Gauge Theory and Langlands Duality

LANGLANDS PROGRAM

In 1940 André Weil was put in jail for his refusal to serve in the army. There, he wrote a letter to his sister Simone Weil (a noted philosopher) in response to her question as to what really interested him in his work [We]. This is a remarkable document, in which Weil tries to explain, in fairly elementary terms (presumably, accessible even to a philosopher), the “big picture” of mathematics, the way he saw it. I think this sets a great example to follow for all of us.

Weil writes about the role of analogy in mathematics, and he illustrates it by the analogy that interested him the most: between Number Theory and Geometry.

On the other side we have Riemann surfaces: smooth compact orientable surfaces equipped with a complex structure, and various geometric objects associated to them: vector bundles, their endomorphisms, connections, etc.

Thus, we find a bridge, or a “turntable” – as Weil calls it – between Number Theory and Geometry, and that is the theory of algebraic curves over finite fields.

In other words, we can talk about three parallel tracks

Weil’s idea is to exploit it in the following way: take a statement in one of the three columns and translate it into statements in the other columns [We]: “my work consists in deciphering a trilingual text; of each of the three columns I have only disparate fragments; I have some ideas about each of the three languages: but I know as well there are great differences in meaning from one column to another, for which nothing has prepared me in advance. In the several years I have worked at it, I have found little pieces of the dictionary.” Weil went on to find one of the most spectacular applications of this “Rosetta stone”: what we now call the Weil conjectures describing analogues of the Riemann Hypothesis in Number Theory in the context of algebraic curves over finite fields.

It is easiest to define ${}^{L}\negthinspace G$ in the case when $G$, defined over a field $k$, is split over $k$, that is, contains a maximal split torus $T$ (which is the product of copies of the multiplicative group $GL_{1}$ over $k$). We associate to $T$ two lattices: the weight lattice $X^{*}(T)$ of homomorphisms $T\to GL_{1}$ and the coweight lattice $X_{*}(T)$ of homomorphisms $GL_{1}\to T$. They contain the sets of roots $\Delta\subset X^{*}(T)$ and coroots $\Delta^{\vee}\subset X_{*}(T)$ of $G$, respectively. The quadruple $(X^{*}(T),X_{*}(T),\Delta,\Delta^{\vee})$ is called the root data for $G$ over $k$. The root data determines the split group $G$ up to an isomorphism.

Let us now exchange the lattices of weights and coweights and the sets of simple roots and coroots. Then we obtain the root data

For other groups the correspondence is expected to be much more subtle; for instance, it is not one-to-one. Homomorphisms from the Weil group of $F$ to ${}^{L}\negthinspace G$ (and more general parameters introduced by J. Arthur, see Section 6.2) should parametrize certain collections of automorphic representations called “$L$-packets.” This has only been proved in a few cases so far.

GEOMETRIC LANGLANDS CORRESPONDENCE

The above discussion corresponds to the middle column in the Weil big picture. What should be its analogue in the right column – that is, for complex curves?

Thus, the geometric counterpart of a (unramified) homomorphism $\operatorname{Gal}(\overline{F}/F)\to{}{}^{L}\negthinspace G$ is a homomorphism $\pi_{1}(X)\to{}{}^{L}\negthinspace G$.

Thus, for complex curves the objects on one side of the Langlands correspondence are equivalence classes of flat (holomorphic or algebraic) ${}^{L}\negthinspace G$-bundles $(E,\nabla)$.

What about the other side? Here the answer is not quite as obvious. I will sketch it briefly referring the reader to Section 3 of [F2] for more details.

More precisely, these sheaves are ${\mathcal{D}}$-modules on $\operatorname{Bun}_{G}$. Recall (see, e.g., [KS, GM]) that a ${\mathcal{D}}$-module on a smooth algebraic variety $Z$ is a sheaf of modules over the sheaf ${\mathcal{D}}_{Z}$ of differential operators on $Z$. An example of a ${\mathcal{D}}$-module is the sheaf of sections of a flat vector bundle on $Z$. The sheaf of functions on $Z$ acts on sections by multiplication, so it is an ${\mathcal{O}}_{Z}$-module. But the flat connection also allows us to act on sections by vector fields on $Z$. This gives rise to an action of the sheaf ${\mathcal{D}}_{Z}$, because it is generated by vector fields and functions. Thus, we obtain the structure of a ${\mathcal{D}}$-module.

In our case, $\operatorname{Bun}_{G}$ is not a variety, but an algebraic stack, but the (derived) category of ${\mathcal{D}}$-modules on it has been defined in [BD]. On this category act the so-called Hecke functors. These are labeled by pairs $(x,V)$, where $x\in X$ and $V$ is a finite-dimensional representation of the dual group ${}^{L}\negthinspace G$, and are defined using certain modifications of $G$-bundles.

More precisely, let ${\mathcal{H}}ecke_{\check{\omega}_{1},x}$ be the moduli stack of pairs $({\mathcal{M}},{\mathcal{M}}^{\prime})$ as above. It defines a correspondence over $\operatorname{Bun}_{n}\times\operatorname{Bun}_{n}$:

Allowing the point $x$ vary, we obtain a correspondence between $\operatorname{Bun}_{G}$ and $X\times\operatorname{Bun}_{G}$ and Hecke functors acting from the category of ${\mathcal{D}}$-modules on $\operatorname{Bun}_{G}$ to the (derived) category of ${\mathcal{D}}$-modules on $X\times\operatorname{Bun}_{G}$, which we denote by $H_{V},V\in\operatorname{Rep}{}^{L}\negthinspace G$.

Now let ${\mathcal{E}}=(E,\nabla)$ be a flat ${}^{L}\negthinspace G$-bundle on $X$. A ${\mathcal{D}}$-module ${\mathcal{F}}$ on $\operatorname{Bun}_{G}$ is called a Hecke eigensheaf with respect to ${\mathcal{E}}$ (or with “eigenvalue” ${\mathcal{E}}$) is we have a collection of isomorphisms

compatible with the tensor product structures. Here

is the flat vector bundle on $X$ associated to ${\mathcal{E}}$ and $V$, viewed as a ${\mathcal{D}}$-module. Thus, in particular, we have a collection of isomorphisms

When we vary the point $x$, the “eigenvalues”, which are all isomorphic to the vector space underlying $V$, combine into the flat vector bundle $V_{\mathcal{E}}$ on $X$.

The geometric Langlands conjecture may be stated as follows: for any flat ${}^{L}\negthinspace G$-bundle ${\mathcal{E}}$ there exists a non-zero ${\mathcal{D}}$-module ${\mathcal{F}}_{\mathcal{E}}$ on $\operatorname{Bun}_{G}$ with eigenvalue ${\mathcal{E}}$.

