The Weil Correspondence and Universal Special Geometry
Sergio Cecotti

Identifying the ground state phases by spin-patterns in the Shastry-Sutherland model
    Hong-Gang Luo
    Received March XX, 2024; accepted XX XX, 2024
====================================================================================










§ INTRODUCTION AND OVERVIEW


In this paper “special geometry”
stands for the holomorphic integrable system[ A holomorphic integral system is an even-dimensional complex manifold 𝒳 equipped with a non-degenerate, holomorphic, closed (2,0)-form Ω. In our set-up 𝒳 is in addition algebraic.] (𝒳,Ω)
associated to the Seiberg-Witten geometry of a 4d =2 QFT <cit.>. To avoid
special cases (which require a slightly different treatment)
we make once and for all the assumption that no subsector of the QFT has a weakly coupled
Lagrangian formulation.[ This is a “rigidity” condition. It is equivalent to the requirement that the geometry is a deformation of a ^×-isoinvariant special geometry with no ^×-isoinvariant deformations.] 
The basic message of this paper is that the traditional approach to special geometry does not completely capture the richness of the theory, and fails to address several issues,
while there is a more elegant geometric framework where these questions are fully dealt with (at least in principle).



There is an on-going program <cit.> to classify all existing =2 SCFT
using special geometry. The holy grail of the program is to construct all =2 SCFTs
with no stringy or Lagrangian construction (or rule out their existence).
While partially successful, this geometric program suffers from a serious drawback: to a given
scale-invariant special geometry there correspond several inequivalent SCFTs with distinct flavor
symmetry groups, central charges, etc. To complete the program one needs
 a geometric understanding of the set of SCFTs described by a 
 a given scale-invariant special geometry. The traditional viewpoint gives no clue on this issue. There are many other features of SCFTs with no geometric intepretation in that set-up. To make a simple example:
 from QFT we know that the relevant chiral
deformations of a SCFT are in one-to-one correspondence with the elements of the chiral ring
of dimension 1<Δ<2. How do we understand this elementary fact in purely geometric terms?
We also know that the mass deformations span the Cartan subalgebra of the flavor symmetry group F.
What is the geometric interpretation of these deformations, and how we may use them to determine geometrically the allowed F's? In this direction an open problem is to determine the maximal  rank f_max of the flavor group for a rank-r SCFT. What is f_max geometrically?



Going to details, there are many aspects of the standard story
which don't look fully satisfactory. One is the Seiberg-Witten differential λ. The traditional viewpoint is that λ is a meromorphic differential
on 𝒳
whose periods are the (local) special coordinates. This looks not to be the complete story on two counts:


  (a) for a SCFT λ is more properly an infinitesimal
symmetry of the special geometry 𝒳. An infinitesimal symmetry of 𝒳
is given by a holomorphic vector field ∈𝔞𝔲𝔱(𝒳). In a symplectic manifold 
vectors and 1-forms are equivalent, and we can write our infinitesimal symmetry as the 1-form λ≡ι_Ω,
hiding its role as the generator of the connected component of 𝖠𝗎𝗍(𝒳).[ We stress that since λ is the generator of 𝖠𝗎𝗍(𝒳)^0, it 
is canonically determined by the underlying symplectic manifold (𝒳,Ω) and not an additional datum.]
 Away from conformality the relation of λ with the geometric symmetries seems to be lost, while one suspects that it must hold in general in the appropriate sense.

  (b) In general one has[ This equation is meant in the sense of cohomology of currents. Note that the (2,0) form Ω represents a (1,1) class not a (2,0) one.] 

    [Ω]=[dλ]=∑_a m_a D_a

where the D_a are polar divisors of λ and the coefficients (residues) m_a are mass parameters.
 It has been observed by Ron Donagi <cit.> that this expression agrees with the
Duistermaat-Heckman (DH) formula <cit.>: the cohomology class of the symplectic form depends linearly on the values of the momentum map. Now, while the parameters m_a are, in a broad sense, momentum maps, the DH
theorem refers to a particular geometric situation namely the Marsden-Weinstein-Meyer (MWM) symplectic quotient <cit.>. In the usual formulation of special geometry the manifold 𝒳 is not a MWM quotient of some bigger integrable system, so something looks missing from the picture.  


Taking seriously Donagi's suggestion, one concludes that there should exist a much bigger integrable system in the form of a universal special geometry (𝒴,Ω) of which the ordinary one (𝒳,Ω) is  a symplectic reduction. The existence of the universal geometry also follows from standard physical considerations, see the paragraph below
about the physical interpretation via spurions.
In the spirit of item (a) one expects that in the universal geometry the differential λ
(or rather its dual vector field ) generates holomorphic automorphisms of 𝒴, so
λ must be canonically determined by the geometry of 𝒴. In particular the polar classes
[D_a] are uniquely fixed by (𝒴,Ω). Since we can read
the flavor symmetry group F from the [D_a]'s, going to the universal geometry (𝒴,Ω) should also
solve (in principle) the issue of a geometric
understanding of flavor symmetry.

The story is a bit more complicated since, in addition to the mass deformations (associated to the flavor symmetry), we have the relevant couplings that also require a geometric interpretation.
They should also be determined
by the universal symmetry ∈𝔞𝔲𝔱(𝒴) of the bigger geometric framework.
In other words, there must be an “universal” geometric theory which encompasses all kinds of physically allowed interactions
and singles out the ones which are consistent at the full quantum level. 



Our proposal for the solution of these issues (and many others) is based on an old mathematical gadget: the “Weil correspondence” that we shall review in a moment after some preliminary. 




  
The fields of definition 𝕂 and 𝕃.
We make the observation that an ordinary special geometry 𝒳 is, in particular, 
a complex model of an Abelian variety (or scheme) X_𝕂 defined over the field 𝕂=(u_1,…,u_r)
of rational functions in r variables (r being the rank i.e. the dimension of the ordinary Coulomb branch 𝒞). This assertion contains a mild but crucial assumption that we now make explicit.
In all known =2 SCFT the Coulomb branch[ Here we look at the Coulomb branch as a mere complex manifold, stripped of all other structures such as metric etc.]
𝒞 is a copy of ^r. However, as pointed out in ref.<cit.>, there are reasons to believe that more general Coulomb branches are also allowed. Since the Coulomb branch 𝒞
is the spectrum of the chiral ring ℛ, 
in any quantum-consistent =2 SCFT 𝒞  is
an irreducible, reduced, normal, affine (complex) variety with an algebraic ^×-action (see section 2 for more details). 
 Which (possibly singular) affine varieties are the Coulomb branch 𝒞 of some physically sound =2 QFT?


The allowed 𝒞's are severely restricted. If you look to a reasonable-looking candidate Coulomb branch ≄^r you almost certainly end up with the conclusion that the dimension of the deformation space of the special geometry is not equal to the number of chiral operators of dimension ≤2 as required
by QFT (and true for 𝒞≃^r <cit.>). The “experimental” evidence indicate that in each dimension r
there are few putative Coulomb branches
consistent with this constraint from the dimension of the deformation space; all of them are expected to have the same field of functions[ In particular in rank-1 there is only one possible Coulomb branch: the plane  <cit.>.] (𝒞)≡(u_1,…,u_r). In this paper we make this empirical suggestion into an assumption.
Under this assumption, while the geometry of the Coulomb branch may be rather subtle, the subtleties are confined in codimension at least two,[ The singularities, if present, are in codimension ≥2 since we are free to assume 𝒞 to be normal by replacing the chiral ring ℛ with its normalization, a procedure which is also suggested by QFT considerations.] while the aspects we are interested in (at this early stage of the program) are typically codimension-1 phenomena.

Since we are mainly interested in codimension-1 issues, we don't bother which specific model of the Abelian variety X_𝕂 yields the actual special geometry 𝒳. At this stage we can work with X_𝕂
which is simply a commutative group (defined on the slightly fancier field 𝕂) and get a lot of mileage from algebraic group theory <cit.>.

However the focus of this paper is not the usual special geometry 𝒳,
 but the much bigger universal integrable system 𝒴. Just as we may replace 𝒳
 with the underlying commutative group X_𝕂 defined over the function field 𝕂=(𝒞), we may replace 𝒴 by a commutative algebraic group Y_𝕃
 defined over the bigger function field 𝕃=(ℬ) which is an extension of
 𝕂 of transcendental degree equal to the complex dimension of the space of mass and relevant couplings.  The group-(scheme) Y_𝕃 is our main object of interest.




  
The Seiberg-Witten differential λ.
In the algebraic group language the Seiberg-Witten differential λ is a differential on X_𝕂
defined over 𝕂. From the QFT point of view λ depends on three 
distinct kinds of variables:[  There are no marginal couplings under our
assumption that theory has no Lagrangian subsectors.]
the Coulomb coordinates (u_1,…,u_r),
the chiral couplings (t_1,…,t_k),
and the mass parameters (m_1,…,m_f). The geometric distinction between the three kinds of parameters is as follows:

    ∂λ/∂ u_i   ∈{differentials of the first kind on X_𝕂}
    ∂λ/∂ t_j   ∈{differentials of the second kind on X_𝕂}/{first kind + exact differentials}
    ∂λ/∂ m_a   ∈{differentials of the third kind on X_𝕂}/{second kind + exact differentials}

The parameters u_i, t_j and m_a exhaust the supply of differentials, so in λ there is no room for couplings of fancier kinds.
This is consistent with the superspace construction of =2 QFT. 
We shall see below that, in addition, only chiral couplings consistent with quantum UV-completeness
are allowed in special geometry.

Eq.(<ref>) suggests that we should treat the parameters u_i, t_j and m_a
on the same footing. Therefore we introduce the universal branch ℬ≃𝒞×𝒫
with global[ To make the masses m_a into global coordinates we may need to go to
a finite cover of the physical coupling space.] coordinates u_i, t_j, m_a.
We write 𝕃=(ℬ)≃(u_i,t_j,m_a) for the rational field in r+k+f
variables. Over ℬ we have the universal Abelian variety
𝒜[r]^(0.3)ϖ   ℬ≃𝒞×𝒫,
i.e. the family of ordinary special geometries parametrized by the coupling space 𝒫.
Again, 𝒜→ℬ is a model of an Abelian variety A_𝕃 defined over 𝕃.
The Seiberg-Witten differential λ is then a third-kind differential on A_𝕃
defined over 𝕃.
 The ordinary special geometry at fixed couplings p≡(t_j, m_a)∈𝒫 is 𝒳≡𝒜|_𝒞× p.






