle _2, the coefficients of the Floer complex are of the following form:
	
    Λ_0^n ⊗⟨α_1 ⟩[rrr]^-[ x_1 · - · x y_1 · - · y       j_1 · ]         Λ_0^n ⊗⟨X̃⟩⊕Λ_0^n ⊗⟨Ỹ⟩⊕Λ_0^r ⊗⟨J̃⟩[rr]^-[ -(y_1 · - · y);    x_1 · - · x;          i_1 · ]      Λ_0^n ⊗⟨α_2 ⟩.

	Notice that the differentials are polynomials and L^ is an object in the subcategory that generated by ^. One can restrict the Complex <ref> to [x,y] and work over the complex coefficients. The resulting complex is the monad of torsion-free sheaves over ^2.




	Given a ζ_-stable framed Lagrangian brane (L^,) over . The cohomology of ^(,)((L^,)) is a torsion-free sheaf of rank r. Furthermore, the monadic complex ^(,)((L^,)) can be extended trivially to ℙ^2 such that the cohomology  gives a framed torsion-free sheaf. 


	By Corollary <ref>, the Maurer-Cartan (deformation) space of (L^,) is equivalent to the representation of the ADHM quiver over Λ_0. Therefore, a ζ_-stable framed Lagrangian brane (L^,) over  is the Lagrangian L^ equipped with a stable framed quiver representation b=(B_1,B_2,i,j) ∈ Rep(Q^ADHM).
	
	As demonstrated in the Chapter 2 of <cit.>, the first differential of the monad is injective at all but finitely many points, while the second differential is surjective for stable framed quiver representations. Consequently, upon restricting to complex coefficients, the cohomology yields a torsion-free sheaf  over 𝔸^2_ℂ. The second statement follows from homogenizing the monad, which trivially extends  to ℂℙ^2.






 §.§ Framed quiver representation as a mirror
 

In this section, we consider the reference framed Lagrangian immersion 𝕃^:=∪ (F,-f_F) and restrict the noncommutative mirror functor ^(^,^) to the subcategory generated by ^. We prove that ^(^,^) gives a framed representation of ADHM quiver Q^ADHM. 



Recall the category of kQ-modules and the category of representation of Q are equivalent. Hence, given a framed Lagrangian brane (L^,) over , one will expect the cohomology of ^(^,^)(L^,) provides a representation of Q^ADHM. Furthermore, this representation should lie in the zero locus of the complex moment map. In other words, it has to satisfy x_0 y_0-y_0 x_0+i_0 j_0=0. An explicit calculation shows this is indeed true. Let's denote the Maurer-Cartan space of (L^,) by , which is the representation space of Q^ADHM as computed above. 





 
	The cohomology of ^(^,^)((L^,)) gives a framed quiver representation. 
 

	L^ is transformed by ^ into the following complex:
	
    (^(^,^)((L^,)),m_1^^): ⟨α_1 ⟩[rrr]^-[ x_1 · - · x_0 y_1 · - · y_0         · i_0         j_1 · ]         ⟨X̃, Ỹ, Ĩ, J̃⟩[rr]^-[ -(y_1 · - · y_0)                0;    x_1 · - · x_0                0;           -· j_0            j_1 ·;            i_1 ·           -· i_0 ]      ⟨α_2, α_3 ⟩,
 
where (m_1^^)^2=0 can be verified directly using unobstructedness of Lagrangians and A_∞-equations. We claim the cohomology concentrates on the highest degree and it provides a framed quiver representation.

In view of the path concatenation relation, we find the only zero divisors are x_0j_0=y_0j_0=0 and i_0x_0= i_0y_0=0. Therefore, ki_0=0  implies that k=0, which means the first order differential is injective.  

We now prove the complex is exact at degree 1. Let a=(a_1,a_2,a_3,a_4)∈ m_1^^  where a_i∈⊗^; more precisely, it means m^^_1(a_1 X̃+ a_2 Ỹ + a_3 Ĩ + a_4 J̃)= 0, thus we have the following equations:  

	 -(y_1a_1-a_1y_0)+ x_1a_2 -a_2x_0 - a_3 j_0 + i_1a_4=0      
	

	j_1a_3-a_4i_0=0     .
 
 Notice that by our convention of coefficients (see Section <ref>), i_0 has no left zero divisor in the coefficients of J̃. From the Equation <ref>, we see that a_3= k i_0, a_4= j_1k for some k∈⊗^. In order to prove the exactness, it suffices to show  a_1=x_1k-kx_0 and a_2=y_1k-ky_0. Let a_1' = a_1-(x_1k-kx_0) and a_2'=a_2-(y_1k-ky_0); subtract a by some multiple  m_1^^(h α_1) of α_1 if necessary, one can assume a_1' has no term whose rightmost path is x_0. Substituting a_1 and a_2 in Equation <ref> by a_1' and a_2' gives rise to the following reduced equation: 
    -(y_1a_1'-a_1'y_0)+x_1a_2'-a_2'x_0 = 0.


 If a_2'=0, a_1' is also zero, since there's no left zero divisor of y_0. Suppose a_1' and a_2' are not zero, then a_1' contains a term that is a multiple of x_0, which contradicts the construction of a_1'. In conclusion, a lies in the image class of k; the complex is exact at degree 1 and the cohomology concentrates on the highest degree.

At the highest degree, the cohomology is the cokernel of the differential. Taking the cokernel of the first row -(y_1 · - · y_0) in the differential matrix means we represent the arrow y_0 by the complex-valued matrix y_1, and similarly for x_0 and x_1. Hence, the cohomology is the framed quiver representation e_1 ⟨α_1 ⟩⊕ e_2 ⟨α_2 ⟩, where the arrows y_0,x_0,j_0, i_0 are represented by the matrices y_1,x_1,j_1,i_1 respectively.





	We will focus on the family of framed Lagrangian brane (L^,) over , but the above result also holds for the family over Λ_0.


Since the morphisms between the trivial vector bundles over the Lagrangians are invariant under gauge change (see Proposition <ref>), the stable quiver representation constructed above descends to Nakajima quiver varieties.


	The ζ_-stable framed Lagrangian branes (L^,) over  are transformed into Nakajima quiver varieties. Namely, the cohomology of ^(𝕃^,^)((L^,)) lands in ℳ(V,W).


Moreover, using the universal twisted complex introduced in <cit.>, we can compare these constructions systematically. On the objects level, it gives a correspondence between the framed quiver representation and the framed torsion-free sheaves, which is a version of ADHM construction.

Let  be the universal twisted complex :=(^(,)(^,^),m_1^,^(-)).  induces a dg-functor ^:= ⊗ - : ^→.

 
	^∘^(^,^)((L^,)) is quasi-isomorphic to ^(,)((L^,)).


	For simplicity, we will omit the deformation parameter in the proof. First, using the same method as in theorem <ref>, we get the following complex:	
    (^(^), m_1^,^): ⟨α_1' ⟩[rrr]^-[ x_0 · - · x y_0 · - · y       j_0 · ]         ⟨X̃', Ỹ', J̃' ⟩[rr]^-[ -(y_0 · - · y);    x_0 · - · x;          i_0 · ]      ⟨α_2' ⟩,

	
	By definition, ^∘^𝕃^(L^)= ^(^) ⊗^^(L^), which can be made into a double complex in the classical way. We will use the spectral sequence to compute the total cohomology.
	
	Let's denote the double complex by C^∙,∙= ⊕_p,qC^p,q:= ⊕_p,q⊗^p(,^) ⊗ (^)^op⊗_^^⊗ CF^q(^,(L^,)), and denote the total complex by Tot(C). Hence, there exists a spectral sequence with the first page E_1^p,q=H^q_ver(C^∙,∙) converging to the total cohomology H^p+q(Tot(C)).
	
	By Theorem <ref>, the cohomology of ^^(L^) concentrates on the top degree. Hence, the first page E_1^p,q consists of a single row. Since the cohomology of ^^(L^) gives a representation of the framed quiver, which represents ^ by the  valued representation, the single row is exactly the monad we obtained in theorem <ref>. Hence, the second page becomes stable, and ^∘^(𝕃^,^)((L^,)) is quasi-isomorphic to ^(,)((L^,)). Moreover, if one extends the complex over ^2_, the stable page has only one nontrivial term, which is the framed torsion-free sheaf.




§ NAKAJIMA QUIVERS AS MIRRORS TO TWO-DIMENSIONAL LAGRANGIAN IMMERSIONS


A Lagrangian immersion and its deformation theory is governed by a quiver algebra with relations (reviewed in Section <ref>).  In this section, we construct the preprojective algebra relations that define the Nakajima quiver varieties by Lagrangian deformation theory of union of (immersed) spheres in dimension two.  Moreover, we transform framed Lagrangian immersions to monadic complexes which are crucial objects in Nakajima's formulation of framed torsion-free sheaves. 





 §.§ Warm-up: quiver algebras without relations

Let us start with complex dimension one.  For each quiver algebra without relation, we can construct a symplectic two-fold together with a Lagrangian immersion whose formal deformation space gives back the quiver algebra.






	
	Given a quiver Q, there exists a symplectic 2-fold M with a Lagrangian immersion 𝕃, such that the associated quiver to 𝕃 is Q. Furthermore, the formal deformation space of  is the free path algebra.