Moreover, if ${\mathcal{E}}$ is irreducible, this ${\mathcal{D}}$-module is supposed to be irreducible (when restricted to each connected component of $\operatorname{Bun}_{G}$) and unique up to an isomorphism (it should also be holonomic and have regular singularities). But if ${\mathcal{E}}$ is not irreducible, we might have a non-trivial (derived) category of Hecke eigensheaves, and the situation becomes more subtle.

Thus, at least for irreducible ${\mathcal{E}}$, we expect the following picture:

CATEGORICAL VERSION

Looking at the correspondence (2.4), we notice that there is an essential asymmetry between the two sides. On the left we have flat ${}^{L}\negthinspace G$-bundles, which are points of a moduli stack $\operatorname{Loc}_{{}^{L}\negthinspace G}$ of flat ${}^{L}\negthinspace G$-bundles (or local systems) on $X$. But on the right we have Hecke eigensheaves, which are objects of a category; namely, the category of ${\mathcal{D}}$-modules on $\operatorname{Bun}_{G}$. Beilinson and Drinfeld have suggested a natural way to formulate it in a more symmetrical way.

This equivalence should send the skyscraper sheaf ${\mathcal{O}}_{\mathcal{E}}$ on $\operatorname{Loc}_{{}^{L}\negthinspace G}$ supported at ${\mathcal{E}}$ to the Hecke eigensheaf ${\mathcal{F}}_{E}$. If this were true, it would mean that Hecke eigensheaves provide a good “basis” in the category of ${\mathcal{D}}$-modules on $\operatorname{Bun}_{G}$, so we would obtain a kind of spectral decomposition of the derived category of ${\mathcal{D}}$-modules on $\operatorname{Bun}_{G}$, like in the Fourier transform. (Recall that under the Fourier transform on the real line the delta-functions $\delta_{x}$, analogues of ${\mathcal{O}}_{\mathcal{E}}$, go to the exponential functions $e^{itx}$, analogues of ${\mathcal{F}}_{\mathcal{E}}$.)

This equivalence has been proved by G. Laumon [Lau2] and M. Rothstein [R] in the abelian case, when $G=GL_{1}$ (or a more general torus). They showed that in this case this is nothing but a version of the Fourier–Mukai transform. Thus, the categorical Langlands correspondence may be viewed as a kind of non-abelian Fourier–Mukai transform (see [F2], Section 4.4).

Unfortunately, a precise formulation of such a correspondence, even as a conjecture, is not so clear because of various subtleties involved. One difficulty is the structure of $\operatorname{Loc}_{{}^{L}\negthinspace G}$. Unlike the case of ${}^{L}\negthinspace G=GL_{1}$, when all flat bundles have the same groups of automorphisms (namely, $GL_{1}$) and $\operatorname{Loc}_{GL_{1}}$ is smooth, for a general group ${}^{L}\negthinspace G$ the groups of automorphisms are different for different flat bundles, and so $\operatorname{Loc}_{{}^{L}\negthinspace G}$ is a complicated stack. For example, if ${}^{L}\negthinspace G$ is a simple Lie group of adjoint type, then a generic flat ${}^{L}\negthinspace G$-bundle has no automorphisms, while the group of automorphisms of the trivial flat bundle is isomorphic to ${}^{L}\negthinspace G$. In addition, unlike $\operatorname{Bun}_{G}$, the stack $\operatorname{Loc}_{{}^{L}\negthinspace G}$ has singularities. All of this has to be reflected on the other side of the correspondence, in ways that have not yet been fully understood.

Nevertheless, the diagram (3.1) gives us a valuable guiding principle to the geometric Langlands correspondence. In particular, it gives us a natural explanation as to why the skyscraper sheaves on $\operatorname{Loc}_{{}^{L}\negthinspace G}$ should correspond to Hecke eigensheaves.

The point is that on the category of ${\mathcal{O}}$-modules on $\operatorname{Loc}_{{}^{L}\negthinspace G}$ we also have a collection of functors $W_{V}$, parametrized by the same data as the Hecke functors $H_{V}$. Following physics terminology, we will call them Wilson functors. These functors act from the category of ${\mathcal{O}}$-modules on $\operatorname{Loc}_{{}^{L}\negthinspace G}$ to the category of sheaves on $X\times\operatorname{Loc}_{{}^{L}\negthinspace G}$, which are ${\mathcal{D}}$-modules along $X$ and ${\mathcal{O}}$-modules along $\operatorname{Loc}_{{}^{L}\negthinspace G}$.

To define them, observe that we have a tautological ${}^{L}\negthinspace G$-bundle ${\mathcal{T}}$ on $X\times\operatorname{Loc}_{{}^{L}\negthinspace G}$, whose restriction to $X\times{\mathcal{E}}$, where ${\mathcal{E}}=(E,\nabla)$, is $E$. Moreover, $\nabla$ gives us a partial connection on ${\mathcal{T}}$ along $X$. For a representation $V$ of ${}^{L}\negthinspace G$, let $V_{\mathcal{T}}$ be the associated vector bundle on $X\times\operatorname{Loc}_{{}^{L}\negthinspace G}$, with a connection along $X$.

Let $p:X\times\operatorname{Loc}_{{}^{L}\negthinspace G}\to\operatorname{Loc}_{{}^{L}\negthinspace G}$ be the projection onto the second factor. By definition,

(note that by construction $V_{\mathcal{T}}$ carries a connection along $X$ and so the right hand side really is a ${\mathcal{D}}$-module along $X$).

Now, the conjectural equivalence (3.1) should be compatible with the Wilson/Hecke functors in the sense that

where $C$ denotes this equivalence (from left to right).

In particular, observe that the skyscraper sheaf ${\mathcal{O}}_{\mathcal{E}}$ at ${\mathcal{E}}\in\operatorname{Loc}_{{}^{L}\negthinspace G}$ is obviously an eigensheaf of the Wilson functors:

Indeed, tensoring a skyscraper sheaf with a vector bundle is the same as tensoring it with the fiber of this vector bundle at the point of support of this skyscraper sheaf. Therefore (3.3) implies that ${\mathcal{F}}_{\mathcal{E}}=C({\mathcal{O}}_{\mathcal{E}})$ must satisfy the Hecke property (2.3). In other words, ${\mathcal{F}}_{\mathcal{E}}$ should be a Hecke eigensheaf on $\operatorname{Bun}_{G}$ with eigenvalue ${\mathcal{E}}$. Thus, we obtain a natural explanation of the Hecke property of ${\mathcal{F}}_{\mathcal{E}}$: it follows from the compatibility of the categorical Langlands correspondence (3.1) with the Wilson/Hecke functors.