  
The Weil correspondence. There are many math gadgets known under the name “Weil correspondence”. The one of interest in this paper is perhaps less known. To put it in the proper perspective, we quote from the book by Mazur and Messing <cit.>:[ The references in the quote 
insides the oval box are: Weil paper <cit.>; Barsotti paper <cit.>, Serre paper <cit.>
and Serre book <cit.>.] 
12pt

460pt
In [27] Weil observed that when working on abelian variety A over an arbitrary field, considerations of extensions of A by a vector group replaces the study of differentials of the second kind, while considerations of extensions of A by a torus replaces the study of differentials of the third kind.
 He attributes these ideas (in the classical case) to Severi.
 Over ℂ, Barsotti in [1 bis] established algebraically the isomorphism-6pt

    𝖤𝗑𝗍(A,𝔾_a)≅differentials of second kind/holomorphic differentials+exact differentials

(See Serre's [24] and [25] for a beautiful account of these ideas)


16pt

Comparing this quotation with eq.(<ref>) we conclude that we may identify 
the couplings allowed by =2 supersymmetry 
with commutative algebraic groups. In particular the ordinary Seiberg-Witten geometry (𝒳,λ) (with mass and relevant deformations switched on)
defines (and is fully determined by) an algebraic group Z_𝕂
 over 𝕂.
To physicists group means
symmetry, and we think the differential λ as describing a symmetry of the geometry. 

Again, to get a more intrinsic and satisfactory picture one has to work with the appropriate commutative extension Y_𝕃
of the universal Abelian group-variety A_𝕃 defined over the field 𝕃.
The physical meaning of Y_𝕃 will be clarified below.
 



  
Anti-affine (quasi-Abelian) groups.
The Weil correspondence replaces the issue of a geometric description of the allowed physical couplings (in particular mass deformations) with the more geometric question of understanding the allowed symmetries.

However not all commutative algebraic groups Y_𝕃 are allowed for the following reasons. 
Consider the mass deformations of a SCFT. Physically they are associated to the flavor Lie group F: the mass parameters take value in the Cartan algebra 𝔠𝔞𝔯(F) of F. 
The deformation space 𝔠𝔞𝔯(F)
comes with a number of specific structures: an action of the Weyl group, an invariant bilinear form, and
 an integral lattice. In addition, we expect that the rank of F is bounded in each fixed rank r≡𝒞. Generic torus extensions do not come with such structures, and their rank has no upper bound.
Hence a general extension is not allowed.

The case of relevant couplings is even sharper.
According to the Weil correspondence,
a SCFT with the relevant couplings switched on (but no mass deformation) corresponds to a group
Y_𝕃 which is a
vector extension of the universal Abelian variety A_𝕃. An Abelian variety has extensions by vector groups of arbitrary dimension. At the classical level the =2 SCFT has indeed infinitely many linearly independent chiral deformations, but almost all of them are inconsistent at the quantum level because they spoil UV completeness. The deformations which are consistent at the full non-perturbative level
are in one-to-one correspondence with the chiral operators of the undeformed theory with scaling dimension 1<Δ<2, in particular their number is at most r. This leads to the


Which algebraic groups Y_𝕃 over 𝕃 are the Weil correspondents
of =2 QFTs which are fully consistent at the quantum level (UV complete)?


The fact that all 4d =2 QFT can be twisted á la Witten <cit.> into a Topological Field Theory (TFT), 
leads to the following (see main text): 

 The algebraic group
Y_𝕃 must be anti-affine (a.k.a. quasi-Abelian), that is,

    Γ(Y_𝕃,_Y_𝕃)≃𝕃.


Here _Y_𝕃 is the structure sheaf of the algebraic variety Y_𝕃 defined over 𝕃.
In this paper we check that (<ref>) exactly matches the conditions from quantum consistency of the QFT. 



  
Universal special geometry. Up to now we just considered (families of) ordinary special geometries, but this is clearly not the full story as the Donagi remark in item (b) indicates. 
There must be a bigger algebraic integrable system with underlying algebraic group Y_𝕃.
The dimension (over 𝕃)
of Y_𝕃 is the total number of parameters r+k+f. A model over  of Y_𝕃 will be a
fibration 
 
    π𝒴→ℬ≃𝒞×𝒫,
   
 where the fibers and the base ℬ  both have complex dimension r+k+f. We claim that 
 the total space 𝒴 is a “symplectic variety” with symplectic form Ω,
 while the fibers of π are Lagrangian submanifolds. 
 We write “symplectic variety” between quotes because we do not have control on singularities in
 codimension ≥2 and it is possible (even expected) that for some =2 QFT these singularities have no crepant resolution.[ A crepant resolution is automatically a symplectic variety.] 

 
Let us make the claim more precise. The Lagrangian fibration  (<ref>) has a ^× group
of automorphisms. Let  be the Euler vector field, that is, the generator of the Lie algebra
𝔞𝔲𝔱(𝒴) normalized so that

    ℒ_Ω=Ω.
 
The dual 1-form λ=ι_Ω is then the universal differential which is holomorphic and canonically defined by the symmetries of the geometry. From (<ref>) dλ=Ω.

The Lagrangian fibration (<ref>) is our universal special geometry. The only difference with respect to an ordinary special geometry is that now the generic (smooth) fiber is a general quasi-Abelian variety instead of an Abelian variety. 



  
Symplectic quotients and SW differentials. The Lagrangian fibration (<ref>) is an (algebraic) Liouville integrable system
whose smooth fibers may be non-compact. Its Hamiltonians in involution are the regular functions on the affine base
ℬ. We can perform the symplectic quotient at fixed values t_j,m_a of the couplings t_j and masses m_a setting

    𝒞={(u_i,t_j,m_a)∈ℬ t_j=t_j, m_a=m_a}⊂ℬ,
    𝒳≡π^-1(𝒞)/H→𝒞,

where H is the group generated by the vector fields dual to the differentials dt_j's and dm_a's.
The reduced fibration 𝒳→𝒞 is an ordinary special geometry over the ordinary Coulomb branch 𝒞 whose
symplectic form class [Ω] depends linearly on the parameters t_j, m_a (in facts only on the masses m_a) according to the Duistermaat-Heckman formula <cit.>.
More in detail: on 𝒴 we have the God given universal Euler differential λ
which induces a differential λ on the symplectic reduction 𝒳
such that dλ=Ω. The differential λ arising from the symplectic quotient
 coincides with the usual
Seiberg-Witten differential: in this framework the Seiberg-Witten differential is constructed out of the symmetries of the problem. The construction of λ is very explicit:
the basic tool is Picard's construction of anti-affine groups. It is easy to check in the examples that λ is the physically correct differential.

The reader may be puzzled. The universal differential λ is perfectly holomorphic. How it happens that λ has now become meromorphic?. The point is that all group extension is
a principal bundle, and to write explicit expressions we need to choose a gauge on this bundle. Just as for the magnetic monopole in ^3, if you insist to use a single coordinate chart you are forced to pick up a singular gauge, and the gauge connection A will look singular in that gauge; but the singularity is a mere gauge artifact: in a regular gauge it looks perfectly regular.
The Seiberg-Witten differential is singular (in presence of non-trivial couplings)
just because we write it in a (convenient) singular gauge. 



  
Physical interpetation: Spurions. Finally we give the physical motivations for the
extension of the special geometry from the ordinary version 𝒳 to the universal one
 𝒴. From the viewpoint of the superspace approach to SUSY
QFTs, we can always see the couplings as =2 chiral superfields which are frozen to their
constant vev's. Such non-dynamical superfields are usually called spurions.
In particular the masses may be thought of as complex scalars in SYM superfields which weakly gauge the flavor symmetry F: physically we may see the flavor symmetry as the zero-coupling limit of a gauge symmetry.
In other words, to switch on the mass deformations we gauge the Cartan subgroup[ Properly speaking one has to gauge the full group F; however in the λ→0 limit the difference becomes inessential.] of F with rank F spurion vector supermultiplets,
adding a kinetic term for them

    1/g_f^2∫(-1/4F_μν  a F^μν  a+∂_μ M_a^* ∂^μ M_a+fermions)d^4x,

and then send g_f→0 while keeping fixed ⟨ M_a⟩=m_a.
In the same way the relevant couplings t_j can be seen as =2 chiral superfields T_j
which are frozen to constant values 
by rescaling their kinetic terms[ The T_i's kinetic terms are irrelevant operators; the situation is
rather similar to the case of (2,2) Landau-Ginzburg in two dimensions <cit.>.] by an overall factor  1/ϵ^2 and then sending ϵ→ 0
while keeping fixed ⟨ T_j⟩ =t_j. The interaction with couplings t_j is written
as an integral over the =2 chiral superspace
∑_j∫d^4 x d^4θ  T_j  ϕ_j+h.c.  ∑_j t_j∫d^4 x d^4θ  ϕ_j+h.c.
where the ϕ_j's are the chiral superfields whose first components are the operators u_j∈ℛ
of dimension 1<Δ_j<2. 


If we add only vector spurions, at finite g_f we have just an ordinary[ More precisely the germ of an ordinary special geometry since after the gauging of the flavor group the theory is typically non UV-complete. Since this is a physicists' argument, we dispense the reader with too many technical pedantries.] special geometry
with a Coulomb branch ℬ of dimension r+f. Sending g_f→0 has two effects:


  (1) it friezes the Coulomb coordinates associated to the flavor group to the values m_a≡⟨ M_a⟩,
thus replacing ℬ with the  subvariety 𝒞={m_a=m_a} i.e. with the ordinary
Coulomb branch;

  (2) the (generic) fiber over 𝒞 degenerates from an Abelian variety of dimension r+f
to a semi-Abelian variety (≡ a torus extension of an Abelian variety). A semi-Abelian variety arises as a semi-stable degeneration of an Abelian variety, so (in a sense) this is the mildest possible degeneration for an ordinary special geometry. 

The situation with relevant coupling is similar except that the dimension of the corresponding operator is Δ >1 so the fiber degeneration is more severe and we get a vector extension (an unstable degeneration). 

Before sending g_f,ϵ→ 0 both the masses m_a and the chiral couplings t_j
are generators of the universal chiral ring 𝒬. In the limit these generators
are replaced by complex numbers, and 𝒬 specializes to the usual chiral ring
ℛ. The base ℬ of the extended geometry is the affine variety 
𝖲𝗉𝖾𝖼 𝒬.




  
Open problems. The Weil correspondence solves many issues
but it also opens new questions. The most important one is to find the maximal rank f(r)
of the flavor group F as a function of the dimension r of the ordinary Coulomb branch.
The overall picture is roughly as follows. The projective closure of (a suitable model of)
X_𝕂 is expected to be a Fano variety  (possibly singular) of dimension 2r
with the property that ∖ K is symplectic  (K a canonical divisor). E.g. for r=1
this condition says that  is a rational elliptic surface <cit.>.
The rank of the flavor group is related to the Picard number of . Hence
the question boils down to an optimal upper bound on the Picard numbers of Fano varieties of dimension 2r
with ∖ K symplectic and only physically admissible singularities.






  
Organization of the paper. The rest of the paper contains detailed explanations, computations, and technicalities
making the above picture more precise. In section 2 we review special geometry from our
abstract viewpoint. In section 3 we discuss the anti-affine algebraic groups, study Lagrangian-fibrations whose general fibers are anti-affine groups, and describe their physical interpretation.
In section 4 we introduce the Mordell-Weil lattices and their Néron-Tate height pairing. We then compare the resulting structures with the physics of flavor symmetry. In section 5 we compute the Seiberg-Witten differential using the Weil correspondence and the Picard explicit construction of quasi-Abelian groups.   