	
	
	

	
	

	The construction is similar to Example <ref>. To each vertex i we associate the graded Lagrangian ⊂ and denote it by L_i.  They are plumbed together according to the directed graph Q to produce .  Then the degree-one immersed generators are one-to-one corresponding to the arrows of Q.  Since there is no degree-two generator, m_0^=0 and hence the formal deformation space of  equals the free path algebra.

	
	
		
	
    

	


	





In the next subsection, we carry out the construction in dimension-two and produce a double quiver with relations.



 §.§ Preprojective quiver algebra coming from Lagrangian Floer theory
 

	 
In this section, we construct framed quiver algebras with relations by using framed Lagrangian immersions. Given a graph D, we show that the (framed) preprojective algebra emerges as the noncommutative deformation space of the (framed) Lagrangian immersion obtained by plumbing of 2-spheres according to D.   In particular, we will show that the unobstructed equations for the (framed) Lagrangian immersions in Theorem <ref> and <ref> give the complex moment map equations for Nakajima quiver varieties. 






Let D:=(I,E) be a graph, where I is the set of vertices and E the set of edges. The corresponding double quiver D^# is the directed quiver with the same vertices and the set of arrows (oriented edges) 

    𝔄:={(e,o(e))| e∈ E, o(e)   is an orientation of  e}.

Thus, each edge e connecting vertices v_i and v_j in D gives rises to two arrows a:v_i→ v_j and a̅:v_j→ v_i in D^#. An orientation Ω of the double quiver is a subset of 𝔄 such that 𝔄=Ω∪Ω̅ and Ω∩Ω̅=∅. The orientation defines a function ϵ:𝔄→{± 1} given by ϵ(a)=1 if a ∈Ω and ϵ(a)=-1 if a ∈Ω̅.

 Similar to Example <ref> and Proposition <ref>, we obtain a Liouville manifold by plumbing 2-spheres according to D. Let  be the  resulting Lagrangian immersion.  Then the associated quiver of  is the double quiver Q=D^# of the graph D. Here we choose a perfect Morse function for each sphere, so the only degree-one generators are given by intersections between adjacent spheres, which appear twice for each transverse intersection of Lagrangian two-spheres. 
 
 For each e∈ E, we label two corresponding degree-one immersed generators  by X_a and X_a̅ , where X_a ∈ CF^1(S^2_t(a),S^2_h(a)) and X_a̅∈ CF^1(S^2_h(a),S^2_t(a)). 
 
 



As in Section <ref>, we will take the completion of Λ_0 Q  by taking the intersection over all the valuations 𝔳 that have
|x_a|_𝔳,|x_a̅|_𝔳≤ 1 and |x_a̅x_a|_𝔳< 1, where x_a is an arrow in Q and x_a̅ is the corresponding arrow with the reverse orientation.  We denote the completed algebra by _free.


 
	 There exists an automorphism on _free such that the noncommutative deformation space  of  is the preprojective algebra of Q, i.e. 
    = _free/J
 where J is the two-sided ideal generated by ∑_t(a)=vϵ(a)x_a̅x_a for v ∈ I for some assignment of signs ϵ.
	
	
	

	
	
	
	
	
	
	Let =∑_a∈𝔄 x_a X_a. 
	We compute the obstruction term m_0^ of . The orientation and spin structure of  correspond to an orientation Ω of the double quiver Q, which defines the function ϵ:𝔄→{± 1}. 
	
	
	
	
	
	
	
	The outputs of the obstruction term m_0^ have degree 2. In this case, the degree-two generators are the minimum points P_v of each sphere S^2_v. Thus, the relations are contributed from pearl trajectories to P_v. Such trajectories are unions of Morse flow lines to P_v and constant polygons (x_a̅x_a)^k for all x_a with t(x_a)=v.  
	Similarly to Theorem <ref>, we obtain  
	
    m_0^b     =∑_k m_2k(∑_a x_a X_a, ∑_a̅ x_a̅ X_a̅, ⋯, ∑_a x_a X_a, ∑_a̅ x_a̅ X_a̅) 
        = ∑_v (∑_t(a)=vϵ(a)x_a̅x_a(1+ ∑_j a_j(x_a̅x_a)^j ))P_v,

  where a_j depends on the Kuranishi perturbation, and (x_a̅x_a)^j is contributed by the constant polygon with corners being the immersed sectors x_a and x_a̅, ordered counterclockwise.
  
 
Because of the anti-symmetric involution in a neighborhood of the transverse intersection point, a pair of arrows contributes to the coefficients x_a̅x_a(1+ ∑_j a_j(x_a̅x_a)^j) at P_t(a) and x_ax_a̅(1+ ∑_j a_j(x_ax_a̅)^j)= x_a (1+ ∑_j a_j(x_a̅x_a)^j)x_a̅ at P_h(a). We define: 
	
    x̃_a= x_a(1+ ∑_j a_j(x_a̅x_a)^j),   x̃_a̅= x_a̅,
 for all a such that ϵ(a)=1.
	
	We will abuse the notation and replace x̃ by x. Thus the noncommutative deformation space of  is := _free/(∑_t(a)=vϵ(a)x_a̅x_a).




Similarly, we compute the weakly unobstructed equation of a framed Lagrangian immersion ^fr:= ⋃_v F_v, which is obtained by extending the construction in Example <ref> to D. The associated quiver will be the framed quiver Q^ of Q. Let i_v (resp. j_v) be the arrow corresponding to the  branch jump I_v from F_v to S^2_v (resp. J_v from S^2_v to F_v). We  take the completion _free^ of Λ_0 Q by taking the intersection over all the valuations 𝔳 that satisfy |x_a|_𝔳,|x_a̅|_𝔳, |i_a|_𝔳, |j_a|_𝔳≤ 1 and |x_a̅x_a|_𝔳, |i_0j_0|_𝔳< 1. 

 
	
	Let ^:=∑_a∈𝔄 x_a X_a + ∑_v∈ I i_vI_v + ∑_v∈ I j_vJ_v. There exists a coordinate change on _free^ such that 
	the formal deformation space of the framed Lagrangian immersion 𝕃^fr equals  
    ^fr= _free^/ J^fr,
  where J^fr is the two-sided ideal generated by (∑_t(a)=vϵ(a) x_a̅ x_a)_v + (ij)_v =0⃗ for v ∈ I.


	The framing Lagrangians are equipped with a Morse function with exactly one maximal point, which has degree zero. Hence, the obstruction term m_0^^ only has outputs at the minimum points on each sphere S_v^2, for v ∈ I. Once we fix the perturbation of Kuranishi structure, we can change the coordinate of i_v at each vertex consistently. The rest of the proof is similar to Theorem <ref> and Theorem <ref>.


Each component _i is equipped with a trivial vector bundle of rank r_i.  We take the vector space M(V,W) ×∏_i (π_1(_i), (r_i,)) and define a family of Fukaya algebras over it.  Since Floer theory, namely the vector space of chains together with the m_k-operators, is equivariant under gauge transformations, the family indeed descends to its quotient by the gauge group ∏_i (r_i,), which in this case (with the above preprojective algebra relations) gives a Nakajima quiver variety by choosing a suitable GIT stability condition.  As a consequence, we have a family of stable Lagrangian deformations of ^ parametrized by a Nakajima quiver variety.


	We briefly recall the equivariant of Floer theory under gauge change below.
	
		The family of Fukaya algebra is independent of choice of basis for each flat bundle. In other words, it's equivariant under the gauge transformation.
	
	
		*** We want to show that the m_k-operations are equivariant under gauge change.
		
		Fix objects (L_1,ℰ_1),⋯, (L_n,ℰ_n) and  intersections q, X_1,⋯, X_n-1. For notational convenience, we add two auxillary points X_0=X_n=q. The parallel transport along the boundary of the holomorphic disk that lies on L_i induces an isomorphism ρ_i∈ Hom(ℰ_i|_X_i-1,ℰ_i|_X_i) and x_i∈ Hom(ℰ_i|_X_i,ℰ_i+1|_X_i) for i∈[n-1]. The composition of these maps gives an element γ in Hom(L_1|_q,L_n|_q) by 
		
    γ = ρ_n· x_n-1·ρ_n-1⋯ρ_2· x_1·ρ_1.

		The gauge group acts on its component by conjugation, and it's easy to see this action is always trivial. 
	






 §.§ ALE spaces: when the Lagrangian immersions are homological tori


From the point of view of Lagrangian deformations, an affine ADE quiver is special because the Euler characteristic of the Floer cohomology of its associated Lagrangian brane is semi-positive definite as a  quadratic form on rank.

First, let us deduce a formula for Euler characteristic of the Floer cohomology of a Lagrangian immersion in general dimensions.


	The Euler characteristic of the Floer cohomology of a Lagrangian immersion  equals
	
    χ^LF() = ∑_i χ(_i) + 2 ∑_e: codim(_e)  is even (-1)^ u_eχ(_e)
where
	_i denotes the normalization of the i-th component of , e are the labeling of the clean intersections _e, u_e is one of the two branch jumps associated with _e (and the parity of u_e is independent of the choice),
	and χ(_i), χ(_e) are the usual Euler characteristics.



First, note that the Euler characteristic of the total cohomology equals that of the chain complex.  It suffices to compute the Euler characteristic of the Floer complex formed by the Morse generators of clean intersections of . 