Let us summarize: the conjectural equivalence (3.1) gives us a natural and convenient framework for the geometric Langlands correspondence. It is this equivalence that Kapustin and Witten have related to the $S$-duality of 4D super–Yang–Mills.

ENTER PHYSICS

We will now add a fourth column to Weil’s big picture, which we will call “Quantum Physics”:

In the context of the Langlands Program, the last column means $S$-duality and Mirror Symmetry of certain 4D and 2D quantum field theories, which we will now briefly describe following [KW].

We start with the pure 4D Yang–Mills (or gauge) theory on a Riemannian four-manifold $M_{4}$. Let $G_{c}$ be a compact connected simple Lie group. The classical (Euclidean) action is a functional on the space of connections on arbitrary principal $G_{c}$-bundles ${\mathcal{P}}$ on $M_{4}$ given by the formula

Here $F_{A}$ is the curvature of the connection $A$ (a ${\mathfrak{g}}$-valued two-form on $M_{4}$), $\star\,$ is the Hodge star operator, and $\operatorname{Tr}$ is the invariant bilinear form on the Lie algebra ${\mathfrak{g}}$ normalized in such a way that the second term is equal to $i\theta k$, where $k$ could be an arbitrary integer, if $G_{c}$ is simply-connected. The second term is equal to $i\theta$ times the second Chern class $c_{2}({\mathcal{P}})$ of the bundle ${\mathcal{P}}$ and hence is topological. Correlation functions are given by path integrals of the form $\int e^{-I}$ over the space of connections modulo gauge transformations. Hence they may be written as Fourier series in $e^{i\theta}$ (or its root if $G_{c}$ is not simply-connected) such that the coefficient in front of $e^{i\theta n}$ is the sum of contributions from bundles ${\mathcal{P}}$ with $c_{2}({\mathcal{P}})=-n$.

It is customary to combine the two parameters, $g$ and $\theta$, into one complex coupling constant

Next, we consider $N=4$ supersymmetric extension of this model. This means that we add fermionic and bosonic fields in such a way that the action of the Lorentz group (we will work in Euclidean signature, where this group becomes $SO(4)$) is extended to an action of an appropriate supergroup (see [KW] or the books [D] for background on supersymmetric quantum field theory).

The $S$-duality of this theory is the statement that the theory with gauge group $G_{c}$ and complex coupling constant $\tau$ is equivalent to the theory with the Langlands dual gauge group ${}^{L}\negthinspace G_{c}$ and coupling constant ${}^{L}\tau=-1/n_{{\mathfrak{g}}}\tau$:

where $n_{{\mathfrak{g}}}$ is the lacing number of the Lie algebra ${\mathfrak{g}}$ (equal to $1$ for simply-laced Lie algebras, $2$ for $B_{n},C_{n}$ and $F_{4}$, and $3$ for $G_{2}$). This is an extension of the duality (0.1) of [MO] discussed in the introduction to non-zero values of $\theta$ (with $g$ normalized in a slightly different way). In addition, for simply-laced $G_{c}$ the path integral is a Fourier series in $e^{i\theta}$, so $\theta$ may be shifted by an integer multiple of $2\pi$ without changing the path integral. Thus, we also have the equivalence

We want to focus next on a “topological sector” of this theory. This means that we pick an element $Q$ in the Lie superalgebra ${\mathfrak{s}}$ of the super-Lorentz group (the supergroup extension of $SO(4)$) such that $Q^{2}=0$, and such that the stress tensor (which is a field responsible for variation of the metric on $M_{4}$) is equal to the commutator of $Q$ and another field. Let us restrict ourselves to those objects (fields, boundary conditions, etc.) in the theory which commute with this $Q$. This is a particular (and relatively small) sector of the full quantum field theory, in which all quantities (such as correlation functions) are topological, that is, metric-independent. This sector is what is usually referred to as Topological Field Theory (TFT).

There is a problem, however. For this $Q\in{\mathfrak{s}}$ to be well-defined on an arbitrary manifold $M_{4}$, it has to be invariant under the action of the Lorentz group $SO(4)$ – more precisely, its double cover $Spin(4)$. Unfortunately, there are no such elements in our Lie superalgebra ${\mathfrak{s}}$. In order to obtain such an element, one uses a trick, called twisting (see, e.g., [Wi6]). Our theory has an additional group of automorphisms commuting with the action of $Spin(4)$, called $R$-symmetry; namely, the group $Spin(6)$. We can use it to modify the action of $Spin(4)$ on the fields of the theory and on the Lie superalgebra ${\mathfrak{s}}$ as follows: define a new action of $Spin(4)$ equal to the old action together with the action coming from a homomorphism $Spin(4)\to Spin(6)$ and the action of $Spin(6)$ by $R$-symmetry. One might then be able to find a differential $Q\in{\mathfrak{s}}$ invariant under this new action of $Spin(4)$.

There are essentially three different choices for doing this, as explained in [VW]. The first two are similar to the twists used in Witten’s construction of a topological field theory that yields Donaldson invariants of four-manifolds (which is a topological twist of an $N=2$ supersymmetric Yang–Mills theory) [Wi1]. It is the third twist, studied in detail in [KW], that is relevant to the geometric Langlands. For this twist there are actually two linearly independent (and anti-commuting with each other) operators, $Q_{l}$ and $Q_{r}$, which square to . We can therefore use any linear combination

Kapustin and Witten assume that the four-manifold $M_{4}$ has the form

where $X$ is a closed Riemann surface (this will be the algebraic curve of the geometric Langlands) and $\Sigma$ is a Riemann surface with a boundary (which we may simply take to be a half-plane). They study the limit of the topological gauge theory on this manifold when $X$ becomes very small (this is called “compactification of the theory on $X$”). In this limit the theory is described by an effective two-dimensional topological field theory on $\Sigma$. In earlier works [BJSV, HMS] the latter theory was identified with the (twisted) topological sigma model on $\Sigma$ with the target manifold ${\mathcal{M}}_{H}(G)$, the Hitchin moduli space of Higgs $G$-bundles on $X$. Moreover, the $S$-duality of the supersymmetric gauge theories on $\Sigma\times X$ (for particular values of $\tau$ and $t$) becomes Mirror Symmetry between the topological sigma models with the targets ${\mathcal{M}}_{H}(G)$ and ${\mathcal{M}}_{H}({}{}^{L}\negthinspace G)$.

Next, we look at the boundary conditions in the gauge theories, which give rise to branes in these sigma models. $S$-duality yields an equivalence of the categories of branes for ${\mathcal{M}}_{H}(G)$ and ${\mathcal{M}}_{H}({}{}^{L}\negthinspace G)$ (also known, after M. Kontsevich, as Homological Mirror Symmetry). Kapustin and Witten have related this equivalence to the categorical geometric Langlands correspondence (3.1). Thus, they establish a link between $S$-duality and geometric Langlands duality. We describe this in more detail in the next section.