§ GENERAL SPECIAL GEOMETRY


General special geometry may be summarized as follows.
We have a polarized, normal algebraic variety 𝒴 over  with a
holomorphic symplectic 2-form Ω. “Polarized” means equipped with an integral  class ω∈ H^2(𝒴,)∩ H^1,1(𝒴) containing a Kähler metric. The crucial property
is that the affinization morphism 

    π𝒴→𝖲𝗉𝖾𝖼 Γ(𝒴,𝒪_𝒴)≡ℬ

is a fibration with (connected) Lagrangian fibers, hence 𝒴 is an algebraic Liouville integrable system whose
first integrals of motion in involution are precisely all the regular functions on the `phase space' 𝒴. The (holomorphic) Hamiltonian vector fields
generate an action G↷𝒴
of a connected commutative group G and π is 
its momentum map. Thus
The relevant geometries are (in particular) algebraic integrable systems whose
momentum map μ coincides with the affinization morphism π.

We write 𝒬≡Γ(𝒴,𝒪_𝒴) and call it the universal ring. The fiber 𝒴_b over a generic point b∈ℬ
is smooth. The locus of points b∈ℬ with non-smooth fiber, iff non-empty, is a divisor 𝒟⊂ℬ called the discriminant. 
When π has a section, eℬ→𝒴,
the smooth locus of a fiber 𝒴_b⊂𝒴_b is an algebraic group
whose connected component has the form G/G_e(b), where G_e(b)⊂ G is the discrete isotropy subgroup. e(b) is then the neutral element of the group 𝒴_b^0, and e is called the zero-section.
We take the existence of a zero-section such that e(ℬ)⊂𝒴 is Lagrangian as part of our definition of special geometry; for more general situations see <cit.>.  Then 𝒴 may be seen as a polarized
commutative group-scheme over the affine scheme ℬ.

The special geometries of interest in this paper
have, in addition, an action ↷𝒴 by conformal-symplectic automorphisms such that
as Ω↦ e^z Ω (z∈). The quotient group which acts faithfully (and algebraically) is a copy
of the multiplicative group ^×≃/2π i m. 
The Lie algebra of ^× is generated by a Euler vector field  such that
ℒ_Ω=Ω
to which it corresponds a differential λ=ι_Ω
with dλ=Ω. 

The smooth fibers 𝒴_b are connected commutative group-varieties over .
The universal cover of all such group-varieties is ^n.
Then, analytically, all smooth fibers 𝒴_b can be written as ^n/Λ_b for some discrete subgroup Λ_b⊂^n. If (x^1,⋯, x^n) are affine coordinates in the covering ^n,
the Euler differential λ  locally takes the Darboux form

    λ=∑_i p_i  dx^i  ⇒  Ω=dp_i∧ dx^i
 
for suitable local functions p_i in ℬ with ℒ_1mu p_i=p_i.






We take the above statements as our basic assumptions (“axioms”).
 



  
Physical justifications. 
All the statements above may be justified (or at least argued)
from the very first principles of quantum physics. In particular the 
equality of the momentum map and the affinization map, eq.(<ref>), follows
from the fact that an =2 QFT can be twisted into a Topological
Field Theory (TFT) á la Witten <cit.>. After the twist, the TFT has the following properties:


  (1) the IR effective theory is exact for topological amplitudes;

  (2) the TFT amplitudes on ^3× S^1 are independent
of the radius R of the circle. As R→∞ we can use the 4d IR effective theory
which is the usual Seiberg-Witten description with Coulomb branch ℬ <cit.>. In the limit R→0 we can use the 3d IR effective theory which is
a σ-model with the hyperKähler target (𝒴,Ω) <cit.>. The results of the two computations should agree,
in particular we must have the same ring of local topological observables 𝒬 in both descriptions.
This is eq.(<ref>).  
 

The basic idea is that we may twist the =2 theory obtained by
promoting the couplings to spurion superfields. For instance, we can see the masses
as the result of very weak gauging of the flavor symmetry and topologically twist this weakly gauged theory. As the gauge coupling goes to zero the Abelian  fibers of the integrable system degenerate into semi-Abelian ones in the well-known way.
 


Abstract special geometry (as defined above) captures all sectors of the =2 model except
for free hypermultiplets which are decoupled from the rest of the theory and hence
do not talk with the vector multiplets which parametrize the special geometry.



This paper is dedicated to understanding
the structure of the geometric objects satisfying the “axioms”.
We start by a review of the ordinary Seiberg-Witten geometries.



 §.§ Review: the case of proper fibers


In all abstract special geometries a generic fiber
𝒴_b is smooth, hence a complex algebraic group with 
 the property that it
has no non-constant regular functions (cf. (<ref>)). The obvious way to get rid of all non-constant
regular functions is to take the
fiber to be compact (proper). In this case the generic fiber 𝒴_b is an Abelian variety
with polarization ω|_𝒴_b. In most applications one assumes the polarization to be principal, but the story makes sense (geometrically as well as physically) for polarizations of any degree. When the generic fiber is proper we use the standard notations and terminology: we write ℛ (resp. 𝒞)
for 𝒬
(resp. for ℬ) which we call the chiral ring
(resp. Coulomb branch); the total space of the geometry will be written 𝒳
with symplectic form Ω.  The dimension r of 𝒞 is the rank of the geometry.





  §.§.§ Superconformal geometries


As already mentioned, we are particularly interested in special geometries whose automorphism group contains ^×. They describe =2 SCFTs where all mass and relevant deformations are switched off.
The Lie algebra of the automorphism group ^× is generated by a complete holomorphic vector field  (the Euler vector) such that

    ℒ_1muΩ=Ω

 which implies

    Ω=dλ_  where  λ_≡ι_Ω.
 
The Euler differential λ_ is the Seiberg-Witten (SW) differential
for SCFT geometries.



The ^×-action on the regular functions induces a ^×-action on 𝒞.
A Hamiltonian h∈ℛ has dimension Δ(h) iff ℒ_ h=Δ(h) h. The identity has dimension 0,
 all other elements of h∈ℛ must have dimension 
 
    Δ(h)≥1

 this fact is known as 
 the unitary bound (see below).  ℛ is then a graded ring of the form

    ℛ=· 1⊕ℛ_+,          ℛ_+=⊕_1≤Δ∈1/nℕℛ_Δ,
       ℛ_Δ·ℛ_Δ^'⊂ℛ_Δ+Δ^'       h∈ℛ_Δ ⇔ ℒ_ h=Δ h,

where we used that n Δ∈ℕ for some integer n since ^× acts algebraically. 
It is known that only finitely many n may appear for a given r <cit.>. The maximal ideal ℛ_+⊂ℛ corresponds to the closed point 0∈𝒞, called the origin, which is the only closed ^×-orbit in 𝒞.
 If we require 𝒞 to be smooth at 0, we get that ℛ≃[u_1,…,u_r]
 is a free polynomial ring, hence 𝒞 is ^r with coordinates u_i
 of dimension Δ_i≥1. However 𝒞 may be singular (cf. Question 1).
 We write ∂_φ^i for the Hamiltonian vector v_u_i. The dual differentials dφ^i
have dimension 
Δ(dφ_i)=-Δ(v_u_i)=1-Δ_i.
Let 𝒳_u be a smooth fiber. Consider the exponential map

    exp_e(u)-5mu(z^i ∂_φ^i)𝖫𝗂𝖾(𝒳_u)≡ T_e(u)𝒳_u⟶𝒳_u,     u∈𝒞.

 and let K_u be its kernel. Clearly

    𝒳_u≃𝖫𝗂𝖾(𝒳_u)/K_u,  that is,  φ^i∼φ^i+ A(u)^ij(m_j+τ(u)_jk n^k),   m^j,n^k∈,

 where τ(u)_ij is the period matrix of the Abelian fiber at u, and 
 ℒ_2muτ_ij=0
 since  acts by automorphisms. The formula (<ref>) holds also when 𝒳_u is not smooth,
 except that 𝖫𝗂𝖾(𝒳_u)/K_u should be identified with the connected component of the smooth locus of the fiber containing e(u).
 Now, 

    ℒ_  A^ij =(1-Δ_i)A^ij  for all j.


 

  
Unitary bound. Let us show eq.(<ref>). All complete vector v satisfies ℒ_ v =Δ(v) v with Δ(v)≥0, while
 for the Hamiltonian vector v_h such that ι_v_hΩ=dh≢0 one has
 
    0≤Δ(v_h)≡Δ(h)-1.
 
The elements u_i∈ℛ which saturate the unitary bound (<ref>), i.e. such that Δ_i=1, have special properties. From eq.(<ref>) we see that Δ_i=1 implies that A^ij is constant in the closure of any ^×-orbit,
 hence constant in 𝒞 because 0 belongs to the closure of all orbits. Since the connected component of the smooth locus of the fiber over 0
must be a group, it follows that
the fiber over 0 contains an Abelian variety of dimension equal to the multiplicity of 1 as a Coulomb dimension, which then is a constant Abelian subvariety B contained in
 all fibers. This Abelian subvariety, describes a free sub-sector which we may decouple without loss. 
 Conversely a constant Abelian subvariety B contained in all the fibers describes a free sector and B=ℛ_1.
 Indeed let A_α, B^β be a symplectic basis of H_1(A_0,). The periods
 a^α=∫_A_α ι_Ω,    a^D_β=∫_B^βι_Ωare global regular functions on 𝒞 of dimension 1.[ Notice that the free subsector has no non-trivial mass or relevant deformations.]
  
 We conclude that we may assume with no loss that the chiral ring has
 the form [u_1,…,u_r], where r≥1 and Δ_i>1 for all i.
 In this situation there is a non-zero divisor 𝒟⊂𝒞, called the discriminant, consisting of the points u∈𝒞 whose fiber 𝒳_u is not smooth. The complement 𝒞≡𝒞∖𝒟 (the “good” locus)
parametrizes a family of Abelian varieties, hence over 𝒞 we have a local system with fiber H^1(𝒳_u,).
 The polarization induces a non-degenerate skew-symmetric pairing of this local system

    ⟨ -,-⟩ H_1(𝒳_u,)× H_1(𝒳_u,)→,    ⟨ξ,η⟩=-⟨η,ξ⟩,

 which is physically interpreted as the Dirac electro-magnetic pairing. The 
 fiber H_1(𝒳_u,) is the lattice of quantized electro-magnetic charges
 in the SUSY preserving vacuum u∈𝒞. The map

    Z_u H_1(𝒳_u,)→,     Z_u(α)=∫_αι_Ω,
 
 is the SUSY central charge of a state with electro-magnetic charge α.
 