The Floer complex is generated by the following.  Codimension zero clean intersections are the connected components _i in the normalization of , which contribute ∑_iχ(_i) in χ^LF().  Additionally, the clean intersections of positive codimension are smooth manifolds _e. Each degree k critical point of _e contributes two generators, which have degree u_e +k and v_e+k respectively, in the Floer complex, where u_e and v_e are the two branch jumps associated with the transverse intersection point. Denote the number of degree k critical points of _e by a_k. The Euler character of  is

    χ^LF()=   ∑_i χ(_i)+ ∑_e ∑_k (-1)^k a_k((-1)^ u_e+(-1)^ v_e) 
    
    	=    ∑_i χ(_i)+ ∑_e (∑_k (-1)^k a_k (-1)^ u_e(1+(-1)^ v_e +  u_e)).

Notice that v_e + u_e  =dim - dim_e. Hence, 
    χ^LF()=   ∑_i χ(_i)+2 ∑_e: codim(_e)  is even(∑_k (-1)^k a_k (-1)^ u_e) 
     
    	=   ∑_i χ(_i) + 2 ∑_e: codim(_e)  is even (-1)^ u_eχ(_e).



Note that  is non-displaceable if χ()≠0.

For a Lagrangian brane (,) of rank r⃗, where r⃗=(r_i)_i ∈ I records the rank of the bundle over each component, the Euler characteristic of its Floer cohomology equals

    χ_^LF(r⃗) = ∑_i χ(_i) · r_i^2 + 2 ∑_e: codim(_e)  is even (-1)^ u_eχ(_e) · r_h(e)r_t(e)

where h(e),t(e) label the two components involved in the intersection _e.  χ_^LF is a quadratic form on integer vectors r⃗.


	A Lagrangian immersion  is said to be (semi-)positive definite if χ_^LF is (semi-)positive definite.


 
	Suppose  is of dimension two and every component of the normalization of  is a sphere.  Then  is positive definite if and only if its intersection graph is of ADE type.   is strictly semi-positive definite if and only if it is of affine ADE type.  Moreover, χ_^LF is always even.



	In this case,
	
    χ_^LF(r⃗) = 2 (∑_i r_i^2 - ∑_e  r_h(e)r_t(e))

	and ∑_i r_i^2 - ∑_e  r_h(e)r_t(e) is the Tits quadratic form on r⃗ associated to the intersection graph with vertices i and edges e.  by the well-known theorems of Gabriel <cit.> and Nazarova <cit.>, the Tits quadratic form is positive definite (resp. semi-positive definite) if and only if the graph is of ADE type (resp. affine ADE type) .



Now consider an affine ADE diagram. Let  be the corresponding Lagrangian immersion.  Then the associated matrix C of χ_^LF(r⃗) is a Cartan matrix of an affine type, which has an one-dimensional kernel. Let δ⃗∈ℤ^I be a vector in the kernel of C such that δ_i >0 and min δ_i=1. This uniquely determines δ⃗. Choose a generic parameter ζ^0 ∈^I such that ζ^0 ·δ⃗=0. As a consequence of Proposition <ref>, the localized mirror of (,) gives an asymptotically locally Euclidean (ALE) space found by <cit.>.





	Let  be the Lagrangian immersion obtained by plumbing according to the affine ADE diagram. Then the moduli space of ζ^0-stable Lagrangian branes (,) of rank δ⃗ over  admits an asymptotically locally Euclidean metric.


    By Theorem <ref>, a Maurer-Cartan element b=∑ B_a X_a ∈^1((,),(,)) is a rank δ⃗ representation of Q over  satisfying the complex moment map equation: 
    (∑_t(a)=vϵ(a) B_a̅ B_a)_v=0⃗,
 where v ∈ I and B_a is the linear map corresponding to the arrow a. 
    
    Let X_ζ^0:= {(B) ∈ Rep(Q,δ⃗) | (∑_t(a)=vϵ(a) B_a̅ B_a)_v=0⃗}_ζ^0 (GL(δ⃗)/^*), where ‘_ζ^0’ means the GIT quotient with respect to the parameter ζ^0. As we assume ζ^0 is generic, X_ζ^0 is the quotient of the set of ζ^0-stable representations by GL(δ⃗)/^*, see <cit.>, which is 1-1 corresponding to the isomorphism class of ζ^0-stable Lagrangian branes (,). Hence, X_ζ^0 is the moduli space of ζ^0-stable Lagrangian branes (,). By the work of Kronheimer <cit.>, X_ζ^0 has a hyperKähler ALE metric.




 

 §.§ Quiver relation implies commutativity of charts for D_4




Before the general construction of framed torsion-free sheaves, let's give an example of how to find an affine chart of a stable family. Using quiver algebroid stack, one can obtain a local-to-global description of the noncommutative deformation spaces.

Let  be the Lagrangian immersion produced by plumbing according to the affine Dynkin diagram D_4. By Theorem <ref>, the quiver Q associated to  is the double quiver of affine D_4, see Figure <ref>. The nc deformation space is the path algebra _0 introduced in Example <ref>.



Let (,) be a family of the Lagrangian branes of rank δ⃗=(2,1,1,1,1) over the noncommutative parameter space , see the end of Section <ref>. In the following, we will show there exists some stable charts and prove that these stable charts are indeed commutative.

 
	Let _0 be the noncommutative deformation space of (,). Take v_1 as the reference vertex. Then there exists some stable affine charts of the stable family _0 over  of rank δ⃗, which can be glued into the minimal resolution of D_4 singularity. In particular, these stable affine charts are commutative.



	By Theorem <ref>, _0 satisfies
	

		∑_i=1^4 a_ib^i=0 and b^ia_i=0 for i=1, ⋯,4.
	
 
	
	In the following, we will write down some affine charts of the stable (_0, δ)-family over  explicitly. 
	
	
	Since the family _0 over  of rank δ⃗ is stable, there's no proper _0-submodule M of ^⊕ (∑δ_i) such that e_1M =. Hence, the representation of a_1 is nonzero.  For the same reason, one of the images of a_2, a_3, a_4 is linearly independent with a_1. Otherwise, there exists a proper _0-submodule violating the definition of stable family. Without loss of generality, we can assume the image of a_2 is linearly independent with a_1. In other words, [ G_20(a_1) G_20(a_2) ] is invertible. Hence, using the automorphism GL_δ_2, one can set [ G_20(a_1) G_20(a_2) ]= [ 1 0; 0 1 ]. Besides, b^2 is also nonzero with b^2a_2=0. Thus, the image of b^2 can be normalized as [ 1 0 ]. Similarly, the restriction on vertices v_3 and v_4 implies that b^3 and b^4 are nonzero. Assume b^3a_1 and b^4a_1 are not zero. Using the weakly unobstructed equation, the representation of _0 over  is determined by three parameters X, Y and Z. Let _2 be the -algebra generated by X, Y and Z. Notice that _2 is a quiver algebra, whose quiver has a single vertex, since  only has a single vertex. One obtain the representation G_20 exactly as shown in Example <ref>.  Conversely, the inverse representation G_02 can be found by localizing a matrix of paths. Keeping track of the automorphisms at each vertex gives rise to the inverse representation G_02 up to gerbe terms. 
	
	Observe that even though _2 is not assumed to be commutative, the existence of such representations G_02 and G_20 forces _2 to be commutative. This is because G_02(XY)=G_02(YX) as shown in Example <ref>. Applying G_20, one has XY=YX. Similarly, one can show [Y,Z]=[X,Z]=0. Thus, _2 has to be a -subalgebra of [X,Y,Z]/(XY-(X+1)Z). In particular, one can take _2 to be [X,Y,Z]/(XY-(X+1)Z) ≅[X,T], where T:=Y-Z. This provides a commutative affine chart of the stable (_0,δ)-family over , which is the affine space _^2. In the following, we will denote the generators of _2 by X_2, Y_2, Z_2, indicating _2 is constructed by localizing _0 at the matrix [ a_1 a_2 ].  
	
	In general, different choices of localization correspond to different local affine charts. If instead of assuming b^3a_1 is nonzero, we assume b^3a_2b^2a_1 is invertible. Denote the corresponding localization of _0 by _0,loc. We have 
	
    G_02': _2':= [X_2',Y_2',Z_2']/(X_2'Y_2'-(X_2'Y_2'^2-1)Z_2') →_0,loc,     G_02':=
    	
    		X_2' ↦ (α_1a_3)(b^3a_2b^2a_1) 
    
    		Y_2' ↦ (b^3a_2 b^2a_1)^-1(b^3a_1)   
    
    		Z_2' ↦ (b^4a_1)^-1b^4a_2b^2a_1

	
    G_2'0: _0,loc→ Mat(_2'),    G_2'0:=
    	
    		a_1 ↦[ 1; 0 ]
    
    		a_2 ↦[ 0; 1 ]
    
    		a_3 ↦[      X_2'; -Y_2'X_2' ]
    
    		a_4 ↦[ -Z_2'(Y_2'X_2'Y_2'-1);        Y_2'X_2'Y_2'-1 ]
    
    		b^1 ↦[                 0 Z_2'X_2'Y_2'-X_2' ]
    
    		b^2 ↦[ 1 0 ]
    
    		b^3 ↦[ Y_2'    1 ]
    
    		b^4 ↦[    1 Z_2' ]
    .

	G_02' and G_2'0 are inverse to each others up to gerbe terms, which can be found in a similar way. Composing G_20 and G_02', one has the transition map G_22':= G_20∘ G_02': _2'→_2:
	
    G_22':=
    	
    		X_2' ↦ (α_1a_3)(b^3a_2b^2a_1) ↦ -X_2Y_2^2
    
    		Y_2' ↦ (b^3a_2 b^2a_1)^-1(b^3a_1) ↦ Y_2^-1
    
    		Z_2' ↦ (b^4a_1)^-1b^4a_2b^2a_1 ↦ Z_2
    	.