MIRROR SYMMETRY OF HITCHIN MODULI SPACES

In [Hi1] N. Hitchin introduced a remarkable hyper-Kähler manifold ${\mathcal{M}}_{H}(G)$ for each smooth projective complex algebraic curve $X$ and reductive Lie group $G$. It is easiest to describe it in its complex structure $I$, in which it is the moduli space of semi-stable Higgs bundles on $X$. Recall that a Higgs $G$-bundle on $X$ is a pair $(E,\phi)$, where $E$ is a (algebraic) $G$-bundle on $X$ and $\phi$ is a Higgs field on it, that is,

where ${\mathfrak{g}}_{E}=E\underset{G}{\times}{\mathfrak{g}}$ is the adjoint vector bundle.

In the complex structure $J$, however, ${\mathcal{M}}_{H}(G)$ is described as the moduli space of semi-stable flat bundles, that is, pairs $(E,\nabla)$, where $E$ is again (algebraic) $G$-bundle on $X$ and $\nabla$ is (algebraic) connection on $E$. To distinguish between it as a complex algebraic variety from the moduli space of Higgs bundles we will denote it by ${\mathcal{Y}}(G)$. The two are isomorphic as real manifolds (this is the statement of non-abelian Hodge theory [Hi1, C, S1]), but not as complex (or algebraic) manifolds.

There are two types of twisted supersymmetric two-dimensional sigma models with Kähler target manifolds: $A$-model and $B$-model (see [Wi2]). The former depends on the symplectic structure on the target manifold and the latter depends on the complex structure.

Kapustin and Witten start with two topological twisted super–Yang–Mills theories on $\Sigma\times X$. One has gauge group $G_{c}$, twisting parameter $t=1$, and $\theta=0$. The other, $S$-dual theory, has gauge group ${}^{L}\negthinspace G_{c}$, the twisting parameter ${}^{L}t=i$, and ${}^{L}\theta=0$ (neither of these topological theories depends on $g$ [KW]).As explained in [KW, Ka], for some other values of parameters one obtains the so-called quantum geometric Langlands correspondence (see [F2], Section 6.3). They show that after compactification on $X$ the first theory becomes the $A$-model with the target manifold ${\mathcal{M}}_{H}(G)$ and the symplectic structure $\omega_{K}$, which is the Kähler form for the complex structure $K$ on ${\mathcal{M}}_{H}(G)$. This symplectic structure has a nice geometric description. Note that the Higgs field $\phi$ is an element of $H^{0}(X,{\mathfrak{g}}_{E}\otimes K_{X})$, which is isomorphic to the cotangent space to $E$, viewed as a point of $\operatorname{Bun}_{G}$, the moduli stack of $G$-bundles on $X$. Thus, ${\mathcal{M}}_{H}(G)$ is almost the cotangent bundle to $\operatorname{Bun}_{G}$; “almost” because we impose the semi-stability condition on the Higgs bundle. The symplectic form $\omega_{K}$ comes from the standard symplectic form on the cotangent bundle (which is the imaginary part of the holomorphic symplectic form).

The second gauge theory becomes, after compactification on $X$, the $B$-model with the target manifold ${\mathcal{Y}}({}{}^{L}\negthinspace G)$; that is, ${\mathcal{M}}_{H}({}{}^{L}\negthinspace G)$ with respect to the complex structure $J$.

After dimensional reduction from 4D to 2D, the $S$-duality of super–Yang–Mills theories becomes Mirror Symmetry between the $A$-model with the target manifold ${\mathcal{M}}_{H}(G)$ (and symplectic structure $\omega_{K}$) and the $B$-model with the target manifold ${\mathcal{Y}}({}{}^{L}\negthinspace G)$ (and complex structure $J$).

Remark. As explained in [KW], there is also Mirror Symmetry between $A$- and $B$-models with respect to other symplectic and complex structures. For instance, there is Mirror Symmetry, studied in [DP, Hi3, Ari], between the $B$-models on ${\mathcal{M}}_{H}(G)$ and ${\mathcal{M}}_{H}({}^{L}\negthinspace G)$ with respect to the complex structures $I$ on both of them. In what follows we will not discuss these additional dualities.

This is slightly non-canonical, because there is no canonical choice of generators $P_{i}$ in general. More canonically, we have a map to

As the result, we obtain two fibrations over the same base ${\mathbf{B}}$:

For generic $b\in{\mathbf{B}}$ (the connected components of) the fibers ${}^{L}{\mathbf{F}}_{b}$ and ${\mathbf{F}}_{b}$ of these Hitchin fibrations are smooth tori, which are in fact isomorphic to abelian varieties in the complex structure $I$. For instance, for $G=SL_{n}$, ${\mathbf{F}}_{b}$ is the generalized Prym variety of the spectral curve associated to $b$, which is a smooth degree $n$ cover of $X$ if $b$ is generic.

We are interested in the study of the $B$-model with the target ${\mathcal{M}}_{H}({}{}^{L}\negthinspace G)$, with respect to the complex structure $J$ (that is, the moduli space ${\mathcal{Y}}({}{}^{L}\negthinspace G)$ of flat bundles), and the $A$-model with the target ${\mathcal{M}}_{H}(G)$, with respect to the symplectic structure $\omega_{K}$. These two topological field theories are expected to be equivalent to each other. Therefore anything we can say about one of them should have a counterpart in the other. For instance, their cohomologies may be interpreted as the spaces of vacua in these field theories, and hence they should be isomorphic. This has indeed been verified by Hausel and Thaddeus [HT] in the case when $G=SL_{n},{}{}^{L}\negthinspace G=PGL_{n}$ (since the Hitchin moduli spaces are non-compact, special care has to be taken to properly define these cohomologies, see [HT]).

To make contact with the geometric Langlands correspondence, Kapustin and Witten study in [KW] the categories of branes in these two topological field theories.

2 Categories of branes

Branes in two-dimensional sigma models are certain generalizations of boundary conditions. When writing path integral for maps $\Phi:\Sigma\to M$, where $\Sigma$ has a boundary, we need to specify boundary conditions for $\Phi$ on $\partial\Sigma$. We may also “couple” the sigma model to another quantum field theory on $\partial\Sigma$ (that is, modify the action by a boundary term) which may be interpreted as a decoration of the boundary condition. In topological field theory these conditions should preserve the supersymmetry, which leads to natural restrictions.