 

  §.§.§ The Chow 𝕂/-trace

 For later reference we rephrase the saturation of the unitary bound in terms of the unerlying Abelian variety X_𝕂 defined over the function field 𝕂. 
 An invariant of an Abelian variety X_𝕂  defined over a complex function field 𝕂≡(𝒞) is its Chow 𝕂/-trace <cit.>
 tr_𝕂/(X_𝕂)≡(B_,τ)
 which is an Abelian variety B_ defined
over  together with a map defined over 𝕂
τB_→X_𝕂
which satisfies the appropriate universal mapping property. 
Here we are interested in the physical meaning of the Chow trace (see also <cit.>).
We claim that the Chow trace of X_𝕂 is the Abelian variety B_ of the free subsector.
Indeed under the isomorphism
Ω𝖫𝗂𝖾(𝒳_u)→T_u^*𝒞≃dℛ/dℛ^2,
which sends Hamiltonian vector fields to the differentials of their Hamiltonians,
the image of the Lie algebra of the Chow trace Ω(𝖫𝗂𝖾(B_))
is contained in dℛ_1 since the Hamiltonian vectors ∂_w tangent to
the Chow-trace have dimension zero. 
Dually all Hamiltonians in ℛ_1 generate constant vertical vector field. 
Thus

 In absence of free subsectors
tr_𝕂/(X_𝕂)=0.




 
 

  §.§.§ Switching on mass and relevant deformations

 
After switching on the mass and relevant deformations the
 geometry 𝒳 remains a holomorphic Lagrangian fibration with section
 over the 
 (same) Coulomb branch 𝒞 with generic Abelian fibers. However the ^×
 symmetry is no longer present, and the Euler holomorphic differential λ_≡ι_Ω
 gets replaced by a meromorphic Seiberg-Witten differential λ_SW
 which depends on the Coulomb branch coordinates u_i, the masses m_a,
 and the relevant couplings t_j.  Eq.(<ref>) still holds with the replacement of λ_ by λ_SW. 
 λ_SW is defined modulo exact forms and 
its dependence on the various parameters follows the rule (<ref>). 
This rule shows that the tangent space to 𝒞 (resp. to the space of
 relevant couplings) injects in H^0(𝒳_u,Ω^1) (resp. H^1(𝒳_u,𝒪)). In particular (as it is clear from the physical side) the space of relevant deformations of a SCFT
 has dimension at most r and is uniquely determined by the ^×-invariant special geometry of the SCFT. The story with the mass deformations is much more involved. This reflects the physical fact that the mass deformation of a SCFT geometry is non-unique, in general,
 that is, there are several distinct flavor symmetries (of different ranks) which are consistent with one and the same SCFT special geometry.
 This non-uniqueness can be seen per tabulas in the explicit classification of the flavor symmetries
 of rank-1 SCFTs in refs.<cit.>. 

 
 
 This concludes our quick review of the ordinary special geometries with compact generic fibers (for more see e.g. <cit.>). Next we consider the general case where the generic fiber may be non-compact. 
 
 

§ GEOMETRIES WITH ANTI-AFFINE FIBERS

 
 The ordinary geometries described above are not the only ones which satisfy our physically motivated “axioms”. The basic condition on the algebraic integrable system that the Liouville
 momentum map is the affinization morphism π has other solutions with group-variety fibers.
 One of the goals of this paper is to present the physical interpretation of these more general geometries.
 We first describe them geometrically, beginning with the structure of a single generic fiber.

 
 
 
 
 
 

 §.§ Anti-affine groups

 An algebraic group is anti-affine <cit.> iff it has no non-constant regular functions; anti-affine groups are also known as quasi-Abelian varieties.
 An anti-affine group is automatically smooth, connected, and commutative <cit.>. Our basic “axiom”,
 eq.(<ref>), says that a general smooth fiber
 𝒴_b is an anti-affine group. This is consistent with the interpretation of the 
 geometry as a complex integrable system: the Lie algebra of the fiber is commutative being generated by Hamiltonians in involution.
 However the fibers of most algebraic integrable systems are not anti-affine.
 
  In the literature there are two distinct notions of “quasi-Abelian variety”.
 A complex variety is quasi-Abelian in the algebraic sense iff it has no non-constant
 regular function. It is quasi-Abelian in the analytic sense iff its underlying complex
 manifold has no non-constant
 holomorphic function. Clearly the analytic notion is stronger, and (as we shall see below)
 there exist
 group-varieties which are quasi-Abelian in the algebraic sense but have non-trivial holomorphic functions, in facts whose underlying complex manifold is Stein. We stress that the notion relevant for our applications is the algebraic one. 
 The existence of non-trivial holomorphic functions which are not algebraic does not spoil the quantum consistency of the QFT associated with the geometry.
 
  
 
 To describe the structure of an anti-affine group over  we start from the structure of
 a general connected, commutative, complex algebraic group G which may or may not be anti-affine. The Barsotti-Chevalley theorem <cit.>
 states that G is an extension of group-varieties of the form

    e→ C G A→ 0,

 where A is an Abelian variety, in facts the Albanese variety of G (𝖺𝗅 is the Albanese map <cit.>),
 and C is a connected commutative affine group with identity e. 
 
 
 
 A connected affine commutative group C is a product of copies of additive and multiplicative groups
 𝔾_a^k×𝔾_m^f; when working over  we identify it with
 the complex Lie group 
 
    ^k× (^×)^f ≡ V× T,  where  V=^k,   T=(^×)^f.

We call T≡𝔾_m^f the torus group and V≡𝔾_a^k the vector group. 
 The inequivalent commutative group-varieties G with given A, k and f
 are then classified by the group

    𝖤𝗑𝗍^1(A,V× T)=𝖤𝗑𝗍^1(A,V)⊕𝖤𝗑𝗍^1(A,T).

 The extension groups are computed in <cit.>

    𝖤𝗑𝗍^1(A,T)   ≃𝖯𝗂𝖼^0(A)^f≡ (A^∨)^f
    𝖤𝗑𝗍^1(A,V)   ≃ V^⊕ A,

where A^∨ is the dual Abelian variety.
 For later reference we sketch the proof of (<ref>),(<ref>).
 
 
 Recall that 𝖯𝗂𝖼^0(A) is the group of divisors on A which are algebraically equivalent to zero modulo linear equivalence,
 while A^∨≅𝖯𝗂𝖼^0(A) is the dual Abelian variety of A (see e.g. . I.8 of <cit.>).
 One shows[ See Proposition 8.4 of <cit.>, or . 5.2 of <cit.>.] that an invertible sheaf (≡ line bundle ≡ linear class of divisors) ∈𝖯𝗂𝖼(A) is algebraically trivial if and only if it is invariant under translations in
 A, that is, if for all a∈ A()
 we have
 
    t_a^*≃,  where t_a b↦ b+a,  a,b∈ A().
 
 We first consider extensions of A by 𝔾_m. Complex analytically, G is a principal bundle over A
 with structure group ^×, which we may see as a bundle ^*→ A where ^* is the complement of the zero section
 in a line bundle → A. The total space of a principal bundle over an Abelian group A with Abelian structure group ^×
 is itself an Abelian group if and only if the line bundle is invariant by translation on A so that the two group operations
  along the base and  the fiber
 commute; by the remark before eq.(<ref>)
  this is equivalent to  being algebraically trivial.
Next we consider extensions of A by 𝔾_a. Complex analytically G is a principal bundle over A with group , and all such principal bundles are commutative groups. Hence
the extension group is the space of isomorphism classes of such bundles H^1(A,_A)≃^ A.
 
 
 We note that
 
    H_1(G,)≃^2 A+ T.


The algebraic group G in eq.(<ref>) is connected and commutative but may or may not be anti-affine. 
 For instance if A is trivial, G is affine; more generally when the extension is trivial, G=C× A,
 the group is not anti-affine, etc. Our next task is to understand which extensions are anti-affine.
We consider first the extension of A by the f-torus T≡ (^×)^f and then the extension by a vector group V≃^k.  
  
  

  §.§.§ Torus extensions of Abelian varieties

  If G is an extension of the Abelian variety A by the torus group T
  
    e→ T→ G A→ 0

the Albanese map G A is a principal T-bundle. We ask when a T-principal bundle over a complex Abelian variety A is a commutative
algebraic group.
Let Λ be the character group of T≃ (^×)^f.
For any T-principal bundle G A we have the “Fourier decomposition”

    α_∗1mu_G =⊕_λ∈Λ_λ,

where _λ→ A are line bundles (invertible sheaves).
Around eq.(<ref>) we saw that the translations in A are
T-automorphisms of the principal bundle G iff
the line bundles _λ→ A are algebraically equivalent to zero,






 that is, _λ∈𝖯𝗂𝖼^0(A) for all λ∈Λ. 
Stated differently:

 The T-principal bundle GA is a group homomorphism with kernel T if and only if the decomposition (<ref>) defines a group homomorphism

    χΛ→𝖯𝗂𝖼^0(A),    χλ↦_λ.
  



Then we have

    Γ(G,_G)= H^0(A,α_∗_G)=⊕_λ∈Λ H^0(A,_λ)

while

    H^0(A,_0)≡ H^0(A,_A)=.
 

Hence G is anti-affine iff H^0(A,_λ)=0 for all  λ≠0. Since 
 H^0(A,)=0 for all line bundle  which is algebraically trivial but
non-trivial, we need that _λ is the zero element of 𝖯𝗂𝖼^0(A) only when
λ=0, that is,

 The group G in eq.(<ref>) is anti-affine if and only if the underlying  group homomorphism
χΛ→𝖯𝗂𝖼^0(A) is injective, i.e. if and only if
Λ is a lattice in the dual Abelian variety A^∨.




An example will clarify the situation.



[Picard 1910] We consider the extensions of the elliptic curve E_τ of period τ
by the one-dimensional torus ^×. We see E_τ as the quotient of its universal cover, ,
by the group generated by the two automorphisms 

    L_1 z↦ z+1  and   L_2 x↦ z+τ,

and we write π→ E_τ≡/⟨ L_1,L_2⟩ for the canonical projection.
Let
p_1×→ be the trivial -bundle. A line bundle _w→ E_τ which is algebraically trivial can always be written as the quotient

    _w= (×)/⟨ L_1,L_2⟩,    L_1    (z,y)↦ (z+1,y)
    
    L_2    (z,y)→ (z+τ, e^2π i wy).

From this expression it is obvious
 that the inequivalent bundles _w∈𝖯𝗂𝖼^0(E_τ)
 are parametrized by the points  w∈ E_τ (which coincides with E_τ^∨≃𝖯𝗂𝖼^0(E_τ)). To get the corresponding ^× extension of E_τ we just cut out the zero section of the bundle 
 
    G_w= (×^×)/⟨ L_1,L_2⟩,    L_1    (z,y)↦ (z+1,y)
    
    L_2    (z,y)→ (z+τ, e^2π i wy).

It may be convenient to write the principal ^×-bundle G_w→ E_τ
in an alternative way which we dub the “singular gauge” ; we parametrize 

    (z,y)=(z, ϑ(z-w,τ)/ϑ(z,τ) x)∈×(ℙ^1∖{0,∞})

where[ The θ-function in the rhs is defined as in chapter 20 of DLMF <cit.>.
By construction ϑ(z,τ) has a single zero at the lattice points z=m+nτ.]

    ϑ(z,τ) def=θ_3(π z-12π -12πτ | τ)

and then take the quotient by the action of the group ⟨L̃_1,L̃_2⟩ acting as

    L̃_1 (z,x)↦ (z+1,x),    L̃_2 (z,x)↦ (z+τ,x),

where x∈^× is now a global (but singular) coordinate along the fiber. 