	This provides the local gluing of one (-2)-curve and the representation of the arrow b^3 serves as the homogeneous coordinate of this exceptional ℙ^1.
	
	By making alternative choices for localization, one can construct other (-2)-curves. If the image of a_3 is linearly independent with a_1, we define  [ α^1; α^3 ] to be the inverse of [ a_1 a_3 ]. Besides, localizing at diag(b^2a_1, b^3 a_1, b^4 a_1), we obtain another commutative chart _3:=[X_3,Y_3,Z_3]/(X_3Z_3-(X_3+1)Y_3). The representations can be found similarly:
	
	
    G_03: _3:= [X_3,Y_3,Z_3]/(X_3Y_3-(X_3+1)Z_3) →_0,loc,     G_03:=
    	
    		X_3 ↦ (b^3a_1)^-1(α_3a_4)(b^4a_1) 
    
    		Y_3 ↦ (b^2 a_1)^-1(b^2a_3)(b^3a_1) 
    
    		Z_3 ↦ (b^4a_1)^-1b^4a_3b^3a_1

	
    G_30: _0,loc→ Mat(_3),    G_30:=
    	
    		a_1 ↦[ 1; 0 ]
    
    		a_2 ↦[ X_3Z_3; -X_3-1 ]
    
    		a_3 ↦[ 0; 1 ]
    
    		a_4 ↦[ -Z_3X_3;     X_3 ]
    
    		b^1 ↦[                  0 X_3Y_3^2-Z_3X_3Y_3 ]
    
    		b^2 ↦[   1 Y_3 ]
    
    		b^3 ↦[ 1 0 ]
    
    		b^4 ↦[   1 Z_3 ]
    .

	If we localize at b^2a_1, b^3a_1, b^4a_3b^3a_1 instead, we obtain the commutative chart _3':=[X_3',Y_3',Z_3']/(X_3'Z_3'-(X_3'(Z_3')^2-1)Y_3'). The transition map between _3 and _3' can be found similarly:
	
    G_33':=
    	
    		X_3' ↦ (α_1a_4)(b^4a_3b^3a_1) ↦ -X_3Z_3^2 
    
    		Y_3' ↦ (b^2a_1)^-1b^2a_3b^3a_1 ↦ Y_3 
    
    		Z_3' ↦ (b^4a_3b^3a_1)^-1b^4a_1 ↦ Z_3^-1

	which provides the second (-2)-curve and the representation of the arrow b^4 serves as the homogeneous coordinate.
	
	In addition, we can also obtain the transition maps between _4:= [X_4,Y_4,Z_4]/(X_4Y_4-(X_4+1)Z_4) and _4':=[X_4',Y_4',Z_4']/(X_4'Y_4'-(X_4'(Y_4')^2-1)Z_4'): 
    G_44':=
    	
    		X_4' ↦ (α_1a_2)(b^2a_4b^4a_1) ↦ -X_4(Y_4)^2 
    
    		Y_4' ↦ (b^2a_4b^4a_1)^-1b^2a_1 ↦ Y_4^-1
    
    		Z_4' ↦ (b^3a_1)^-1b^3a_4b^4a_1 ↦ Z_4

	where _4 (resp. _4') comes from the normalization of _0 after localizing at [ a_1 a_4 ] and diag[ b^2a_1 b^3a_1 b^4a_1 ] (resp. [ a_1 a_4 ] and diag[ b^2a_4b^4a_1       b^3a_1       b^4a_1 ]). This gives the third (-2)-curve. 
	
	The fourth exceptional curve, which intersects the previous three, comes from the transition maps G_32: _2 →_3:
	
    G_32:=
    	
    		X_2 ↦ (b^2a_1)^-1(α_2 a_3)(b^3a_1) ↦ -(X_3+1)^-1
    
    		Y_2 ↦ (b^3a_1)^-1(b^3a_2)(b^2a_1) ↦ -Y_3(X_3+1) 
    
    		Z_2 ↦ (b^4a_1)^-1(b^4a_2)(b^2a_1) ↦ Z_3
    	.
 Notice that gluing between _2 and _3 happens at the open subset X_3 ≠ -1. Thus, the closure of Y_3=0 indeed provides the fourth (-2)-curve. The dual graph of these four (-2)-curves form the D_4 Dynkin  diagram. The stable family _0 over  of rank δ⃗ provides a quiver algebroid stack corresponding to the minimal resolution of D_4-singularity.	



	As in Example <ref>, _k ≅[X,T] for k=2,3,4. Besides, _k'≅[X',Y']_(X'Y'^2-1), since X_k'Y_k'^2-1=0 would imply X_k'Y_k'=0 in _k', which is a contradiction. All these charts are open subsets of _^2.




 §.§ Mirror functor and monadic complexes
 

In this section, we compute the images of framed Lagrangian branes under the mirror functor and show that they coincide with the monadic complexes algebraically constructed by Nakajima <cit.>.  Restricting to the affine ADE cases, we show that they produce the framed torsion-free sheaves over the mirror asymptotically locally Euclidean (ALE) spaces.









	The quiver Q associated to  is the affine ADE quiver. Up to a change of coordinate, the noncommutative deformation space  of  is 
    = _free/J,
 where J is the two-sided ideal generated by ∑_t(a)=vϵ(a)x_a̅x_a, for v ∈ I.









Let  (resp. ^) be a (framed) Lagrangian immersion obtained by applying 2-sphere plumbing construction as in Section <ref>. In order to obtain the monad of framed torsion-free sheaves, we consider the family of framed Lagrangian branes (^fr,) of rank r⃗.  By replacing the formal universal deformations (^fr,^) by the family of branes (^fr,) in Proposition <ref> and using the same method, we obtain the Maurer-Cartan space of (^,) as below.



 
	Let b=∑_a B_a X_a + ∑_v i_v I_v + ∑_v j_v J_v where a denotes an arrow and v denotes a vertex; B_a, i_v, j_v are matrices of corresponding ranks.  The unobstructed equation for the family (^fr,) is 
	
    ∑_t(a)=vϵ(a) B_a̅ B_a + i_vj_v =0

	for all v ∈ I. In particular, in the subcategory generated of ^, the Maurer-Cartan space of the family  (^fr,) over Λ_0 is the stack [M(Q^,r⃗)/GL(r⃗)], where M(Q^,r⃗) ⊂ Rep(Q^,r⃗) is the subvariety of -representations.



	The above unobstructed equation exactly coincides with the complex moment map equation.  Thus, below we consider the Nakajima quiver variety in zero complex moment map level.  The real parameters of Nakajima quiver variety correspond to choices of stability conditions.



	


Next, we restrict the functor ^(,) to the branes (^,) and produce the monadic complex of the corresponding framed double quiver. We write =∑_a x_a X_a for (,) where a are arrows of Q.  The deformation parameter for (^,) is denoted as b given in Lemma <ref>.







 
	
	^(,) transforms the framed Lagrangian branes (^fr,) into the following complexes:
	
    (^(,)(^fr,),m_1^):  L^0((,),(,)) → E((,),(,)) ⊕ L^1((,), (F,)) → L^2((,),(,)),

which coincides with the monad constructed algebraically by Nakajima <cit.>. 
The complex is spanned by Floer generators M_v ∈ L^0((,),(,)), P_v ∈ L^2((,),(,)), and the immersed sectors J_v ∈ L^1((,), (F,)), X_a ∈ E((,),(,)) for all vertices v and arrows a of Q. Here, M_v, P_v are the maximal points and minimum points of the Morse function f_v on S^2_v respectively, while X_a ∈^1(S^2_t(a),S^2_h(a)) and J_v ∈^1(_v, F_v) are degree 1 immersed sectors. 
Furthermore, the first differential takes the form 
    m_1^(η M_v)= ∑_h(a̅)=v (B_a̅η) X_a̅ + ∑_h(a)=v ( η x_a )X_a + j_vη J_v,
 and the second one is 
    m_1^(η' X_a)=  η' x_a̅ P_h(a)+ B_a̅η' P_t(a);     m_1^(η” J_v) = i_v η” P_v.




	
	
    
	
	
	
	Recall that = ⋃_v ∈ I S^2_v and ^fr= ⋃_v ∈ I (S^2_v ∪ T^*_p_vS^2_v). By the construction of the localized mirror functor, 
    ^(,)(^fr,)= ⊗_Λ^⊕_0^∙(, (,)) ⊕⊗_Λ^⊕_0^∙(, (,)),
 where ^∙(, (,))=⊕_v∈ IΛ_0 ⊗|_J_v⟨ J_v ⟩ and ^∙(, (,))= ⊕_a ∈𝔄Λ_0 ⊗|_X_a⟨ X_a ⟩⊕_v ∈ I⊕_Y ∈ Crit(f_v)Λ_0 ⊗|_Y ⟨ Y ⟩ is generated by X_a, M_v and P_v using the Morse model. Hence, ^(,)(^fr,)  has the form as the complex <ref>. More conventions about the notations can be found in Section <ref>.
	
	
	
	It remains to compute the differential. For simplicity, we will omit the sign, which depends on the choices of orientations and spin structure.  
	