A typical example of a boundary condition is specifying that $\Phi(\partial\Sigma)$ belongs to a submanifold $M^{\prime}\subset M$. In the $B$-model the target manifold $M$ is a complex manifold, and in order to preserve the supersymmetry $M^{\prime}$ has to be a complex submanifold. In the $A$-model, $M$ is a symplectic manifold and $M^{\prime}$ should be Lagrangian. Coupling to field theories on $\partial\Sigma$ allows us to introduce into the picture a holomorphic vector bundle on $M^{\prime}$ in the case of $B$-model, and a flat unitary vector bundle on $M^{\prime}$ in the case of $A$-model.

More generally, the category of branes in the $B$-model with a complex target manifold $M$ (called $B$-branes) is the (derived) category of coherent sheaves on $M$, something that is fairly well understood mathematically. The category of branes in the $A$-model with a symplectic target manifold $M$ (called $A$-branes) is less understood. It is believed to contain what mathematicians call the Fukaya category, typical objects of which are pairs $(L,\nabla)$, where $L\subset M$ is a Lagrangian submanifold and $\nabla$ is a flat unitary vector bundle on $L$. However (and this turns out to be crucial for applications to the geometric Langlands), it also contains more general objects, such as coisotropic submanifolds of $M$ equipped with vector bundles with unitary connection.

Under the Mirror Symmetry between the sigma models with the target manifolds ${\mathcal{Y}}({}{}^{L}\negthinspace G)$ and ${\mathcal{M}}_{H}(G)$ we therefore expect to have the following equivalence of (derived) categories of branes (often referred to, after Kontsevich, as Homological Mirror Symmetry):

It is this equivalence that Kapustin and Witten have related to the categorical Langlands correspondence (3.1). The category on the left in (5.2) is the (derived) category of coherent sheaves on ${\mathcal{Y}}({}^{L}\negthinspace G)$, which is the moduli space of semi-stable flat ${}^{L}\negthinspace G$-bundles on $X$. It is closely related to the (derived) category of coherent sheaves (or, equivalently, ${\mathcal{O}}$-modules) on $\operatorname{Loc}_{{}^{L}\negthinspace G}$, which appears on the left of (3.1). The difference is that, first of all, $\operatorname{Loc}_{{}^{L}\negthinspace G}$ is the moduli stack of flat ${}^{L}\negthinspace G$-bundles on $X$, whereas ${\mathcal{Y}}({}^{L}\negthinspace G)$ is the moduli space of semi-stable ones. Second, from the physics perspective it is more natural to consider coherent sheaves on ${\mathcal{Y}}({}^{L}\negthinspace G)$ with respect to its complex analytic rather than algebraic structure, whereas in (3.1) we consider algebraic ${\mathcal{O}}$-modules on $\operatorname{Loc}_{{}^{L}\negthinspace G}$. These differences aside, these two categories are very similar to each other. They certainly share many objects, such as skyscraper sheaves supported at points corresponding to stable flat ${}^{L}\negthinspace G$-bundles which we will discuss momentarily.

3 Triangle of equivalences

The categories on the right in (3.1) and (5.2) appear at first glance to be quite different. But Kapustin and Witten have suggested that they should be equivalent to each other as well. Thus, we obtain the following triangle of derived categories:

\textstyle{\boxed{\text{B-branes on }{\mathcal{Y}}({}^{L}\negthinspace G)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$\textstyle{\boxed{{\mathcal{D}}\text{-modules on }\operatorname{Bun}_{G}}}$ (5.3) The upper arrow represents Homological Mirror Symmetry (5.2) whereas the lower arrow represents the categorical Langlands correspondence (3.1).

According to [KW], Section 11, the vertical arrow is another equivalence that has nothing to do with either Mirror Symmetry or geometric Langlands. It should be a general statement linking the (derived) category of ${\mathcal{D}}$-modules on a variety $M$ and the (derived) category of $A$-branes on its cotangent bundle $T^{*}M$ (recall that ${\mathcal{M}}_{H}(G)$ is almost equal to $T^{*}\operatorname{Bun}_{G}$). Kapustin and Witten have proposed the following functor from the category of $A$-branes on $T^{*}M$ with respect to the symplectic structure ${\rm Im}\,\Omega$ (where $\Omega$ is the holomorphic symplectic form on $T^{*}M$) to the category of ${\mathcal{D}}$-modules on $M$:

where ${\mathcal{A}}_{cc}$ is a “canonical coisotropic brane” on $T^{*}M$. This is $T^{*}M$ itself (viewed as a coisotropic submanifold) equipped with a line bundle with connection satisfying special properties. They argued on physical grounds that the right hand side of (5.4) may be “sheafified” along $M$, and moreover that the corresponding sheaf of rings $\operatorname{Hom}({\mathcal{A}}_{cc},{\mathcal{A}}_{\operatorname{cc}})$ is nothing but the sheaf of differential operators on ${\mathcal{M}}_{H}(G)$.More precisely, it is the sheaf of differential operators acting on a square root of the canonical line bundle on $\operatorname{Bun}_{G}$, but we will ignore this subtlety here. Hence the (sheafified) right hand side of (5.4) should be a ${\mathcal{D}}$-module. While this argument has not yet been made mathematically rigorous, it allows one to describe important characteristics of the ${\mathcal{D}}$-module associated to an $A$-brane, such as its reducibility, the open subset of $M$ where it is represented by a local system, the rank of this local systems, and even its monodromy (see Section 4 of [FW]).

An alternative (and mathematically rigorous) approach to establishing an equivalence between the categories of $A$-branes and ${\mathcal{D}}$-modules has also been proposed by D. Nadler and E. Zaslow [NZ, Nad].

Though a lot of work still needs to be done to distill this connection and reconcile different approaches, this is clearly a very important and beautiful idea on its own right.

Thus, according to Kapustin and Witten, the (categorical) geometric Langlands correspondence (3.1) may be obtained in two steps. The first step is the Homological Mirror Symmetry (5.2) of the Hitchin moduli spaces for two dual groups, and the second step is the above link between the $A$-branes and ${\mathcal{D}}$-modules.

There is actually more structure in the triangle (5.3). On each of these three categories we have an action of certain functors, and all equivalences between them are supposed to commute with these functors. We have already described the functors on two of these categories in Section 3: these are the Wilson and Hecke functors. The functors acting on the categories of $A$-branes, were introduced in [KW] as the two-dimensional shadows of the ’t Hooft loop operators in 4D super–Yang–Mills theory. Like the Hecke functors, they are defined using modifications of $G$-bundles, but only those modifications which preserve the Higgs field. The $S$-duality of super–Yang–Mills theories is supposed to exchange the ’t Hooft operators and the Wilson operators (whose two-dimensional shadows are the functors described in Section 3), and this is the reason why we expect the equivalence (5.2) to commute with the action of these functors.