It is clear that the groups extensions G_w
are parametrized by degree-zero divisors of the form

    (ϑ(z-w,τ)/ϑ(z,τ))=w-0,

i.e. by points of E_τ^∨≅ E_τ.
We conclude:
 Modulo isomorphism we have one connected, commutative,
 algebraic group (over ), G_w, which fits in the exact sequence

    1→^×→ G_w→ E_τ→ 0

per point w∈ E_τ≅𝖯𝗂𝖼^0(E_τ).

The Corollary is just eq.(<ref>) with f=1.
Now let us see for which points w∈ E_τ the commutative
 algebraic group G_w is anti-affine. Suppose that the point w∈ E_τ which defines the group G_w is torsion, that is,
m w=0 in E_τ for some m∈, i.e. 

    m  w= a+b τ in   with a,b∈.
 
The function on ×^×

    f(z,y)≡ e^-2π i b z y^m
 is invariant under L_1, L_2 so f(z,y) is a non-constant global holomorphic function and G_w is not
anti-affine. However, when w is not torsion G_w is manifestly anti-affine.[ It is also quasi-Abelian in the analytic sense.] This
conclusion is in agreement with Lemma <ref> (see <cit.>
for more details).

The dual of the Lie algebra of the 2-dimensional group G_w is generated by two 
holomorphic differentials dz and dy/2π i y. H_1(G_w,)≃^3 is generated by the three  cycles

    A   ={(z(s),y(s))=(s,1)},        B={(z(s),y(s))=(τ s, e^2π i s w)},
    
    C   ={(z(s),y(s))=(0, e^2π is)},      where  0≤ s≤ 1

with period matrix

    ∫_A dz   =1,    ∫_B dz   =τ,    ∫_C dz   =0,
    ∫_A dy/2π i y   =0,    ∫_B dy/2π iy   =w,    ∫_C dy/2π i y   =1,
  
and the groups G_w are distinguished by their period of dy/2π i y on the B-cycle.
G_w is anti-affine (quasi-Abelian) when this period cannot be written as a +bτ for a,b∈ℚ, i.e. when the three non-zero periods (<ref>) are linearly independent over ℚ.


Torus extensions of Abelian varieties are also called semi-Abelian varieties.
They may be seen as semistable degenerations of Abelian varieites.



  §.§.§ Vector extensions of Abelian varieties


For A an Abelian variety over  we have

    𝖤𝗑𝗍^1-2mu(A,)≃𝖫𝗂𝖾(A),

so that an extension of A by a vector space V of dimension k is specified by a k-tuple of elements of the 
Lie algebra of A. More precisely

[see <cit.> Proposition 2.3] All extension G of an Abelian variety A
by a vector group V fits in a unique commutative diagram

    0[r]     H^1(A,_A)^∨[d]_γ[r]     E(A)[d] [r]     A@=[d][r]    0
    
    0 [r]    V [r]     G [r]     A[r]    0

where E(A) is the universal vector extension of A <cit.> (which is an anti-affine group).
Then G is anti-affine if and only if γ is surjective. The transpose map

    γ^t V^∨→ H^1(A,_A)

is then injective and we conclude that the anti-affine groups over A
obtained as vector extensions are classified by subspaces of the -space 

    H^1(A,_A)≃ H^0(A,Ω^1_A)^∨≃ (T^*_eA)^∨≡ T_eA ≃𝖫𝗂𝖾(A)

where the first isomorphism is given by the polarization, the second one by translation invariance
in the Abelian group A.


This statement is crucial for the physical interpretation of vector extensions in the context of =2 QFT.
We have non-trivial group extensions of an Abelian variety A by any vector space ^k
of arbitrary large dimension k but if we insist that the extension group should be
anti-affine, we conclude that k cannot be larger than r≡ A.

 The vector extensions are quasi-Abelian varieties in the algebraic
sense but not in the analytic sense. In other words, while they have no non-constant
regular functions they have plenty of non-constant holomorphic functions, indeed the underlying complex manifold of the universal vector extension is Stein see the Example below. 




We consider the additive version of the  Picard construction for the group

    0[r]    [r]     H_η[r]     A[r]     0


    H_η=( ×)/⟨ L_1,L_2⟩where
    L_1 (z,y)→ (z+ω_1,y+ξ_1),     L_2 (z,y)→ (z+ω_2, y+ξ_2).

H_1(H_η,) is generated by the two cycles A={s ω_1,s ξ_1} and B={s ω_2,s ξ_2}.
We define

    η≡1/2π i(∫_A dz∫_B dy-∫_A dy∫_B dz)=ω_1 ξ_2-ω_2 ξ_1/2π i,

Again, formulae look simpler in a “singular gauge”. We write

    (z,y)=(z, x+η ζ(z))

subject to the identifications
(z,x)∼ (z+ω_1,x)∼ (z+ω_2,x).
In eq.(<ref>) ζ(z) is the Weierstrass ζ-function which satisfies

    d/dzζ(z)= -℘(z)

with ℘(z) the Weierstrass function on E_τ of periods ω_1, ω_2
and τ=ω_2/ω_1.
The condition that H_η is anti-affine in the algebraic sense is
η≠0: indeed in this case 
no polynomial in y with coefficients holomorphic in z is invariant under L_1, L_2. However 
we have a ring of (non-algebraic) non-trivial global holomorphic functions

    ψ_m,n(z y)=exp[m(ξ_1 z-ω_1y)+n(ξ_2 z-ω_2y)/η],  
    m,n∈.

One checks that these functions separate points in 
H_η: indeed as a complex manifold H_η is known to be Stein.

From eq.(<ref>) it is clear that η measures the projection in H^0,1(E)
of the cohomology class of the second-kind differential dy, in agreement with Proposition <ref>
which says that the extension by  of E are classified by H^1(E,_E)≃ H^0,1(E).  






 §.§ General anti-affine Lagrangian fibrations


Our next task is to study Lagrangian fibration with general (algebraic) quasi-Abelian fibers.

We first consider a fibration 

π𝒴→ℬ
whose
general fiber 𝒴_b is a general (algebraic) connected commutative
complex group, not necessarily anti-affine, which is a Lagrangian submanifold, Ω|_𝒴_b=0. As in . <ref>  in addition
 we assume the existence of
a complete Euler vector  which generates a group ^× of
automorphisms of all the relevant structures with ℒ_ Ω=Ω.
As before this leads to an action on the zero-section ℬ and, just as in . <ref>,
when ℬ is smooth it is the spectrum of a polynomial ring 𝒬=[x_1,…,x_ℓ]
with ℓ=𝒴_b. So ℬ=𝖲𝗉𝖾𝖼 𝒬
and ℒ_  x_s= Δ(x_s) x_s with Δ(x_s)>0 for all s. 




By the Barsotti-Chevalley theorem, 
the generic fiber 𝒴_b is the extension of
an Abelian variety 𝒜_b of dimension r by a vector group V of dimension k
and a torus group T of dimension f (ℓ=r+k+f). The Lie algebra of the fiber has the form

    𝖫𝗂𝖾(𝒴_b)≡ T_e(b)𝒴_b= 𝖫𝗂𝖾(𝒜_b)⊕ V⊕𝖫𝗂𝖾(T),

while the symplectic form yields an inverse pair of isomorphisms
T_e(b)𝒴_b @/^1pc/[rr]^Ω        @/^1pc/[ll]^℧ T_b^*ℬ@=[r]   ⊕_s   dx_s
which define three Pfaff systems Ω 𝖫𝗂𝖾(A_s),
Ω  V and Ω 𝖫𝗂𝖾(T) in Ω^1(ℬ)
which are integrable since the fibers' Lie algebra is commutative. Then 
we have 

    (x_1,…,x_ℓ)=(t_1,…,t_k,m_1,…,m_f,u_1,…,u_r)

with

    Ω  V=⊕  dt_j,  Ω 𝖫𝗂𝖾(T)=⊕  dm_a
      Ω 𝖫𝗂𝖾(A_s)=⊕  du_i.

and the symplectic form may be written as

    Ω= dα^j ∧ dt_j+ dy^a/2π i y^a∧ dm_a+d z^i∧ du_i,

where α^j, y^a and z^i are coordinates along the orbits of the dual Hamiltonian vector fields.   
As before we may assume Δ(u_i)>1 without loss. This is all we can say without further restrictions on the commutative groups which are allowed as fibers. However, as we argued, physics requires in addition
the fibers to be anti-affine. 



  §.§.§ Vector extension vs. relevant interactions

We  consider the vector extension first and then the torus one.
We write ∂_z^i for the vector fields ℧  du_i which generate the Lie algebra of 𝒜_b. One has

    Δ(∂_z^i)=Δ(u_i)-1.

 H^0(𝒜_b,Ω^1) is spanned by the dual differentials dz^i with

    Δ(dz^i)=1-Δ(u_i).

By Proposition <ref> the summand V≡⊕ ∂_α_j in the rhs of (<ref>) 
is a quotient of
H^1(A,_A)^∨≃⊕_i  dz^i.
Let dz^i(j) be
 a representative of ∂_α_j; we have

    Δ(∂_α_j)= Δ(d z^i(j))=1-Δ(u_i(j))

and the dimension of the coordinate t_j such that dt_j=Ω ∂/∂α_j is

    Δ(t_j)= 1+Δ(∂_α_j)=2-Δ(u_i(j))

that is,

    Δ(t_j)+Δ(u_i(j))=2.

Thus the condition that the fiber is anti-affine associates to each summand ∂_α_j in V
a distinct operator u_i(j) with the dimension (<ref>) predicted by physics (this conclusion will be confirmed below by the explicit construction of the SW differential).
Since the ring 𝒬 is positively graded, Δ(t_j)>0 and we see that 
the additive extension is at most given by one new extra coordinate t_j
(coupling) per chiral operator u_i of the undeformed SCFT with dimension 
1< Δ(u_i) <2.
A chiral operator ϕ∈ℛ 
with dimension in this range is automatically a generator of the polynomial ring,[ Under our standing assumption that no element of ℛ has dimension 1.] i.e. one of the u_i's. 
We stress that this correspondence between the deformation parameters t_j
and “relevant operators” in ℛ follows from the requirement
that the fiber is anti-affine. Without the condition of anti-affinity,
no relation between chiral operators and coupling parameters would exist and we would get physically meaningless geometries.

 The conclusion holds whether the Coulomb branch is smooth or not.




  §.§.§ Torus extensions vs. flavor symmetry



Let us now turn to the subtler torus extensions. The Lie algebra of ^× is generated by y ∂_y,
so the Hamiltonian vector fields  naturally have dimension zero. The corresponding Hamiltonians m_a, i.e. coordinates on the base ℬ, then have Δ(m_a)=1, which is consistent
with their interpretation as mass parameters. The
physical masses take value in the complexified dual of the Cartan subalgebra of the
flavor symmetry group F. In particular f≡ T should coincide with the rank of the flavor group F. 