	Observe that M_v is the fundamental class of the sphere S^2_v. We have m_2(∑_a X_a, M_v)=∑_ h(a)=v m_2(X_a,M_v)=∑_h(a)=v X_a, m_2( M_v,∑_a X_a)=m_2(M_v,∑_t(a)=v X_a)=∑_t(a)=vX_a and m_2( M_v, ∑_k J_k)=m_2(M_v,J_v)=J_v. Besides, m_k(⋯, M, ⋯) vanishes for k≥ 3. Hence, for the first differential, we have 
    m_1^(η M_v)= m_2(∑_h(a)=v x_a X_a ,η M_v) + m_2(η M_v, ∑_t(a)=v B_aX_a)+ m_2(η M_v, j_vJ_v)= ∑_h(a̅)=v (B_a̅η) X_a̅ + ∑_h(a)=v ( η x_a )X_a + j_vη J_v,
which counts the bigons that pre- and post-compose formal boundary deformations of compact components. 
	
	For the second differential, we should use the fact that m_1^ (X)=∑ m_k(, , ⋯, X, ,⋯,) is the same as ∂_x m_0^. 
	Using the coordinate change in Theorem <ref>, the second differential is the same as differential of the weakly unobstructed relations. In other words, we have 
    m_1^(η' X_a)=   m_2(η' X_a, B_a̅X_a̅) + m_2(x_a̅ X_a̅,η' X_a)= η' x_a̅ P_h(a)+ B_a̅η' P_t(a)

	
    m_1^(η” J_v)= m_2(η” J_v , i_vI_v)=i_v η” P_v.

	Due to the unobstructed equations, (m_1^)^2 is zero, yielding the Floer complex <ref>. Notice that the projective -module e_v in the coefficient corresponds to the tautological bundle geometrically, see Section 4.4 of <cit.> for such correspondence. Therefore, ^(,)(^fr,) is the monad demonstrated in <cit.>. 

Now, let's restrict to the affine ADE case so that χ_^LF(r⃗) is strictly semi-positive definite.
To construct framed torsion-free sheaves, let's first recall the chamber structure of the stability parameters.  Readers are refered to <cit.> for more details. Let D=(I,E) be an affine ADE Dynkin diagram. Let A be the adjacency matrix of the graph. Then C=2I-A is a Cartan matrix of an affine type (which is also the matrix associated to χ_^LF(r⃗)). Fix a dimension vector v. Let 

    R_+     :={θ=(θ_i) ∈_≥ 0^I |θ^t C θ≤ 2}∖{0},
    
    	R_+(v)     :={θ=(θ_i) ∈ R_+ |θ_i ≤_ V_i  for all i},
    
    	D_θ    :={x=(x_i)∈ R^I | x·θ=0} for  θ∈ R_+.
 For the graph of affine type, R_+ is the set of positive roots of the corresponding Dynkin diagram, and D_θ is the wall defined by the root θ. 

As in Corollary <ref>, we pick the primitive vector δ⃗ in the kernel of 𝐂 and a generic parameter ζ^0 ∈^I. Namely, ζ^0 is not contained in any D_θ, where D_θ is the hyperplane defined by a real root. We take a parameter ζ_ from the chamber containing -ζ^0 in its closure with ζ_·δ⃗ < 0.



 
	Let D be the affine ADE Dynkin diagram.  We take the stable family of Lagrangian branes (,') of rank δ⃗ over the coordinate ring R of the subvariety M(Q,δ⃗)⊂Rep(Q,δ⃗) of -representations.  Then ^(,') transforms the ζ_-stable framed Lagrangian branes (^,) into the monad of the torsion-free sheaves over the mirror.



	Since m_k-operators are GL(δ⃗)-equivariant, the Floer complex ^(,')((^,)) descends to the ALE space if (,') is a stable family of rank δ⃗ over R. The coefficients in the resulted Floer complex are the tensor product of the tautological bundles and the trivial representation bundles. Hence, this Floer complex coincides with the monad of torsion-free sheaves constructed algebraically by Nakajima <cit.>. Furthermore, one can homogenizing the monad and get the framed torsion-free sheaves by trivial extension.


In the affine A_n case, we have the following nice property of the framed Lagrangian immersion ^.

	Let (^fr,) be the framed Lagrangian immersion in affine A_n. Then the cohomology of ^(^fr,)(^fr,) gives a framed quiver representation.



	In this proof, we will write the formal boundary deformation of (^fr,^) by ^=∑ x_k X_k + y_k Y_k +i_k I_k +j_k J_k, while the deformation of (^,) by b^=∑ x_k^c X_k + y_k^c Y_k+ i_k^c I_k+ j_k^c J_k. Here X_k ∈^1(^2_k, ^2_k+1), Y_k ∈^1(^2_k+1, ^2_k), I_k ∈^1(F_k, ^2_k) and J_k ∈^1(^2_k, F_k) are the degree 1 immersed sectors for all k mod n+1. Besides, we will disregard the sign of the complex, which depends on the choice of orientation and spin structure. As before, x_k, y_k, i_k, j_k are arrows in Q^, while x_k^c, y_k^c,i_k^c,j_k^c are matrices of corresponding ranks. 
	

	
	
	
	The complex (^(^fr,^)(^fr,), m_1^^) has the following form:
	
    L^0((,),(,)) → E((,),(,)) ⊕ L^1((,), (F,)) ⊕ L^1((F,),(,)) → L^2((^,),(^,)),
 which can be computed as in Theorem <ref>. The main difference is that we have additional degree 1 immersed sectors I_k and the degree 2 point classes of the framing components, denoted by F_k. 

    
    We will prove that the cohomology concentrates on the highest degree and it provides a framed quiver representation. The idea is to make use of the symmetry of the affine A_n diagram and reduce to Theorem <ref>.  
    
    First, we observe that the first differential is injective. The output of m_1^(∑η_k M_k) at I_v is given by m_2(i_v I_v, ∑η_k M_k)= η_v i_v I_v for all vertices v. Since η_v ∈|_M_v⊗ e_v and i_v have no left zero divisors in |_M_v⊗ e_v,we have  η_v i_v=0 if and only if η_v=0 for all v.
    
    Let's show the complex is exact at degree 1. Let U:= ∑_k∈ I a_k X_k + b_k Y_k+ c_k J_k + d_k I_k be an element in the kernel. The outputs at P_k and F_k are zero for k=0, 1, ⋯ n. In other words, 
    
    	(y_k^c a_k + b_k x_k + a_k-1y_k-1+ x_k-1^c b_k-1+i_k^c c_k+ d_k j_k)P_k=0      
    	

    	(c_k i_k + j_k^c d_k)F_k=0.     
    
    
    Notice that i_k has no left zero divisors. Thus c_k is a multiple of j_k^c, while d_k is a multiple of i_k. Let's write c_k=j_k^c d_k' and d_k= d_k' i_k for some d_k' and k=0, ⋯, n. Hence the Equation (<ref>) becomes 
    
    y_k^c a_k + b_k x_k + a_k-1y_k-1+ x_k-1^c b_k-1+i_k^c c_k+ d_k j_k
    
        		=    y_k^c a_k + b_k x_k + a_k-1y_k-1+ x_k-1^c b_k-1+ (y_k^c x_k^c+ x^c_k-1y^c_k-1)d_k'+ d_k'(y_kx_k+ x_k-1y_k-1)=0,
 where we used the ubostructed equations.

     In order to prove the exactness, it suffices to show  a_k=x_k^c d_k'+ d_k+1' x_k and b_k=d_k' y_k+ y_k^c d_k+1' for k=1, ⋯, n. Let a_k =  a_k'+(x_k^c d_k'+ d_k+1' x_k) and b_k=b_k'+(d_k' y_k+ y_k^c d_k+1'). One can assume a_k' has no terms with rightmost path being x_k by adding m_1^^(h_k+1 M_k+1) for some h_k+1 if necessary. In other words, one can replace U by U+ m_1^^(h_k+1 M_k+1). The new U is also an element in the kernel. 
     
     Substituting a_k and b_k in Equation ((<ref>)) by the above expressions, we have the following reduced equation:
     
    y_k^c a_k'+ b_k'x_k+ a_k-1'y_k-1 + x_k-1^c b_k-1'=0
 for k=0, 1, ⋯, n. 
 
     By the convention of coefficients, h(a_k')=v_k+1 (resp. h(b_k')=v_k), while t(a_k')=v_k (resp. t(b_k')=v_k+1) for k=0,1, ⋯, n. Therefore, the sum of these n+1 Equations (<ref>) is equal to 
     
    (∑_l y_l^c)(∑_k a_k')+ (∑_k b_k')(∑_l x_l) + (∑_k a_k')(∑_l y_l)+ (∑_l x_l^c)(∑_k b_k')=0,
 since the product is zero if the tail and head vertices do not match.
 
     Besides, if ∑_k a_k'=0, a_k' is zero for all k, since a_k' has tails at different vertex for each k. Similarly, b_k' also possesses this property. Furthermore, if ∑_k b_k'=0, the Equation (<ref>) would force ∑_k a_k' to be zero. Suppose ∑_k b_k' ≠ 0,  ∑_k a_k' should be a multiple of ∑_l x_l, since ∑_l x_l has no left zero divisors. However, by the construction, a_k' doesn't contain a term which is a multiple of x_k. Thus, ∑_k a_k' is not a multiple of ∑_k x_k, which is a contradiction. 
    

