The Weil Correspondence and
Universal Special Geometry

We make the observation that an ordinary special geometry $\mathscr{X}$ is, in particular, a complex model of an Abelian variety (or scheme) $X_{\mathbb{K}}$ defined over the field $\mathbb{K}={\mathbb{C}}(u_{1},\dots,u_{r})$ of rational functions in $r$ variables ($r$ being the rank i.e.​ the dimension of the ordinary Coulomb branch $\mathscr{C}$). This assertion contains a mild but crucial assumption that we now make explicit. In all known ${\cal N}=2$ SCFT the Coulomb branch⁵⁵5 Here we look at the Coulomb branch as a mere complex manifold, stripped of all other structures such as metric etc. $\mathscr{C}$ is a copy of ${\mathbb{C}}^{r}$. However, as pointed out in ref.​[A8], there are reasons to believe that more general Coulomb branches are also allowed. Since the Coulomb branch $\mathscr{C}$ is the spectrum of the chiral ring $\mathscr{R}$, in any quantum-consistent ${\cal N}=2$ SCFT $\mathscr{C}$ is an irreducible, reduced, normal, affine (complex) variety with an algebraic ${\mathbb{C}}^{\times}$-action (see section 2 for more details).

The allowed $\mathscr{C}$’s are severely restricted. If you look to a reasonable-looking candidate Coulomb branch $\not\simeq{\mathbb{C}}^{r}$ 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 $\leq 2$ as required by QFT (and true for $\mathscr{C}\simeq{\mathbb{C}}^{r}$ [A23]). 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 functions⁶⁶6 In particular in rank-$1$ there is only one possible Coulomb branch: the plane ${\mathbb{C}}$ [A11]. ${\mathbb{C}}(\mathscr{C})\equiv{\mathbb{C}}(u_{1},\dots,u_{r})$. 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,⁷⁷7 The singularities, if present, are in codimension $\geq 2$ since we are free to assume $\mathscr{C}$ to be normal by replacing the chiral ring $\mathscr{R}$ 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 $X_{\mathbb{K}}$ yields the actual special geometry $\mathscr{X}$. At this stage we can work with $X_{\mathbb{K}}$ which is simply a commutative group (defined on the slightly fancier field $\mathbb{K}$) and get a lot of mileage from algebraic group theory [milne].

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

In the algebraic group language the Seiberg-Witten differential $\lambda$ is a differential on $X_{\mathbb{K}}$ defined over $\mathbb{K}$. From the QFT point of view $\lambda$ depends on three distinct kinds of variables:⁸⁸8  There are no marginal couplings under our assumption that theory has no Lagrangian subsectors. the Coulomb coordinates $(u_{1},\dots,u_{r})$, the chiral couplings $(t_{1},\dots,t_{k})$, and the mass parameters $(m_{1},\dots,m_{f})$. The geometric distinction between the three kinds of parameters is as follows:

The parameters $u_{i}$, $t_{j}$ and $m_{a}$ exhaust the supply of differentials, so in $\lambda$ there is no room for couplings of fancier kinds. This is consistent with the superspace construction of ${\cal N}=2$ QFT. We shall see below that, in addition, only chiral couplings consistent with quantum UV-completeness are allowed in special geometry.

Eq.(1.2) suggests that we should treat the parameters $u_{i}$, $t_{j}$ and $m_{a}$ on the same footing. Therefore we introduce the universal branch $\mathscr{B}\simeq\mathscr{C}\times\mathscr{P}$ with global⁹⁹9 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 $\mathbb{L}={\mathbb{C}}(\mathscr{B})\simeq{\mathbb{C}}(u_{i},t_{j},m_{a})$ for the rational field in $r+k+f$ variables. Over $\mathscr{B}$ we have the universal Abelian variety

[r]^(0.3)ϖ&B≃C×P, i.e.​ the family of ordinary special geometries parametrized by the coupling space $\mathscr{P}$. Again, $\mathscr{A}\to\mathscr{B}$ is a model of an Abelian variety $A_{\mathbb{L}}$ defined over $\mathbb{L}$. The Seiberg-Witten differential $\lambda$ is then a third-kind differential on $A_{\mathbb{L}}$ defined over $\mathbb{L}$. The ordinary special geometry at fixed couplings $p\equiv(\underline{t}_{j},\underline{m}_{a})\in\mathscr{P}$ is $\mathscr{X}\equiv\mathscr{A}|_{\mathscr{C}\times p}$.