As explained above, the central objects in the geometric Langlands correspondence are Hecke eigensheaves attached to flat ${}^{L}\negthinspace G$-bundles. Recall that the Hecke eigensheaf ${\mathcal{F}}_{\mathcal{E}}$ is the ${\mathcal{D}}$-module attached to the skyscraper sheaf ${\mathcal{O}}_{\mathcal{E}}$ supported at a point ${\mathcal{E}}$ of $\operatorname{Loc}_{{}^{L}\negthinspace G}$ under the conjectural equivalence (3.1). These ${\mathcal{D}}$-modules have very complicated structure. What can we learn about them from the point of view of Mirror Symmetry?

Let us assume first that ${\mathcal{E}}$ has no automorphisms other than those coming from the center of ${}^{L}\negthinspace G$. Then it is a smooth point of ${\mathcal{Y}}({}^{L}\negthinspace G)$. These skyscraper sheaves ${\mathcal{O}}_{\mathcal{E}}$ are the simplest examples of $B$-branes on ${\mathcal{Y}}({}^{L}\negthinspace G)$ (called -branes). What is the corresponding $A$-brane on ${\mathcal{M}}_{H}(G)$?

The answer is surprisingly simple. Let $b\in{\mathbf{B}}$ be the projection of ${\mathcal{E}}$ to the base of the Hitchin fibration. For ${\mathcal{E}}$ satisfying the above conditions the Hitchin fiber ${}^{L}{\mathbf{F}}_{b}$ is a smooth torus (it is actually an abelian variety in the complex structure $I$, but now we look at it from the point of view of complex structure $J$, so it is just a smooth torus). It is identified (possibly, up to a choice of the square root of the canonical line bundle $K_{X}$, see the footnote on page 10) with the moduli space of flat unitary line bundles on the dual Hitchin fiber ${\mathbf{F}}_{b}$, which happens to be a Lagrangian submanifold of ${\mathcal{M}}_{H}(G)$. The Mirror Symmetry sends the $B$-brane ${\mathcal{O}}_{\mathcal{E}}$ to the $A$-brane which is the pair $({\mathbf{F}}_{b},\nabla_{\mathcal{E}})$, the Lagrangian submanifold ${\mathbf{F}}_{b}$ of ${\mathcal{M}}_{H}(G)$, together with the flat unitary line bundle on it corresponding to ${\mathcal{E}}$:

Since ${\mathcal{O}}_{\mathcal{E}}$ is obviously an eigenbrane of the Wilson functors (as we discussed in Section 3), the $A$-brane $({\mathbf{F}}_{b},\nabla_{\mathcal{E}})$ should be an eigenbrane of the ’t Hooft functors. This may in fact be made into a precise mathematical conjecture, and Kapustin and Witten have verified it explicitly in some cases.

Thus, the $A$-branes associated to the simplest $B$-branes turn out to be very nice and simple. This is in sharp contrast with the structure of the corresponding ${\mathcal{D}}$-modules, which is notoriously complicated in the non-abelian case. Therefore the formalism of $A$-branes developed in [KW] has clear advantages. It replaces ${\mathcal{D}}$-modules with $A$-branes that are much easier to “observe experimentally” and to analyze explicitly. One can hope to use this new language in order to gain insights into the structure of the geometric Langlands correspondence. It has already been used in [FW] for understanding what happens in the endoscopic case as explained in the next section.

4 Ramification

Up to now we have considered the unramified case of the geometric Langlands correspondence, in which the objects on the Galois side of the correspondence are holomorphic ${}^{L}\negthinspace G$-bundles on our curve $X$ with a holomorphic connection. These flat bundles give rise to homomorphisms $\pi_{1}(X)\to{}^{L}\negthinspace G$. In the classical Langlands correspondence one looks at more general homomorphisms $\pi_{1}(X\backslash\{x_{1},\ldots,x_{n}\})\to{}^{L}\negthinspace G$. Thus, we look at holomorphic ${}^{L}\negthinspace G$-bundles on $X$ with meromorphic connections which have poles at finitely many points of $X$. The connections with poles of order one (regular singularities) correspond to tame ramification in the classical Langlands Program. Those with poles of orders higher than one (irregular singularities) correspond to wild ramification.

Mathematically, the ramified geometric Langlands correspondence has been studied in [FGa] and follow-up papers (see [F3] for an exposition), using the affine Kac–Moody algebras of critical level and generalizing the Beilinson–Drinfeld approach [BD] to allow ramification.

S. Gukov and E. Witten [GW1] have explained how to include tame ramification in the $S$-duality picture. Physicists have a general way of including into a quantum field theory on a manifold $M$ objects supported on submanifolds of $M$. An example of this is the surface operators in 4D super–Yang–Mills theory, supported on two-dimensional submanifolds of the four-manifold $M_{4}$. If we include such an operator, we obtain a certain modification of the theory. Let us again take $M_{4}=\Sigma\times X$ and take this submanifold to be of the form $\Sigma\times x,x\in X$. Gukov and Witten show that for a particular class of surface operators the dimensional reduction of the resulting theory is the sigma model on $\Sigma$ with a different target manifold ${\mathcal{M}}_{H}(G,x)$, the moduli space of semi-stable Higgs bundles with regular singularity at $x\in X$. It is again hyper-Kähler, and in the complex structure $J$ it has a different incarnation as the moduli space of semi-stable bundles with a connections having regular singularity (see [S]).

The moduli space ${\mathcal{M}}_{H}(G,x)$ has parameters $(\alpha,\beta,\gamma)$, which lie in the (compact) Cartan subalgebra of ${\mathfrak{g}}$ (see [S]). For generic parameters, this moduli space parametrizes semi-stable triples $(E,\phi,{\mathcal{L}})$, where $E$ is a (holomorphic) $G$-bundle, $\phi$ is a Higgs field which has a pole at $x$ of order one whose residue belongs to the regular semi-simple conjugacy class of $\frac{1}{2}(\beta+i\gamma)$, and ${\mathcal{L}}$ is a flag in the fiber of $E$ at $x$ which is preserved by this residue (the remaining parameter $\alpha$ determines the flag). Various degenerations of parameters give rise to similar moduli spaces in which the residues of the Higgs fields could take arbitrary values. The moduli spaces ${\mathcal{M}}_{H}(G,x)$ and ${\mathcal{M}}_{H}({}^{L}\negthinspace G,x)$ (with matching parameters) are equipped with a pair of mirror dual Hitchin fibrations, and the Mirror Symmetry between them is again realized as fiberwise $T$-duality (for generic fibers which are again smooth dual tori).