The anti-affine condition requires the character group of the torus T
to embed as a lattice χ(T) in the Picard variety 𝒜_b^∨ for general b∈ℬ. We wish to interpret the lattice χ(T)
as the weight/root lattice of
the flavor symmetry F. Just as for the relevant deformations, we expect that in each given rank r (≡ the dimension of the Albanese variety of the general fiber)
 there is
a maximal dimension f of the extending torus T which is consistent with the fiber being an anti-affine algebraic group.
In facts, we expect that a ^×-isoinvariant
 special geometry in the standard sense describes finitely many
SCFTs, each one with its own flavor symmetry, so that only finitely many maximal torus extensions are allowed for a given ordinary special geometry, and hence there if a maximal dimension f_max which depends on
the invariants of the ordinary special geometry to be deformed. To make this idea concrete
we use the Mordell-Weil group <cit.>, generalizing to arbitrary r the discussion in ref.<cit.>
for r=1. We shall return to the flavor issue after
reviewing the Mordell-Weil lattice. 




§ THE MORDELL-WEIL LATTICE


Our fibration (<ref>) is a model of an anti-affine group variety Y_𝕃 defined over the function field 𝕃≡(ℬ).
Its Albanese variety A_𝕃 has a model over , 𝒜→ℬ,
which is the universal special geometry of the =2 QFT parametrized by the space 𝒫 of couplings t_j,m_a, i.e.

𝒜[rr][dr]       ℬ≃𝒞×𝒫@/^1pc/[ld][r]    𝒞

    𝒫

A_𝕃 is polarized, and as customary we assume the polarization to be principal for simplicity. This assumption is not important. Anyhow it allows us to identify A_𝕃
and its dual A_𝕃^∨. Y_𝕃 is the extension of A_𝕃
by the affine commutative group 𝔾_m^f×𝔾_a^k. We focus on the multiplicative part of the story since the vector extensions are already fully understood, so set k=0.
Since Y_𝕃 is anti-affine we have a group embedding, defined over 𝕃,

χC→A_𝕃^∨(𝕃)≃A_𝕃(𝕃)
of the character group ≃^f of 𝔾_m^f into the group A_𝕃(𝕃) of 𝕃-valued
points of A_𝕃.
The image χ(C) is a rank-f sub-lattice contained
in the Mordell-Weil lattice 
𝖬𝖶𝖫(A_𝕃) def=  A_𝕃(𝕃)/ A_𝕃(𝕃)_torsion
of the Abelian variety A_𝕃. 


In the language of the complex model ϖ𝒜→ℬ of A_𝕃, the generic smooth fiber 𝒜_b
 is a polarized Abelian variety of complex dimension r, and we have
 a rank-f sublattice χ(C)⊂𝒜_b. 
Varying b in ℬ each element of C defines a section of ϖ.
 The Mordell-Weil group
 is then realized as a group of global sections of 𝒜→ℬ <cit.>,
 and χ(C)⊂𝒜 is a group of sections over the base ℬ. 
 


 §.§ The Chow 𝕃/-trace and the Lang-Néron theorem
An important invariant of an Abelian variety A_𝕃  defined over a complex function field 𝕃≡(ℬ) is its Chow 𝕃/-trace <cit.>: tr_𝕃/(A_𝕃)≡(B_,τ) is an Abelian variety B_ defined
over  together with a map defined over 𝕃
τB_→A_𝕃
which satisfies the appropriate universal mapping property. 
Here we are interested in the physical meaning of the Chow trace (see also <cit.>).
We specialize the geometry 
 to a point p∈𝒫  in parameters' space getting a
a Lagrangian fibration 
𝒳≡𝒜_p→𝒞
which is the special geometry of the =2 QFT with fixed couplings p (see eq.(<ref>)).
We focus on the underlying Abelian variety A_𝕂 over the function field 𝕂≡(𝒞) and
 identify the generic fiber 𝒳_u with A_𝕂.
Then
𝒳_u contains the Chow 𝕂/-trace B_ of A_𝕂
as an Abelian subvariety. The total space of 𝒳 is a symplectic variety with symplectic form 
Ω and smooth fibers complex groups. Under the isomorphism
Ω𝖫𝗂𝖾(𝒳_u)→T_u^*𝒞≃dℛ/dℛ^2,
which sends Hamiltonian vector fields to the differentials of their Hamiltonians,
the image of the Lie algebra of the Chow trace Ω(𝖫𝗂𝖾(B_))
is contained in dℛ_1 since the Hamiltonian vectors ∂_w tangent to
the Chow-trace have dimension zero: 
ℒ_∂_w=0.
Indeed the image is precisely dℛ_1. Since the elements of ℛ_1
correspond to the free subsector of the QFT, we conclude that the Chow trace is the fiber
of the free subsector.[ Notice that the free subsector has no non-trivial mass or relevant deformations.]  Since we are assuming (with no loss) that our geometry does not
contain free subsectors, i.e. ℛ_1=0, the Chow trace is automatically trivial. 


When the Chow trace tr_𝕃/(A_𝕃) is trivial, 
 the Lang-Néron theorem <cit.> guarantees that 
the Mordell-Weil group 𝖬𝖶(A_𝕃) is a finitely generated group. Its free part is the Mordell-Weil lattice <cit.> 
𝖬𝖶𝖫(A_𝕃)=𝖬𝖶(A_𝕃)/𝖬𝖶(A_𝕃)_torsion.


Comparing with the discussion around eq.(<ref>) we conclude:
 An anti-affine extension Y_𝕃 of the Abelian variety A_𝕃
defined on the function field 𝕃=(ℬ) with tr_𝕃/(A_𝕃)=0 (≡ no free subsectors) by the torus group 𝔾_m^f
corresponds to a rank-f sublattice
Λ⊂A(𝕃)/A(𝕃)_tor≡𝖬𝖶𝖫(A_𝕃)
of the Mordell-Weil lattice of
A_𝕃. In particular the rank f of the torus extension is at most equal to rank 𝖬𝖶𝖫(A_𝕃). 


We see that the anti-affine condition goes in the desired direction of putting an upper bound on the rank of the extension (which we wish to identify with the rank of the flavor group F).



  §.§.§ Néron-Tate height


The lattice 𝖬𝖶𝖫(A_𝕃) is endowed with a non-degenerate,
positive-definite, symmetric bilinear pairing called the Néron-Tate height
(or canonical height) <cit.>.

 Let Γ be a finitely generated subgroup of

    A_𝕃(𝕃)/(τ B_()+A_𝕃(𝕃)_ torsion).

The canonical height is non-degenerate on Γ, and its extension to
⊗ A_𝕃(𝕃^ al)/τ B_() is a positive definite quadratic form.
(𝕃^ al is an algebraic closure of 𝕃).


As a corollary the lattice Λ is
 endowed with a symmetric positive-definite pairing

    ⟨ -,-⟩Λ×Λ→.
  

The physical intuition says that the rank-f lattice Λ equipped with the Néron-Tate (NT) quadratic form should
be identified with the lattice of weights of the (visible) flavor Lie group F associated to the mass parameters
in the space 𝒫. This was shown to be the case when r=1 <cit.>.
The weight lattice of F carries a positive-definite quadratic form
(invariant under the action of the Weyl group) which we wish to identify with the Néron-Tate form
(possibly up to normalization) <cit.>. 



 §.§ The rank of the torus extension


We wish to say more on the rank of the Mordell-Weil lattice of A_𝕃. In the classical case, i.e. in dimension r=1, the Mordell-Weil rank is given by the Shioda-Tate formula <cit.>. There is a generalization
of this formula valid in higher dimension
due to Oguiso <cit.> (see also <cit.>).

Th Oguiso theorem applies in the following situation:
we have a fibration (with section) π W→ Z, with W,Z normal projective, such that
the generic fiber A_(Z)≡ W_η is an Abelian variety (of positive dimension), defined over (Z),
with the properties:


  (1) W, Z have only ℚ-factorial canonical singularities;

  (2) there is no prime divisor D⊂ W such that
π(D)≤ Z-2;

  (3) h^1(W,_W)=h^1(Z,_Z)=0.

In our case Z is typically ℙ^2mur while W is a suitable projective variety
such that 𝒜=W∖ E for a divisor E. The condition (3) is automatically satisfied in absence of
free subsectors (i.e. A_(Z) has zero Chow (Z)/-trace) which is a necessary condition for the Mordell-Weil group A_(Z)((Z)) to be finitely generated. (1),(2) are mild regularity conditions that we assume to hold (recall that in this paper we ignore all singularities as long as they are confined in codimension at least 2).

Let ⊂ Y be the discriminant locus, =∑_i D_i
and 

    π^*(D_i)_red=∑_j=0^m_i-1 D_ij

be the respective decompositions into irreducible components.
m_i is the number of prime divisors on W over D_i⊂ Z.
Then

 Under the assumptions (1)-(3), the Mordell-Weil group 
𝖬𝖶(A_(Z)) is a finitely generated Abelian group of rank

    rank 𝖬𝖶(A_(Z))=ρ(W)-ρ(Z)-rank 𝖭𝖲(A_(Z))-∑_i=1^k(m_i-1)

Here ρ(W) (resp. ρ(Z)) is the Picard number of W (resp. of Z)
and 𝖭𝖲(A_(Z)) is the Néron-Severi group of A_(Z). In particular,

    𝗋𝖺𝗇𝗄 𝖬𝖶(A_(Z))≤ρ(W)-2.
 

This result gives us an upper bound on the rank of the Mordell-Weil lattice, hence
on the rank of the flavor group of a QFT whose universal ordinary special geometry
is given by 𝒜=W∖ E. If we can find an upper bound on the Picard
numbers of the W's which may appear at a given rank r (≡_(ℤ)A_(Z))
we would have an uniform bound on the flavor symmetry valid for all SCFT of rank r. 

Not all fibrations W→ Z whose generic fibers are dimension-r Abelian varieties are allowed in special geometry.
First of all we have severe conditions on the base; when smooth Z is a normal projective
compactification of ^k.
In addition we have the condition that if we specialize to a generic value of the couplings p≡ (t_j,m_a)
the resulting fibration 𝒳≡𝒜_p→𝒞 is Lagrangian. 
By restriction, a section of 𝒜→ℬ defines a section of 𝒳→𝒞,
so if the couplings p are sufficiently generic we have an embedding

    𝖬𝖶(𝒜)⊂𝖬𝖶(𝒳).

For the purpose of getting an upper bound, we can then apply Oguiso's Shioda-Tate formula (<ref>) we can replace the geometry 𝒜→ℬ with the
ordinary special geometry 𝒳→𝒞 at a generic p∈𝒫. There is a pair (M,S) with M projective
such that 𝒳=M∖ S. Since 𝒳 is symplectic, M
is log-symplectic, and the anti-canonical -K_M is  Therefore we expect that there is an upper bound for the Picard number ρ(M) of all 2r-dimensional projective varieties which are
fibered in Abelian varieties of dimension r and are log-symplectic. This upper bound minus 2 will yield an uniform upper bound for the rank of the flavor symmetry group of a rank r SCFT. While finding the actual bound looks hard in general, this line of reasoning suggests that a bound on the flavor exists (as expected by physicists) and has a very deep mathematical origin. 
Without the condition of anti-affineness of the underlying algebraic group no upper bound would arise.