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 [MM]:¹⁰¹⁰10 The references in the quote insides the oval box are: Weil paper [Weil]; Barsotti paper [barsotti], Serre paper [serre1] and Serre book [serre2].

Comparing this quotation with eq.(1.2) we conclude that we may identify the couplings allowed by ${\cal N}=2$ supersymmetry with commutative algebraic groups. In particular the ordinary Seiberg-Witten geometry $(\mathscr{X},\lambda)$ (with mass and relevant deformations switched on) defines (and is fully determined by) an algebraic group $Z_{\mathbb{K}}$ over $\mathbb{K}$. To physicists group means symmetry, and we think the differential $\lambda$ 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 $Y_{\mathbb{L}}$ of the universal Abelian group-variety $A_{\mathbb{L}}$ defined over the field $\mathbb{L}$. The physical meaning of $Y_{\mathbb{L}}$ will be clarified below.

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 $Y_{\mathbb{L}}$ 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 $\mathfrak{car}(F)$ of $F$. The deformation space $\mathfrak{car}(F)$ 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 $r\equiv\dim\mathscr{C}$. 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 $Y_{\mathbb{L}}$ which is a vector extension of the universal Abelian variety $A_{\mathbb{L}}$. An Abelian variety has extensions by vector groups of arbitrary dimension. At the classical level the ${\cal N}=2$ 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 $1<\Delta<2$, in particular their number is at most $r$. This leads to the

The fact that all 4d ${\cal N}=2$ QFT can be twisted á la Witten [ttf1, witten, marino] into a Topological Field Theory (TFT), leads to the following (see main text):

Here ${\cal O}_{Y_{\mathbb{L}}}$ is the structure sheaf of the algebraic variety $Y_{\mathbb{L}}$ defined over $\mathbb{L}$. In this paper we check that (1.4) exactly matches the conditions from quantum consistency of the QFT.

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 $Y_{\mathbb{L}}$. The dimension (over $\mathbb{L}$) of $Y_{\mathbb{L}}$ is the total number of parameters $r+k+f$. A model over ${\mathbb{C}}$ of $Y_{\mathbb{L}}$ will be a fibration

where the fibers and the base $\mathscr{B}$ both have complex dimension $r+k+f$. We claim that the total space $\mathscr{Y}$ is a “symplectic variety” with symplectic form $\boldsymbol{\Omega}$, while the fibers of $\pi$ are Lagrangian submanifolds. We write “symplectic variety” between quotes because we do not have control on singularities in codimension $\geq 2$ and it is possible (even expected) that for some ${\cal N}=2$ QFT these singularities have no crepant resolution.¹¹¹¹11 A crepant resolution is automatically a symplectic variety. Let us make the claim more precise. The Lagrangian fibration (1.5) has a ${\mathbb{C}}^{\times}$ group of automorphisms. Let ${\cal E}$ be the Euler vector field, that is, the generator of the Lie algebra $\mathfrak{aut}(\mathscr{Y})$ normalized so that

The dual 1-form $\boldsymbol{\lambda}=\iota_{\cal E}\boldsymbol{\Omega}$ is then the universal differential which is holomorphic and canonically defined by the symmetries of the geometry. From (1.6) $d\boldsymbol{\lambda}=\boldsymbol{\Omega}$. The Lagrangian fibration (1.5) 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.

The Lagrangian fibration (1.5) 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 $\mathscr{B}$. We can perform the symplectic quotient at fixed values $\underline{t}_{j},\underline{m}_{a}$ of the couplings $t_{j}$ and masses $m_{a}$ setting

where $H$ is the group generated by the vector fields dual to the differentials $dt_{j}$’s and $dm_{a}$’s. The reduced fibration $\mathscr{X}\to\mathscr{C}$ is an ordinary special geometry over the ordinary Coulomb branch $\mathscr{C}$ whose symplectic form class $[\Omega]$ depends linearly on the parameters $t_{j}$, $m_{a}$ (in facts only on the masses $m_{a}$) according to the Duistermaat-Heckman formula [DH]. More in detail: on $\mathscr{Y}$ we have the God given universal Euler differential $\boldsymbol{\lambda}$ which induces a differential $\lambda$ on the symplectic reduction $\mathscr{X}$ such that $d\lambda=\Omega$. The differential $\lambda$ 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 $\lambda$ is very explicit: the basic tool is Picard’s construction of anti-affine groups. It is easy to check in the examples that $\lambda$ is the physically correct differential. The reader may be puzzled. The universal differential $\boldsymbol{\lambda}$ is perfectly holomorphic. How it happens that $\lambda$ 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 ${\mathbb{R}}^{3}$, 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.