    
    
    	
    

   
    
   
     
    
   
    
    
    
    

     

    

    
    

    
    
    
    
    
    Hence, ∑_k a_k'= ∑_k b_k'=0, which implies a_k'=b_k'=0 for all k. So the complex is exact in degree one. The only nontrivial cohomology appears at the top degree, which gives the framed equiver representation. In particular, this implies Theorem <ref>, the self-plumbing case, when k=0. 
    
    
    
    
	



	In the affine A_n case, ^∘^(^fr,)(^fr,) is quasi-isomorphic to ^(,)((^fr,)).


	The proof is similar to the Corollary <ref>.




	
		
		
		
	
		
		
		  
		  
		  
		  
		
		
	
	
		
		  
	




 §.§ Stable family of affine A_n Lagrangian immersions and special Lagrangians

The affine A_n surface is particularly well-studied.  This local model admits a special Lagrangian fibration constructed by ^1-symplectic reduction <cit.>.  Its Fukaya category was known by <cit.> and stability conditions were studied by <cit.>.  SYZ and homological mirror symmetry in this case is well-known <cit.>.  An explicit expression of the non-Archimedean dual fibration was recently found by <cit.>.

In this subsection, we revisit mirror symmetry and stability conditions for the affine A_n surface.  The new discovery is the gluing relation for the affine A_n Lagrangian skeleton (which is a union of spheres) with SYZ fibers via the language of quiver algebroid stacks, and the identification between its stable deformations and special Lagrangian torus objects.










Fix ϵ∈ℝ_>0. We consider the surface 

    X={(x,y,z)∈ℂ^3| xy = p(z)  and z≠ϵ}
 where p(z)=∏_k=1^n+1(z-a_k) is a monic polynomial and -∞ < a_n+1<⋯ < a_1<0 are distinct real numbers. This is a smoothing of the A_n singularity. X is equipped with the Kähler form inherited from the standard form on ℂ^3. As is known, the projection map π: (x,y,z)↦ z can be viewd as a Lefschetz fibration. One may regard Example <ref> as its special case. It admits a natural Hamiltonian ^1-action given by 
    e^it·(x,y,z) =  (e^it x, e^-it y, z)

whose associated moment map is μ(x,y,z)=1/2(|x|^2-|y|^2). 

X admits a special Lagrangian torus fibration 

    F:X →^2,   F(x,y,z)=(μ(x,y,z), ln  |z|^2).
 
We denote the singular fibers F^-1(0, ln |a_i|^2) by _i, and the Lagrangian tori F^-1(0, ln  |r|^2) by T_r,0 for r ≠ a_i and i=1,⋯, n+1.  In particular, we have the families of special Lagrangian tori T_r,0 for |a_i|<r<|a_i+1|.  We also denote these families by T_i for simplicity.

Let C_q={(x,y,z)∈ X|μ(x,y,z) = 0  and z= q}.  Consider := ⋃_q∈ cC_q, where c is a loop passing through all the zeros of p(z) as shown in Figure <ref>.  is a union of n+1 Lagrangian spheres ⋃_i=1^n+1^2_i.   corresponds to the affine A_n quiver shown in Figure <ref>.




The generators of ^1(,) are the immersed sectors U_i^L ∈^1(^2_i, ^2_i+1), V_i^L ∈^1(^2_i+1, ^2_i); the generators of ^1(_i,_i) are the self-immersed sectors U_i,V_i for i=1,⋯ n+1 as shown in Figure <ref>. Besides, the holonomy variables of T_i will be denoted by x_i, y_i. Let's denote the deformation parameter of  (resp. _i) by _0= ∑_i u_i^LU_i^L+ v_i^LV_i^L (resp. _i= u_iU_i + v_iV_i) and the formal deformation space by _0 (resp. _i).

We find that there exists a quiver algebroid stack , over which the Fukaya isomorphism equations between (,_0) and immersed spheres (_i,_i) as well as (T_i,∇^(x_i,y_i)) can be solved. This induces a map from the Maurer-Cartan space of  to that of _i and T_i.







	There exist preisomorphism pairs between (,_0) and (_i,_i),i=1,⋯, n+1:
	
    α_i ∈__0(U_0i)⊗_i((,_0),(_i,_i)),  β_i ∈__i ⊗_0(U_0i)((_i,_i),(,_0))

	and a quiver stack  corresponding to the minimal resolution of A_n-singularity over the Novikov field Λ, whose charts are _0 and _i,i=1,⋯,n+1, that solves the isomorphism equations for (α_i,β_i):
	
    m_1,^_0,_i(α_i) = 0,       m_1,^_i,_0(β_i) = 0;
    
    		m_2,^_0,_i,_0(α_i,β_i)  = _,       m_2,^_i,_0,_i(β_i,α_i) = __i.

	In above, _0(U_0i) is the localization of _0 at the set of arrows
	{v_1^L,⋯, v_i-1^L, u_i+1^L, ⋯, u_n+1^L} for i=1,⋯,n+1 respectively. In particular, the existence of the isomorphism pairs imposes commutativity on the local affine charts of _0.



	Let A_i and A_i'  denote areas of the polygons with vertices α_i^1,u_i+1^L,⋯ ,u_n+1^L,β_i^1 and α_i^2,v_i-1^L,⋯, v_1^L,β_i^1. The isomorphism pairs are defined using normalized immersed sectors α_i^1,α_i^2 and β_i^1(see Figure <ref>), 
	
    α_i= T^-A_i(u_n+1^L⋯ u_i+1^L)^-1α_i^1+ T^-A_i' (v_1^L⋯ v_i-1^L)^-1α_i^2,   β_i = β_i^1.
 
	It's easy to check they satisfy Equations (<ref>) and <ref>.
	
	For simplicity, we assume that the area terms A_i=A_i'=0. The only difference this assumption makes is that the transition map will be defined up to a constant T^-A. However, this assumption suffices for the purpose of constructing the quiver algebroid stack. It is important to keep track of these area terms when considering the maps between Maurer-Cartan spaces.
	
	We now construct a quiver stack 𝒴̂ as follows:
	
		
  * The underlying topological space of 𝒴̂ is the dual polyhedral set  B of the fan of A_n-resolution.
		
		
		We endow B with a topology {∅,B, F_1^c,⋯, F_n+1^c}  where F_i^c is the complement of the i-th facet F_i. We take {U_i=F_0^c ∩⋯∩ F_i-2^c∩ F_i+1^c∩⋯ F_n+1^c }_1≤ i ≤ n+1 together with U_0:=B,  which serve as the open cover of the quiver stack. 


		
  * We associate each open set U_i a sheaf by localizing the quiver algebras 𝔸_0 for i=0, and 𝒜_i for 1≤ i≤ n+1. So for convenience, if there is no ambiguity we will use 𝔸_0 or 𝒜_i to denote the sheaves. Then 𝔸_0(U_0∩ U_i) is defined by localizing 𝔸_0 at {v_1^L,⋯, v_i-1^L,u_i+1^L,⋯, u_n+1^L}; on the intersection it's the algebra localized at the union of corresponding elements.  
		
		
  * The representation maps  G_i0:𝔸_0|_U_0i→𝒜_i|_U_0i for 1≤ i ≤ n+1 and G_0i are defined as 
    G_i0:=  
    			u_j^L→  v_iu_i   ∀ 1≤ j ≤ i-1 
    
    			u_i^L→ u_i   
     
    			u_j^L→ 1    ∀ i+1 ≤ j ≤ n+1  
    
    			v_j^L→ 1    ∀ 1 ≤ j ≤ i-1  
    
    			v_i^L→ v_i   
    
    			v_j^L →  u_iv_i   ∀ i+1 ≤ j ≤ n+1 
    			,   G_0i:=
    			u_i→ (u_n+1^L⋯ u_i+1^L)u_i^L(v_1^L⋯ v_i-1^L)^-1   
    
    			v_i→ (v_1^L⋯ v_i-1^L)v_i^L(u_n+1^L⋯ u_i+1^L)^-1.

		
  * The gerbe terms of the form c_0i0  is defined by  
    c_0i0(h(u_j^L))= 
    			v_1^L⋯ v_j^L     1≤ j ≤ i-1   
    
    			u_n+1^L⋯ u_i+1^L      j=i  
    
    			u_n+1^L⋯ u_j+1^L     i+1≤ j≤ n+1 
    			   c_0i0(t(u_j^L))=
    			v_1^L⋯ v_j-1^L     1≤ j ≤ i-1  
    
    			v_1^L⋯ v_i-1^L     j=i 
    
    			u_n+1^L⋯ u_j^L     i+1 ≤ j≤ n+1 
    			.
Since h(u_j^L) =t(v_j^L)  and t(u_j^L)=h(v_j^L),  we have the following identities c_0i0(h(u_j^L)) =c_0i0(t(v_j^L))  and c_0i0(t(u_j^L))=c_0i0(h(v_j^L)). 
			
			
  * We define the transition maps between i-th chart and k-th chart to be G_ik(x):=G_i0∘ G_0k(x), so the corresponding gerbe terms are always trivial.
	
	In particular, the transition map between i-th chart and (i+1)-th chart is the following:
	
    G_i,i+1=G_i0∘ G_0,i+1=
    			u_i+1→ v_i^-1   
    
    			v_i+1→ v_iu_iv_i=u_iv_i^2
    		,

	which are exactly the transition functions of the minimal resolution of A_n-singularity as a complex variety.