 Consider r=1. In this case M is an elliptic surface such that M minus a fiber is
symplectic, i.e. such that the canonical class is minus the class of a fiber <cit.>.
Then M is a rational elliptic surface fibered over ℙ^1.
The invariant entering in the Oguiso formula (<ref>) are
ρ(W)=b_2(W)=10,  ρ(Z)≡ρ(ℙ^1)=1,  rank  𝖭𝖲(A_(Z))=1
so that in

    rank (flavor group)≤ 10-2-∑_i(m_i-1)≡ 8-∑_i(m_i-1)

which is (of course) the original Shioda-Tate formula <cit.>. Thus in rank 1
we have f≤ 8
an inequality saturated by the Minahan-Nemeshanski E_8 model <cit.>.





 §.§ Flavor symmetries. Ambiguities


In rank 1 the theory of the flavor symmetry from the viewpoint of the Mordell-Weil
group was worked out in full detail in ref.<cit.>. Basically it turned out that the 
Mordell-Weil lattice 𝖬𝖶𝖫(A)
 should be identified with the root lattice of the flavor group F and the Nerón-Tate
height with the Weyl invariant pairing on the root lattice. 
However there remain some ambiguity since
these two data cannot distinguish, say, the flavor groups SU(3) and G_2.
To eliminate all ambiguity one has to construct the actual root system
i.e. describe the finitely many elements of 𝖬𝖶𝖫(A) which are roots
of the flavor Lie algebra. This was done in <cit.> using the finite-subset of integral points
in 𝖬𝖶𝖫(A_(Z))
(as contrasted to the rational ones) and some long calculations.

The root lattices and Weyl invariant pairings are expected to work in the same
way in higher dimension: they can be read directly from the Mordell-Weil lattice.
Finding the actual root system, however, goes beyond “Diophantine geometry over
function fields”
since it requires the knowledge of  the specific model
𝒳 of the Abelian variety X_𝕂
which describes the special geometry. In dimension 1 we had a preferred model, the Néron-Kodaira one
<cit.>, but the situation in higher dimension is not so clear.    




§ THE SW DIFFERENTIALS AND THE WEIL CORRESPONDENCE


Our proposal is that the universal special geometry is the Lagrangian fibration which describes a SCFT with all possible
couplings switched on and promoted from complex parameters
to non-propagating superfields (spurions).
From the extended special geometry one gets back the
ordinary Seiberg-Witten geometry of the theory with given masses m_a and relevant couplings
t_j by taking the symplectic quotient. In detail: one considers the level manifold
iM↪𝒴
where the spurions' vevs are kept fixed 
t_j=t_j and m_a=m_a; M is fibered over the submanifold
𝒞≡{(t_1,…,t_l,m_1,…,m_f,u_1,…,u_r)}⊂ℬ,
which we identify with the Coulomb branch 𝒞≡𝖲𝗉𝖾𝖼 ℂ[u_1,…,u_r], 
M≡π^-1(𝒞).
Then one takes the quotient M/H where H is the complex Lie group generated by the
Hamiltonian vector fields ∂_α^j and 2π i  y^a∂_y^a (cf. (<ref>)) of the Hamiltonians t_j and m_a, respectively.
If ϖ M→ M/H is the natural projection, the reduced
integrable system is (M/H, Ω_red) where the symplectic form  Ω_red
is related to the original one Ω in 𝒴 by the equation

ϖ^*Ω_red=i^*Ω.
The fibration
M/H→𝒞 is then a non-^×-isoinvariant 
special geometry namely the integrable system of the SW geometry for the
SCFT deformed by the mass and relevant couplings m_a=m_a, t_j=t_j.
When these parameters are set to zero one gets back the ^×-isoinvariant geometry of the undeformed SCFT.

It remains to determine the SW differential λ_SW for general values of the couplings.
In view of (<ref>) λ_SW should be induced by the (holomorphic) universal Euler differential λ_ via the symplectic
reduction on the locus t_j=t_j, m_a=m_a, that is, 

ϖ^*λ_SW=i^*λ_+exact.
Taking the exterior derivative of this equation we get back the symplectic quotient formula
(<ref>). Now we compute explicitly λ_SW case by case, and check that it has the expected properties for a physical Seiberg-Witten differential. Comparing (<ref>), (<ref>) we see that the differential behaves according to the Duistermaat-Heckman theorem as predicted by <cit.>.



 §.§ Torus extensions: rank-1

We start from the very simplest case. We consider the extension of a rank-1 geometry by a f=1 torus.
 The mass parameter m is the vev of a vector superfield which as g_f→0 gets frozen to the constant value m (as determined by the the choice of boundary conditions at infinity),
and we consider the reduced geometry at a fixed value m of m

    𝒳≡𝒳_m  def={y∈𝒴 m(y)=m}/^×.
 
i.e. the ordinary rank-1 special geometry over 𝒞≃ with mass m.
We have the anti-affine extension

    𝒴[dr]_Π[rr]^      𝒜[dl]^π
       ℬ

where the generic fiber 𝒜_b is an elliptic curve E_τ(b) of period τ(b)
while the generic fiber 𝒴_b is an anti-affine group of the form 
G_w(b)→ E_τ(b) as in
 Example <ref>. Thus the extension is specified by 
two distinct sections
s_0,s_1ℬ→𝒜 where s_0 is the zero section and
s_1 is a generator of the rank-1 lattice Λ in the Mordell-Weil group of
𝒜. Explicitly 

s_1b↦w(b)∈E_τ(b),
and w(b) is not torsion for generic b.
We write

    χ(z,w|τ) def=θ_3(π z-π w-12π -12πτ | τ)/θ_3(π z-12π -12πτ | τ)

and identify (θ∈)

    G_w{(z,χ(z,w|τ) e^iθ)∈×^×}/(z∼ z+ m+n τ).

As in Example <ref> we write y for the covering fibers' coordinate,

    y≡χ(z,w|τ) e^iθ.




Let U⊂ℬ be a sufficiently small coordinate patch with special coordinates (m,a) where m is a mass parameter
and a is an usual local special coordinate along the Coulomb branch 𝒞.
The restriction 𝒴|_U of the extended geometry to U  has the form

    (m,a, z, χ(z,w(m,a)|τ(m,a)) e^iθ)∈ U××

modulo the obvious identifications

    z∼ z+r + s τ(m,a),   r,s∈
    θ∼θ+2π  k,   k∈.

 From eq.(<ref>) we see that the Euler differential in 𝒴|_U
has the local form

    λ_=m dy/2π i  y+ a dz= m dθ/2π +m/2π i dχ(z,w|τ)/χ(z,w|τ)+ a dz.

Comparing with eq.(<ref>) we see that the periods of λ_ are

    A-period:      a
       B-period:      τ a+ m w
       C-period:      m.
 




  §.§.§ The reduced SW differential.
 

In terms of the local coordinates in eq.(<ref>) the (complex)
 time evolution of the phase space 𝒴 generated by the Hamiltonian m is
just a linear shift of the coordinate θ canonically conjugate to the momentum m

    θθ+t,     t∈.


Hence the Legendre differential (alias SW differential) of the reduced integrable system is obtained by simply dropping
in eq.(<ref>) the term containing the dθ∼ dt. Then the SW differential
takes the local form 

    λ_SW=m/2π i dχ(z,w|τ)/χ(z,w|τ)+ a dz.

The `naive' symplectic structure 

    Ω=dλ_SW=da∧ dz  (locally)

  is still holomorphic (as it should be in special geometry) but the SW differential
  λ_SW is now just meromorphic. In facts from the explicit expression (<ref>)
  and (<ref>) we see
that a mass-deformed ordinary special geometry has a meromorphic SW differential
  λ_SW with constant residues equal to  linear combinations of the mass parameters with integral coefficients.
 We stress that the singularities in λ_SW are an “artifact” of the 
  MWM reduction procedure. The parent differential λ_
  is perfectly holomorphic.
 
  


  §.§.§ The SW differential as a current
  There is a less naive point of view <cit.>.
 The proper way to understand λ_SW is to see it as a (1,0)-current
 not as a (1,0)-form. The space of (1,0)-currents
 on a complex n-fold M is the topological dual to the space C^∞_c-1.5mu(M)^n-1,n of compactly supported, smooth differential forms
 of type
 (n-1,n)  (see e.g. <cit.>).
 We write T_SW for λ_SW seen as a current i.e.
 
    ∫ T_SW ϕ=∫_M λ_SW∧ϕ,    ϕ∈ C^∞_c-1.5mu(M)^n-1,n.

 By general theory (see page 368 of <cit.>) the exterior derivative of a meromorphic differential seen as a current
 differs from its `naive' differential by a `residue' term, that is, by a current with a codimension-1 support; then eq.(<ref>) should be replaced by
 
    dT_SW =Ω +“residue”.
 
The residue is easily computed with the help of the Poincaré-Lelong formula.
 
 

  
The Poincaré-Lelong formula. Let D≡ (f)⊂ M be a smooth irreducible divisor.
 The Poincaré-Lelong formula is the equality of (1,1)-currents  (cf. page 388 in <cit.>) 
 
    d(df/2π i f)=δ_D

 where δ_D is the “Dirac” (1,1)-current defined by
 
    ∫δ_D ϕ =∫_D ϕ,    ϕ∈ C^∞_c-1.5mu(M)^n-1,n-1.

 More generally, if D=∑_i a_i  D_i, with D_i irreducible, we set
 
    δ_D=∑_i a_i  δ_D_i.

 Clearly,  in the sense of the cohomology of currents, which is isomorphic to the usual de Rham cohomology of M (see page 385 in <cit.>),
 we have
 
    [δ_D]≡ [D]

 where [D] is the fundamental class of D.
 
 

  
The differential of the SW current.
 Using the Poincaré-Lelong formula it is easy to see that, locally in 𝒴|_U, we have the equality of currents
 
    dT_SW=Ω+ m δ_(z-w(u))-m δ_(z).

The corresponding global statement is
 
    dT_SW= Ω+m(δ_s_1-δ_s_0)

 where s_0 is the zero-section and the section s_1 is defined in (<ref>).
 Therefore, in the cohomology of currents ≡ de Rham cohomology, we have 
 
    [Ω]=m (δ_s_0-δ_s_1),

as expected in <cit.>.
 