The $S$-duality of the super–Yang–Mills theories, associated to the dual groups $G_{c}$ and ${}^{L}\negthinspace G_{c}$, with surface operators gives rise to an equivalence of categories of $A$- and $B$-branes on ${\mathcal{M}}_{H}(G,x)$ and ${\mathcal{M}}_{H}({}^{L}\negthinspace G,x)$. The mirror dual for a generic 0-brane on ${\mathcal{M}}_{H}({}^{L}\negthinspace G,x)$ is the $A$-brane consisting of a Hitchin fiber and a flat unitary line bundle on it, as in the unramified case.

The analysis of [GW1] leads to many of the same conclusions as those obtained in [FGa] by using representations of affine Kac–Moody algebras and two-dimensional conformal field theory.

Gukov and Witten also considered [GW2] more general surface operators associated to coadjoint orbits in ${\mathfrak{g}}$ and ${}^{L}{\mathfrak{g}}$. The $S$-duality between these surface operators leads to some non-trivial and unexpected relations between these orbits. Gukov and Witten present many interesting examples of this in [GW2] drawing connections with earlier work done by mathematicians.

In [Wi3], Witten has generalized the analysis of [GW1, GW2] to the case of wild ramification.

MORE GENERAL BRANES

In the previous section we discussed applications of Mirror Symmetry of the dual Hitchin fibrations to the geometric Langlands correspondence. We saw that the $A$-branes associated to the $B$-branes supported at the generic flat ${}^{L}\negthinspace G$-bundles have very simple description: these are the Hitchin fibers equipped with flat unitary line bundles. But what about the $B$-branes supported at more general flat ${}^{L}\negthinspace G$-bundles? Can we describe explicitly the $A$-branes dual to them?

This question goes to the heart of the subtle interplay between physics and mathematics of Langlands duality. Trying to answer this question, we will see the limitations of the above analysis and a way for its generalization incorporating more general branes. This will lead us to surprising physical interpretation of deep mathematical concepts such as endoscopy and Arthur’s $SL_{2}$.

The generic flat ${}^{L}\negthinspace G$-bundles, for which Mirror Symmetry works so nicely, are the ones that have no automorphisms (apart from those coming from the center of ${}^{L}\negthinspace G$). They correspond to smooth points of ${\mathcal{Y}}({}^{L}\negthinspace G)$ such that the corresponding Hitchin fiber is also smooth. We should consider next the singularities of ${\mathcal{Y}}({}^{L}\negthinspace G)$. The simplest of those are the orbifold singularities. We will discuss them, and their connection to endoscopy, in the next subsection, following [FW]. We will then talk about more general singularities corresponding to flat ${}^{L}\negthinspace G$-bundles with continuous groups of automorphisms, and what we can learn about the corresponding categories from physics.

We start with the mildest possible singularities in ${\mathcal{Y}}({}^{L}\negthinspace G)$; namely, the orbifold singular points. The corresponding flat ${}^{L}\negthinspace G$-bundles are those having finite groups of automorphisms (modulo the center). In the classical Langlands correspondence the analogous Galois representations are called endoscopic. They, and the corresponding automorphic representations, play an important role in the stabilization of the trace formula.

This was analyzed very explicitly in [FW] in the case when $X$ is an elliptic curve. Here we allow a single point of tame ramification (along the lines of Section 5.4) – this turns out to be better for our purposes than the unramified case. The corresponding Hitchin moduli spaces are two-dimensional. They fiber over the same one-dimensional vector space, and the fibers over all but three points in the base are smooth elliptic curves. The three pairs of dual singular fibers look as follows:

Singular Hitchin fiber in Singular Hitchin fiber in

the $A$-model, $G=SL_{2}$. the $B$-model, ${}^{L}\negthinspace G=SO_{3}$.

The mirror dual statement, shown in [FW], is that we have a similar formula for the action of the corresponding ’t Hooft operators on the ${\mathcal{A}}$-branes ${\mathcal{A}}_{1}$ and ${\mathcal{A}}_{2}$. Again, only their sum (or union), which gives the entire Hitchin fiber, is an eigenbrane of the ’t Hoof operators.

Since an eigenbrane ${\mathcal{A}}$ decomposes into two irreducible branes ${\mathcal{A}}_{1}$ and ${\mathcal{A}}_{2}$, the corresponding Hecke eigensheaf ${\mathcal{F}}$ on $\operatorname{Bun}_{G}$ should also decompose as a direct sum of two ${\mathcal{D}}$-modules, ${\mathcal{F}}_{1}$ and ${\mathcal{F}}_{2}$, corresponding to ${\mathcal{A}}_{1}$ and ${\mathcal{A}}_{2}$, respectively. Furthermore, these two ${\mathcal{D}}$-modules should then separately satisfy an analogue of formula (6.2), which is a natural modification of the standard Hecke property. We called it in [FW] the fractional Hecke property, and the ${\mathcal{D}}$-modules ${\mathcal{F}}_{1}$ and ${\mathcal{F}}_{2}$ fractional Hecke eigensheaves. We have also generalized this notion to other groups in [FW].

The upshot of all this is that by analyzing the categories of $A$-branes supported on the singular Hitchin fibers, we learn many things about the geometric Langlands correspondence (and even the Langlands correspondence for curves over finite fields) which would have been very difficult to see directly using the conventional formalism of ${\mathcal{D}}$-modules. This is a good illustration of the power of this new method.

2 S𝑆S-duality of more general boundary conditions

More general flat ${}^{L}\negthinspace G$-bundles have continuous groups of automorphisms. For instance, generic flat bundles reduced to a Cartan subalgebra ${}^{L}\negthinspace H$ have the group of automorphisms ${}^{L}\negthinspace H$. Or consider the trivial flat ${}^{L}\negthinspace G$-bundle, whose group of automorphisms is ${}^{L}\negthinspace G$ itself. What are the $A$-branes corresponding to these flat ${}^{L}\negthinspace G$-bundles?

The picture of two dual Hitchin fibrations discussed above is too naive to answer this question. The reason is that even if the flat bundles with continuous groups of automorphisms are semi-stable (which is not necessarily the case), they correspond to points of ${\mathcal{Y}}({}^{L}\negthinspace G)$ with singularity so severe that the category of $B$-branes corresponding to it cannot be described solely in terms of the moduli space ${\mathcal{Y}}({}^{L}\negthinspace G)$. In fact, the definition of the sigma model itself is problematic for singular target manifolds.

In order to understand better what is going on we should go back to the four-dimensional gauge theory and look more closely at the $S$-duality of boundary conditions there. From the physics perspective, this is the “master duality” and everything should follow from it. The Mirror Symmetry of the Hitchin fibrations is but the first approximation to the $S$-duality when we compactify the theory to two dimensions.