    An explicit computation shows the above data indeed defines a quiver algebroid stack over P. We take the composition G_0i∘ G_i0 as an example: 
	
		
  * 1≤ j ≤ i-1:  Using the substitution, v_j^Lu_j^L = u_j-1^Lv_j-1^L, we find 
    G_0i∘ G_i0(u_j^L)    =(v_1^L⋯ v_i-1^L)v_i^Lu_i^L(v_1^L⋯ v_i-1^L)^-1
        = v_1^L⋯ v_j^Lu_j^L (v_j-1^L)^-1⋯ (v_1^L)^-1
        = (v_1^L⋯ v_j^L)u_j^L(v_1^L⋯ v_j-1^L)^-1.
 
		
  * j=i: 
    G_0i∘ G_i0(u_i^L)    =(u_n+1^L⋯ u_i+1^L)u_i^L(v_1^L⋯ v_i-1^L)^-1.

		
  * i+1≤ j ≤ n+1: 
    G_0i∘ G_i0(u_j^L)    =1 
        = (u_n+1^L⋯ u_j+1^L)u_j^L(u_n+1^L⋯ u_j^L)^-1.

	
	Therefore, they satisfy the equality G_0i∘ G_i0(x) = c_0i0(h(x))· G_00(x)· c_0i0^-1(t(x)). 
	
	In the above construction, we didn't assume _i is commutative. However, The existence of representations G_0i imposes commutativity on _i. This arises because G_0i possesses a left inverse and they satisfy G_0i(u_iv_i)=G_0i(v_iu_i) for all i.  



Furthermore, there exists an isomorphism pair between the family of special Lagrangian tori (T_i, ∇^(x_i,y_i)) and (,_0) over the same quiver algebroid stack .


	There exist preisomorphism pairs between the family of special Lagrangian tori (T_i,_i:=∇^(x_i,y_i)),i=1,⋯, n and (,_0):
	
    α_i ∈__0(U_0i')⊗_i((T_i,_i),(,_0)),β_i ∈__i ⊗_0(U_0i')((,_0),(T_i,_i ))

	
	that solves the Fukaya isomorphism equations  over the quiver algebroid stack  corresponding to the minimal resolution of A_n singularity (see Theorem <ref>):
	
    m_1,^_i,_0(α_i) =    0, m_1,^_0,_i(β_i) = 0;
    
    		m_2,^_i,_0,_i(α_i,β_i) =   _T_i, m_2,^_0,_i,_0(β_i,α_i) = _,

	where U_0i':= U_0i∩ U_0(i+1), and _0(U_0i') is the localization of _0 at the set of arrows
	
    {v_1^L,⋯, v_i^L, u_i+1^L, u_i+2^L, ⋯, u_n+1^L}
 
	for i=1,⋯,n respectively. 




	In this proof, we will use the same notations for the quiver algebroid stack as in Theorem <ref> and denote the deformation space of the tori T_i by _i:=Λ_0[x_i,x_i^-1,y_i,y_i^-1].
	
	The torus T_i cleanly intersects  at two connected circles. Using Morse model, the critical points of the Morse functions generate the Lagrangian Floer complexes ((T_i,_i),(,_0)) and ((,_0),(T_i,_i)), among which the maximal points α_i^1 ∈^0((T_i,_i),(,_0)) and β_i^1 ∈^0((,_0),(T_i,_i)) have degree zero. We observe that the normalized intersection points form a Fukaya isomorphism pair: 
    α_i:= T^-A_i' (v_1^L ⋯ v_i^L)^-1α_i^1,   β_i:= β_i^1.

	
	Similar as before, we set A_i=A_i'=0. The quiver algebroid stack  obtained in theorem <ref> also provides a solution to the Fukaya isomorphism equations:
	
		
  * The underlying topological space of 𝒴̂ is the polyhedral set P of the minimal resolution of A_n singularity, which is the same as in Theorem <ref>. But in this proof, we will focus on U_0i':= U_0i∩ U_0(i+1). 
		
  * We associate each open set U_i a sheaf of path algebras as above, for i=0,1,⋯, n. In addition, we also assign the sheaf _̃ĩ associated to _i to U_0i'. 
		
  * The transition maps  G_i0:𝔸_0|_U_0i'→_i|_U_0i' and G_0i: _i|_U_0i'→𝔸_0|_U_0i' for 1≤ i ≤ n  are defined by 
    G_i0:=  
    			u_j^L→  x_i-1    ∀ 1≤ j ≤ i 
    
    			u_i+1^L→ y_i^-1   
     
    			u_j^L→ 1    ∀ i+2 ≤ j ≤ n+1  
    
    			v_j^L→ 1    ∀ 1 ≤ j ≤ i  
    
    			v_i+1^L→ (x_i-1)y_i    
    
    			v_j^L →  x_i-1   ∀ i+2 ≤ j ≤ n +1
    		,   G_0i:=
    			x_i→ (v_1^L⋯ v_i+1^L)u_i+1^L(v_1^L⋯ v_i^L)^-1 + e_1    
    
    			y_i→ (v_1^L⋯ v_i^L)(u_n+1^L⋯ u_i+1^L)^-1.
 Recall that (,_0) is weakly unobstructed. Namely, we have v_j^Lu_j^L= u_j+1^L v_j+1^L  ∀ 1 ≤ j ≤ n and u_n+1^L v_n+1^L=v_1^L u_1^L.  Hence, G_0i(x_i)=(v_1^L⋯ v_i+1^L)u_i+1^L(v_1^L⋯ v_i^L)^-1 + e_1= v_1^L u_1^L+e_1 by the weakly unobstructed equations.
		
  * The gerbe terms c_0i0 at the vertices of Q  is defined by  
    c_0i0(a_j)= 
    			e_1     j=1
    
    			v_1^L⋯ v_j-1^L    ∀ 2≤ j ≤ i+1   
    
    			u_n^L⋯ u_j^L    ∀ i+2 ≤ j ≤ n+1
 where a_j stands for the j-th vertex of Q. The other gerbe terms are trivial.
	
Notice that G_0i is well-defined in the sense that G_0i is intertwining, i.e. G_0i(x_iy_i)=G_0i(y_ix_i). We have 
    G_0i(x_iy_i)=    (v_1^L⋯ v_i+1^L) (u_n+1^L ⋯ u_i+2^L)^-1 + (v_1^L ⋯ v_i^L)(u_n+1^L ⋯ u_i+1^L)^-1.

On the other hand, 
    G_0i(y_ix_i)=    (v_1^L⋯ v_i^L) (u_n+1^L ⋯ u_i+1^L)^-1v_1^L u_1^L + (v_1^L ⋯ v_i^L)(u_n+1^L ⋯ u_i+1^L)^-1
    
    	=    (v_1^L⋯ v_i^L) (u_n+1^L ⋯ u_i+1^L)^-1 u_n+1^L v_n+1^L + (v_1^L⋯ v_i^L) (u_n+1^L ⋯ u_i+1^L)^-1
    
    	=    (v_1^L⋯ v_i^L) (u_n-1^L ⋯ u_i+1^L)^-1 (u_n^L)^-1 v_n+1^L + (v_1^L⋯ v_i^L) (u_n+1^L ⋯ u_i+1^L)^-1
    
    	=    (v_1^L⋯ v_i^L) (u_n-1^L ⋯ u_i+1^L)^-1 v_n^L (u_n+1^L)^-1 + (v_1^L⋯ v_i^L) (u_n+1^L ⋯ u_i+1^L)^-1
    
    	=    (v_1^L⋯ v_i+1^L) (u_n+1^L ⋯ u_i+2^L)^-1 + (v_1^L⋯ v_i^L) (u_n+1^L ⋯ u_i+1^L)^-1.
 On the first line, we use another expression of G_0i(x_i). On the fifth line, we apply the weakly unobstructed equations until it gets to the desired form. 

One can check that G_0i∘ G_i0(x) = c_0i0(h(x))· G_00(x)· c_0i0^-1(t(x)) for all x.  For instance, 
    G_0i∘ G_i0(u_i+1^L)= (u_n+1^L ⋯ u_i+2^L) u_i+1^L(v_1^L ⋯ v_i^L)^-1= c_0i0(a_i+2) u_i+1^L c_0i0(a_i+1).


Furthermore, we can solve the isomorphism equations for (α_i,β_i) over the quiver stack explicitly. More precisely, we get 
    m_1,^_0,_i(β_i)    = ((v_1^L ⋯ v_i^L)+ y_i(u_n+1^L ⋯ u_i+1^L))Y̅_i^1 + ((v_1^Lu_1^L+1)+x_i) β_i^2


    m_2,^_0,_i,_0(β_i,α_i)   =(v_1^L ⋯ v_i^L (v_1^L ⋯ v_i^L)^-1)_^2_1+ ⋯ + (v_i^L(v_1^L ⋯ v_i^L)^-1v_1^L ⋯ v_i-1^L)_^2_i
       ((v_1^L ⋯ v_i^L)^-1y_i(u_n+1^L ⋯ u_i+1^L))_^2_i+1 + ⋯ + ((u_n^L ⋯ u_i+1^L)(v_1^L ⋯ v_i^L)^-1y_iu_n+1^L)_^2_n+1.

Using the transition representations G_0i: _i|_U_0i'→𝔸_0|_U_0i', 

    m_1,^_0,_i(β_i)    = ((v_1^L ⋯ v_i^L)+ y_i(u_n+1^L ⋯ u_i+1^L))Y̅_i^1 + ((v_1^Lu_1^L+1)+x_i) β_i^2=0


    m_2,^_0,_i,_0(β_i,α_i)   = _^2_1 + ⋯_^2_i-1+ _^2_i + ⋯_^2_n+1= _.