 §.§ General torus extension
 
We now extend the above result to arbitrary ranks r and f but still with all
relevant deformation switched off.
After the reduction to
an ordinary special geometry over the usual Coulomb branch 𝒞, the polar locus of the 
meromorphic differential λ_SW defines a divisor
in 𝒳 whose
irreducible components E_i⊂𝒳 are complex submanifolds of dimension
2r-1, which are “horizontal”  in the sense that their intersection
with each fiber, E_i∩𝒳_u  defines a divisor inside the fiber 𝒳_u. More intrinsically
they are divisors of the underlying Abelian variety X_𝕂 defined over the function field 𝕂, that is,
𝕂-valued points of the dual Abelian variety X_𝕂^∨
which is isomorphic to X_𝕂 (since we assume the polarization to be principal). 
The derivative

    ∂/∂ m^aλ_SW

is a meromorphic differential whose polar divisor D_a
intersects each fiber 𝒳_u along a 
 divisor D_a∩𝒳_u⊂𝒳_u which is required to satisfy two conditions:
 
 
  (a) is algebraically trivial in 𝒳_u
 by eq.(<ref>);
 
  (b) is non-torsion (in order the group to be anti-affine).
 
 
 Therefore, in the general case we
 have
 
    dT_SW=Ω+∑_a m_a δ_D_a
    Ω=-∑_a m_a [D_a].

 
At first sight it may seem that the situation for higher r
is quite different from the one we saw in rank-1, where
the polar divisor of λ_SW
is a linear combination (with integral coefficients)
of sections of π (submanifolds of dimension r)
whereas for general r we have divisors i.e. submanifolds of dimension
2r-1≠ r.   However the difference is
only apparent. Indeed, 
an algebraically trivial “horizontal” divisor corresponds to a 𝕂-point in X_𝕂^∨
while the principal polarization yields an isomorphism X_𝕂^∨→ X_𝕂, so 
these divisors may be seen as points in X_𝕂(𝕂) i.e. sections of π𝒳→ℂ.
Let ϕ X_𝕂(𝕂)→ X_𝕂(𝕂)^∨ be the isomorphism sending
 sections (≡ points in X_𝕂(𝕂)) into “horizontal” divisors. One has
 dT_SW=Ω+∑_a m_a δ_ϕ(s_a)
 This formula makes clear that the mass parameters m_a
take value in a vector space
whose dual contains a natural 
lattice, generated by the δ_ϕ(s_a), which carries
a natural symmetric product which is positive-definite.





  §.§.§ Relation to the Duistermaat-Heckman theorem

This is the point emphasized by Donagi <cit.>.
The issue is to understand the deep reason
why the residues of λ_SW are constants linear in the mass parameters m^a.
In view of eqn.(<ref>) this amounts to asking why the
cohomology class of the symplectic form [Ω] depends linearly on the masses.

This fact follows from our construction of the mass-deformed special geometry as 
a symplectic (Hamiltonian) reduction of the extended geometry associated to the zero-coupling
gauged flavor symmetry model. The Duistermaat-Heckman theorem <cit.> holds for the symplectic quotient by the action of
any compact group F (acting symplectically) <cit.>. For easy of exposition we take F=U(1),
the generalization to arbitrary compact flavor groups being obvious (see <cit.> or
<cit.>).
The theorem is usually stated for real symplectic manifolds,
but everything remains valid for complex symplectic manifolds.
In this case U(1) is replaced by its complexification ^×≡ GL(1,).



We have the extended symplectic manifold (𝒴,Ω) with a Hamiltonian action of U(1)
generated by the Hamiltonian vector field v with momentum map

    μ𝒴→𝔤𝔩(1)^∨,   ι_vΩ=dμ.

We assume that the U(1) action is free on the level set 𝒴_m≡μ^-1(m). 
The reduced phase space
(≡ total space of the ordinary special geometry with mass parameter m set to m_0) is

    𝒳_m=μ^-1(m)/GL(1,)

so that 𝒴_m→𝒳_m is a principal GL(1,) bundle. 
We fix a connection 1-form α on this principal bundle:
by definition one has 

    ι_v α=1.

By the Marsden-Weinstein-Meyer theorem,
𝒳_m is a symplectic manifold
with symplectic form ω_m.
 We consider a tubular neighborhood T_m of 𝒴_m in 𝒴, say

    T_m={y∈𝒴m-ϵ < μ(y)≡ m < m+ϵ}⊂𝒴.

Let Ω_m≡Ω|_𝒴_m and π_m T_m→𝒴_m a projection. One checks that for small ϵ the form

    π_m^∗2muΩ_m-d((m-m) α)

is a symplectic form on T_m. Hence, for m-m small enough,
the symplectic form on 𝒳_m≡μ^-1(m)/GL(1,) is

    ω_m=ω_m-(m-m)F,

where F is the curvature form of the U(1) connection defined by 

    π^* F=dα.

This construction depends on several choices; however all choices are homotopically equivalent, 
and hence the above equation is intrinsic in cohomology  <cit.>.
Therefore, passing in cohomology, we get

    [ω_m]=[ω_m]+(m-m)c_1

where c_1 is the first Chern class of the line bundle 𝒴_m_0→𝒳_m_0. 
Formula (<ref>) is the Duistermaat-Heckman theorem. We see that the class of the symplectic form
is linear in m, as predicted by the previous arguments.
In facts there is much more: taking the derivative of the Duistermaat-Heckman formula we get

    ∂/∂ m[ω_m]=c_1()

where  is the line bundle which defines the extension
of the dimension r Abelian variety A_u,m by ^×.
This is exactly the same as the derivative of eqn.(<ref>)
since from our explicit construction (e.g. (<ref>) for r=1)
we see that

    =(-D)  ⇒   c_1()=-[D].
  
























 



 §.§ Relevant deformations and SW differentials
 The derivatives
 
    ∂/∂ t_αλ_SW|_X_u

  are (generically)
meromorphic differentials of the second kind, that is with higher order poles
and zero residues (see <cit.>).
The relevant deformations produce higher singularities of the SW differential
than mass deformation. This agrees with the physical intuition: relevant perturbations
are less soft than mass perturbations, so produce stronger deviations from the
superconformal situation (which corresponds to a regular λ_SW). 
This is reflected in the Weil correspondence in the fact that mass deformations lead to
“tame” semi-stable degenerations of Abelian varieties (i.e. torus extensions) while
 relevant deformations lead to un-stable degenerations.
We limit ourselves to present
a an explicit example which illustrates all the issues involved.

 We consider the family of extension H_η of an elliptic curve E
 by the additive group 

    0→→ H_η→ E → 0

which form a one-parameter family parametrized by η∈.
The extension group H_η is anti-affine iff η∈^×. A family of such extension groups 
describes the extended geometry of a rank-1 QFT with one relevant operator and trivial flavor group.

To pass from the multiplicative (torus extension) to the additive case (vector extension) 
we replace the automorphic factor
we used in Example <ref> with the analogue automorphic summand.
We write the elliptic curve E in terms of its Weierstrass equation

    y^2=4 x^3-g_2 x-g_3,

where g_2 and g_3 are the usual functions of the two periods (2ω_1,2ω_2) of
the elliptic curve 

    E≡/(z∼ z+2mω_1+2nω_2).






The universal cover of H_η is × whose coordinates we write in the form
(cf. eq.(<ref>))

    (z, x-κ ζ(z))

subject to the identification

    z∼ z+2m ω_1+2n ω_2.

In eqn.(<ref>) ζ(z) is the Weierstrass ζ-function which satisfies (<ref>).














Now assume that we have an extended geometry

    𝒴→ℬ

 of rank (r,f,k)=(1,0,1), so that the smooth
fibers 𝒴_b are groups of the form H_κ(b) for some elliptic curve E_τ(b). 
We use standard local coordinates (t, a) on ℬ
of dimensions {2-Δ,1} where Δ>1 is the dimension of the global
coordinate u on the ordinary Coulomb branch .
Then locally the canonical 
Euler differential has the form

    λ_= a  dz+ t  d(x-η ζ(z)).

After the MWM reduction to the ordinary special geometry where the relevant coupling t is frozen to the value t, we have a SW differential defined as
λ_SW= λ_|_t=tdx=0+exact
so that

    λ_SW= (a+t η ℘(z))dz.

λ_SW is now a meromorphic differential of the second kind
<cit.> with poles of the second order at the lattice points and zero residues in agreement with the Weil correspondence.
On the other hand the (reduced) symplectic form

    Ω≡ dλ_SW= da∧ dz,

is a holomorphic (2,0) form.


 We see that when η≠0 we can reabsorb it in the definition of
the relevant coupling t, so that we may set η to a convenient constant with no loss.




 §.§ Example: Argyres-Douglas QFT of type A_2
 The above example corresponds to
an actual =2 QFT namely the Argyres-Douglas model of type A_2.
It is convenient to geometrically engineer this QFT in F-theory, see e.g. ref.<cit.>.
From the F-theory construction, one gets the so-called “SW curve” in the form

    y^2= 4 x^3-t x-u,

where (t, u) are the global coordinates in ℬ≅^2, with u the coordinate on the
ordinary Coulomb branch 𝒞. 
The elliptic curve (<ref>) is the Albanese variety A_𝕃 of the universal special geometry of the A_2 SCFT, where 𝕃≡(ℬ)≡(λ,u). 
The SW differential is then

    λ_SW≡ y dx

which, by definition, has dimension 1

    Δ(y dx)≡Δ(y)+Δ(x)=1.

Comparing with the scaling of the equation (<ref>) we get

    Δ(y)=3/5,  Δ(x)=2/5,  Δ(t)=4/5,  Δ(u)=6/5.

Note that these numbers have the two properties expected from grand geometry:
Δ(u) is an allowed dimension for a rank-1 SCFT <cit.>
and the universal geometric relation (<ref>) is satisfied:

    Δ(t)+Δ(u)=2.

The (locally defined) special coordinate a  is (up to a multiplicative constant)

    a= u^5/6.



From eq.(<ref>) we read 
the SW differential on A_𝕃 

    λ_SW≡√(4 x^3-t  x-u) dx+exact

And (neglecting the irrelevant exact term)

    ∂/∂ aλ_SW   =6/5 u^1/5∂/∂ uλ_SW=
    -3/5 u^1/5 dx/√(4 x^3-t  x-u)
       ≡ -3/5 u^1/5 dx/y≡ -3/5 u^1/5 dz≡ω+exact,
    ∂/∂ tλ_SW   =
    1/2 x dx/√(4 x^3-t  x-u)= 1/2 ℘(z) dz≡ -1/2  dζ(z),

where ω is the differential of the first kind in the elliptic fiber normalized so that its periods
are ^×-invariant. 
We see that the derivative of the 
 the SW differential on the universal ordinary special geometry A_𝕃
 exactly matches the form (<ref>) (with the coupling normalization 
 η=1/2) which is expected
on the grounds of the theory of anti-affine group schemes (i.e. from the Weil correspondence).
The holomorphic differential λ_ on the total space of the universal geometry 𝒴 is (cf. eq.(<ref>))
λ_=u  dz+t dy ≡u  dz+t  d(x-1/2ζ(z)).
While the ordinary special geometry is not ^×-isoinvariant when t≠0,
the universal one is still ^×-isoinvariant. This amounts to saying that the period τ(t,u) 
of the Albanese variety of the
fiber 𝒴_(t,u) over (t,u)∈ℬ depends only on the scale invariant combination

    v= u/t^3/2,    ℒ_ v=0  ⇒ℒ_ τ=0.

This is a well-known property of the elliptic curve A_𝕃 in eq.(<ref>).

























































































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