Finally we give the physical motivations for the extension of the special geometry from the ordinary version $\mathscr{X}$ to the universal one $\mathscr{Y}$. From the viewpoint of the superspace approach to SUSY QFTs, we can always see the couplings as ${\cal N}=2$ 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 subgroup¹²¹²12 Properly speaking one has to gauge the full group $F$; however in the $\lambda\to 0$ limit the difference becomes inessential. of $F$ with $\mathrm{rank}\,F$ spurion vector supermultiplets, adding a kinetic term for them

and then send $g_{f}\to 0$ while keeping fixed $\langle M_{a}\rangle=\underline{m}_{a}$. In the same way the relevant couplings $t_{j}$ can be seen as ${\cal N}=2$ chiral superfields $T_{j}$ which are frozen to constant values by rescaling their kinetic terms¹³¹³13 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 [ttstar]. by an overall factor $1/\epsilon^{2}$ and then sending $\epsilon\to 0$ while keeping fixed $\langle T_{j}\rangle=\underline{t}_{j}$. The interaction with couplings $t_{j}$ is written as an integral over the ${\cal N}=2$ chiral superspace

where the $\phi_{j}$’s are the chiral superfields whose first components are the operators $u_{j}\in\mathscr{R}$ of dimension $1<\Delta_{j}<2$. If we add only vector spurions, at finite $g_{f}$ we have just an ordinary¹⁴¹⁴14 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 $\mathscr{B}$ of dimension $r+f$. Sending $g_{f}\to 0$ has two effects:

• (1)

  it friezes the Coulomb coordinates associated to the flavor group to the values $\underline{m}_{a}\equiv\langle M_{a}\rangle$, thus replacing $\mathscr{B}$ with the subvariety $\mathscr{C}=\{m_{a}=\underline{m}_{a}\}$ i.e.​ with the ordinary Coulomb branch;

• (2)

  the (generic) fiber over $\mathscr{C}$ degenerates from an Abelian variety of dimension $r+f$ to a semi-Abelian variety ($\equiv$ 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 $\Delta>1$ so the fiber degeneration is more severe and we get a vector extension (an unstable degeneration). Before sending $g_{f},\epsilon\to 0$ both the masses $m_{a}$ and the chiral couplings $t_{j}$ are generators of the universal chiral ring $\mathscr{Q}$. In the limit these generators are replaced by complex numbers, and $\mathscr{Q}$ specializes to the usual chiral ring $\mathscr{R}$. The base $\mathscr{B}$ of the extended geometry is the affine variety $\mathsf{Spec}\,\mathscr{Q}$.

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) $X_{\mathbb{K}}$ is expected to be a Fano variety ${\cal F}$ (possibly singular) of dimension $2r$ with the property that ${\cal F}\setminus K$ is symplectic ($K$ a canonical divisor). E.g.​ for $r=1$ this condition says that ${\cal F}$ is a rational elliptic surface [A11]. The rank of the flavor group is related to the Picard number of ${\cal F}$. Hence the question boils down to an optimal upper bound on the Picard numbers of Fano varieties of dimension $2r$ with ${\cal F}\setminus K$ symplectic and only physically admissible singularities.

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.

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.(2.1), follows from the fact that an ${\cal N}=2$ QFT can be twisted into a Topological Field Theory (TFT) á la Witten [ttf1, witten, marino]. After the twist, the TFT has the following properties:

• (1)

  the IR effective theory is exact for topological amplitudes;

• (2)

  the TFT amplitudes on ${\mathbb{R}}^{3}\times S^{1}$ are independent of the radius $R$ of the circle. As $R\to\infty$ we can use the 4d IR effective theory which is the usual Seiberg-Witten description with Coulomb branch $\mathscr{B}$ [wittenM]. In the limit $R\to 0$ we can use the 3d IR effective theory which is a $\sigma$-model with the hyperKähler target $(\mathscr{Y},\boldsymbol{\Omega})$ [gaiotto2]. The results of the two computations should agree, in particular we must have the same ring of local topological observables $\mathscr{Q}$ in both descriptions. This is eq.(2.1).

The basic idea is that we may twist the ${\cal N}=2$ 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 ${\cal N}=2$ 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.