The computations of the remaining isomorphism equations are similar. Hence, (α_i,β_i) forms a Fukaya isomorphism pair. 





	
	





By the above theorem, the transition maps for the Fukaya isomorphism pair (α_i,β_i) give a map between (subsets of) the Maurer-Cartan spaces of the Lagrangian branes  and special Lagrangian tori T_i.  Note that it is important to take localizations for the Maurer-Cartan space of  for existence of such maps.  Indeed, these localizations come from a stability condition.  Below, we make this more explicit.

Implicitly the Lagrangian brane (,) has rank δ⃗=(1,1,⋯,1). Let ζ∈^n+1 be a generic character such that ζ·δ⃗=0 and only ζ_1 is positive. In addition, recall that the subvariety of _0-representations in Rep(Q,δ⃗) is denoted by M(Q,δ⃗) .

Let's denote the moduli space of ζ-stable deformation of (,) by Def(), which is a subset of MC((,)). More precisely, Def() is defined as a coset with Λ_0-values


	Def():= {b ∈ M(Q,δ⃗) | b is ζ-stable }/GL(δ⃗). 

 

Let's define Λ_U:= ^×⊕Λ_+. Λ_U is a valuation zero subset of Λ which forms a multiplicative group. Besides, we will consider the following subsets of the Maurer-Cartan spaces:

    Def(S_1):=   {(u_1,v_1) ∈ (Λ_0)^2 | (u_1) ≥ A_1 }
    Def(S_j):=   {(u_j,v_j) ∈ (Λ_0)^2 | (u_j), (v_j) ≥max{|A_j-A_j'|,0} and (u_jv_j)>0} for  j=2, ⋯, n
    Def(S_n+1):=   {(u_n+1,v_n+1) ∈ (Λ_0)^2 | (v_n+1) ≥ A_n+1' }
    Def(T_i):=   {(x_i,y_i) ∈ (Λ_U)^2 |  x_i ∈ -1+ Λ_≥ |A_i-A_i'|}.



Here we use the same notations as in the quiver algebroid stack , but the variables stand for the Λ_0-valued solutions of the unobstructed equations.



	
	There exists a correspondence between the isomorphism class of ζ-stable deformation of  and the deformation spaces of special Lagrangians T_r,0. In other words, 
    Def() ≅⋃_r ∈ [a_1,a_n+1]Def (T_r,0)
 In particular, when r= a_i, T_r,0 is the immersed sphere _i for i=1, ⋯, n+1. 



	Let U_i:=U_0i be the open subsets introduced in the quiver algebroid stack . Notice that Def() can be covered by 

	 Def(,U_i):={(u^L,v^L)∈ M(Q,δ⃗)|  v_1^L,⋯, v_i-1^L, u_i+1^L, ⋯, u_n+1^L∈Λ_0^∗}/GL(δ⃗),
	
 for i=1, ⋯,n+1, because the representation is ζ-stable. Thus, given an element [b] ∈Def(), there exists some i such that [b] ∈Def(,U_i). Using the group action, [b] has a representative satisfies v_1^L=⋯=v_i-1^L=u_i+1^L=⋯= u_n+1^L=1. In the following, we will use the restriction of transition maps, found in Theorem <ref>, to give a correspondence between the isomorphism classes of Maurer-Cartan subspaces.
	
	First, since (u_i^L v_i^L)>0, Def(,U_i) ∩Def(,U_j)= ∅ for j ≠ i-1,i,i+1.   We can define
	the correspondence as follow. Given [b] ∈Def(,U_i) ∖ (Def(,U_i-1) ∪Def(,U_i+1)) for i=1, ⋯, n+1, the correspondence G is defined as the restriction of G_i0: 𝔸_0|_U_0i→𝒜_i|_U_0i, where 𝒜_i is the formal deformation space of _i. Namely,  
    G_i0:=  
    		u_j^L↦ v_iu_i   ∀ 1≤ j ≤ i-1 
    
    		u_i^L↦ T^A_i'-A_iu_i   
     
    		u_j^L↦ 1    ∀ i+1 ≤ j ≤ n+1  
    
    		v_j^L↦ 1    ∀ 1 ≤ j ≤ i-1  
    
    		v_i^L↦ T^A_i-A_i'v_i   
    
    		v_j^L ↦  u_iv_i   ∀ i+1 ≤ j ≤ n+1 
    	,    G_0i:=
    		u_i↦ T^A_i-A_i'(u_n+1^L⋯ u_i+1^L)u_i^L(v_1^L⋯ v_i-1^L)^-1   
    
    		v_i↦ T^A_i'-A_i(v_1^L⋯ v_i-1^L)v_i^L(u_n+1^L⋯ u_i+1^L)^-1.


For [b] ∈Def(,U_i) ∩Def(,U_i+1), the correspondence G is defined as the restriction G_i0:𝔸_0|_U_0i'→_i|_U_0i', where _i is the formal deformation space of T_i for i=1,⋯,n. This gives a well-defined map from Def() to ⋃_r ∈ [a_1,a_n+1]Def (T_r,0). Notice that G_0i∘ G_0i= id, G_0i∘ G_i0(a)= c_0i0(h(a)) · a · c_0i0^-1(t(a)) and the gerbe terms are elements in GL(δ⃗) when considering Λ_0-values deformations. Hence, G_0i and G_i0 descend to well-defined mutually inverse representations. This provides the desired correspondence. 



	The transition map, as constructed in Theorems <ref> and <ref>, identifies the ζ-stable deformations of  with special Lagrangian tori on the moment map level zero with a flat connection. By restricting to complex-valued deformations, the moduli space of ζ-stable Lagrangian branes (,) forms a minimal resolution of A_n-singularity.  If we enlarge the deformation space of  over Λ, the transition map identifies the ζ-stable deformations of  with special Lagrangian tori in other levels as well.







 §.§ Extended Lagrangian deformations and derived Nakajima schemes


In <cit.>, the noncommutative mirror functor was extended to the space of formal deformations in general degrees for a Lagrangian immersion.  The resulting mirror is a curved dg algebra (,d,W), where  is a quiver algebra that has arrows in non-positive degrees (while the quiver appeared in previous sections has arrows in degree 0), W ∈ is the extended disc potential, and d: → is a derivation of degree 1, which satisfy the equalities 

    dW = 0  and  d^2 = [W,·].


We apply this construction to the two-dimensional framed Lagrangian immersion in Section <ref>.  Our situation is graded, and hence W̃=0.  The result is similar to <cit.> for CY3 Ginzburg algebras.  In <cit.>, dg algebras were constructed for Legendrian cohomology.  In our situation, the Lagrangians are framed.  Moreover, constant disc bubbles with more than two inputs of immersed sectors occur in the dimension-two situation, and we have to perform a change of coordinates as in Section <ref> to standardize the resulting dg algebra.

Recall that the dg algebra is constructed from the generators of ^∙(^fr) in all degrees, other than the unit of each component.  In our case, we have the degree one generators X_a, X_a̅, I_v, J_v, and the degree two generators T_v (the minimal points of the Morse functions on the sphere components).  Note that the Morse function on the framing ^2 only has a single maximal point and does not have any degree two generators.  The resulting quiver Q̃ is the framed framed Nakajima quiver added with one self loop θ_v of degree (-1) at each unframed vertex v.  Figure <ref> shows Q̃ in the case of ADHM quiver.



Let

    b̃ = ∑_a (x_a X_a + x_a̅X_a̅) + ∑_v t_v T_v

be the formal deformations.  Formally b̃ is made to be degree one by setting x_a =  x_a̅ = 0 and t_v = -1.  
Then m_0^b̃ is in degree 2 and takes the form

    m_0^b̃ = ∑_v p_v T_v.

Note that _L_v and X_a,X_a̅ do not appear in the above expression by degree reason.  Moreover, p_v must have degree 0 and hence are series only in x_a,x_a̅,i,j.

The derivation d is defined by d x_a = d x_a̅ = 0 and d t_v = p_v, and then extended by Leibniz rule.  The A_∞ equations imply that(ΛQ̃, d) is a dg algebra.

The only non-trivial term dt_v = p_v counts Floer contributions to the minimal point of the Morse function in each sphere component.  Its expression follows from definition and the change of coordinates in Section <ref>.  Thus we can summarize as follows.

 
	Let ^fr be the framed Lagrangian immersion as in Section <ref>, whose (possibly immersed) sphere components are equipped with perfect Morse functions.   Then its extended localized mirror is (Q̃,d), where Q̃ is the corresponding framed Nakajima quiver with one more self loop t_v of degree (-1) attached to each unframed vertex v; 
	
    d: ΛQ̃→ΛQ̃
 
	is the differential, which sends the degree-zero generator x̃_a,x̃_a̅, i_v,j_v of the path algebra to 0 and
	
    d t_v = ∑_h(x̃_a)=vϵ(a) x̃_a x̃_a̅ + i_vj_v

	where x̃_a, x̃_a̅ are given in Equation (<ref>) as a change of coordinates on x_a,x_a̅.


In <cit.>, the dga (Q̃,d) was used as an explicit cofibrant resolution of the affine Nakajima variety and serves as a derived representation scheme model.  In a general dimension d, (2-d)-shifted symplectic structures <cit.> are important in this subject.














alpha 