It is instructive to recall how one obtains the Hitchin moduli spaces in the first place: Each of the $S$-dual gauge theories has a differential $Q$ such that $Q^{2}=0$, and we study the corresponding topological field theories. In the topological theory the path integral localizes on the moduli space of solutions to the “BPS equations”, which read $Q\cdot\Psi=0$, for all fermionic fields $\Psi$ of our theory (since $Q$ is fermionic and we want the equations on the bosonic degrees of freedom). After that we make dimensional reduction of these equations. This means that we assume that our four-manifold has the form $\Sigma\times X$ and the fields on $\Sigma$ vary “slowly” along $\Sigma$. The corresponding equations have been written in [KW]:

where $d_{A}$ is the exterior derivative corresponding to the connection $A$, and $\star$ is the Hodge star operator. These are precisely the Hitchin equations [Hi1] describing the moduli spaces ${\mathcal{M}}_{H}(G)$ or ${\mathcal{M}}_{H}({}^{L}\negthinspace G)$ (depending on which side of $S$-duality we are on). For example, points of ${\mathcal{M}}_{H}({}^{L}\negthinspace G)$ in the complex structure $J$ are semi-stable flat ${}^{L}\negthinspace G$-bundles on $X$. The flat connection on this bundle is given by the formula $\nabla=A+i\phi$ (the flatness of $\nabla$ is a corollary of (6.3)). This is how the ($B$-twisted) sigma model on $\Sigma$ with values in ${\mathcal{M}}_{H}({}^{L}\negthinspace G)$ appears in this story. One obtains the ($A$-twisted) sigma model with target ${\mathcal{M}}_{H}(G)$ in the complex structure $I$ in a similar way.

However, as Kapustin and Witten explain in [KW], this derivation breaks down when we encounter singularities of the Hitchin moduli spaces. Thus, the sigma models with the targets ${\mathcal{M}}_{H}({}^{L}\negthinspace G)$ and ${\mathcal{M}}_{H}(G)$ are only approximations to the true physical theory. To understand what happens at the singularities we have to go back to the four-dimensional theory and analyze it more carefully (for more on this, see [Wi4]).

There is also another problem: in the above derivation we have not taken into account all the fields of the super–Yang–Mills theory. In fact, there are additional scalar fields, denoted by $\sigma$ and $\overline{\sigma}$ in [KW], which we have ignored so far. The field $\sigma$ is a section of the adjoint bundle ${\mathfrak{g}}_{E}$ on $X$, and $\overline{\sigma}$ is its complex conjugate. On the $B$-model side, which we have so far approximated by the sigma model with the target ${\mathcal{M}}_{H}({}^{L}\negthinspace G)$, we obtain from the BPS equations that $\sigma$ is annihilated by the flat connection $\nabla=A+i\phi$, that is, $\nabla\cdot\sigma=0$. In other words, $\sigma$ belongs to the Lie algebra of infinitesimal automorphisms of the flat bundle $(E,\nabla)$.

Up to now we have considered generic flat ${}^{L}\negthinspace G$-bundles which have no non-trivial infinitesimal automorphisms. For such flat bundles we therefore have $\sigma\equiv 0$, and so we could safely ignore it. But for flat bundles with continuous automorphisms this field starts playing an important role.

The upshot of this discussion is that when we consider most general flat bundles there are new degrees of freedom that have to be taken into account. In order to find a physical interpretation of the geometric Langlands correspondence for such flat bundles we need to consider the $S$-duality of boundary conditions in the four-dimensional gauge theory with these degrees of freedom included.

A detailed study of $S$-duality of these boundary conditions has been undertaken by Gaiotto and Witten [GaW1, GaW2]. We will only mention two important aspects of this work.

First of all, Gaiotto and Witten show that in the non-abelian gauge theory the $S$-dual of the Neumann boundary condition is not the usual Dirichlet boundary condition as one might naively hope, but a more complicated boundary condition in which the field $\sigma$ has a pole at the boundary. This boundary condition corresponds to a solution of the Nahm equations, which is in turn determined by an embedding of the Lie algebra ${\mathfrak{s}}{\mathfrak{l}}_{2}$ into ${}^{L}{\mathfrak{g}}$ (see [Wi6]).

In the geometric Langlands correspondence, Arthur’s $SL_{2}$ may be observed in the following way. The analogue of the trivial representation is the constant sheaf ${\mathbf{C}}$ on $\operatorname{Bun}_{G}$. It is a Hecke eigensheaf, but the “eigenvalues” are complexes of vector spaces with cohomological grading coming from the Cartan subalgebra of the principal $SL_{2}$ in ${}^{L}\negthinspace G$. For example, consider the case of $G=GL_{n}$ and let us apply the Hecke operators $H_{\check{\omega}_{1},x}$ defined by formula (2.2) to the constant sheaf. It follows from the definition that

Thus, the eigenvalue is a graded $n$-dimensional vector space with one-dimensional pieces in cohomological degrees $0,2,4,\ldots,2(n-1)$. In the standard normalization of the Hecke operator the cohomological grading is shifted to $-(n-1),-(n-3),\ldots,(n-1)$ – as in the grading on the $n$-dimensional representation of $GL_{n}$ coming from the principal $SL_{2}$.

The $A$-brane corresponding to the constant sheaf on $\operatorname{Bun}_{G}$ is the Lagrangian submanifold of ${\mathcal{M}}_{H}(G)$ defined by the equation $\phi=0$ (this is the zero section of the cotangent bundle to the moduli space of semi-stable $G$-bundles inside ${\mathcal{M}}_{H}(G)$). This $A$-brane corresponds to a Neumann boundary condition in the 4D gauge theory with gauge group $G_{c}$. According to Gaiotto and Witten, the dual boundary condition (in the theory with gauge group ${}^{L}\negthinspace G_{c}$) is a generalization of the Dirichlet boundary condition, in which the field $\sigma$ has a pole at the boundary solving the Nahm equations corresponding to the principal $SL_{2}$ embedding into ${}^{L}\negthinspace G$. Thus, we obtain a beautiful interpretation of Arthur’s $SL_{2}$ from the point of view of $S$-duality of boundary conditions in gauge theory. For more on this, see [Wi6, FGu].

Another important feature discovered in [GaW1, GaW2] is that the $S$-duals of the general Dirichlet boundary conditions involve coupling the 4D super–Yang-Mills to 3D superconformal QFTs at the boundary. This means that there are some additional degrees of freedom that we have to include to describe the geometric Langlands correspondence.

What we learn from all this is that the true moduli spaces arising in the $S$-duality picture are not the Hitchin moduli spaces ${\mathcal{M}}_{H}(G)$ and ${\mathcal{M}}_{H}({}{}^{L}\negthinspace G)$, but some enhanced versions $\widetilde{\mathcal{M}}_{H}(G)$ and $\widetilde{\mathcal{M}}_{H}({}{}^{L}\negthinspace G)$ thereof, including, in addition to the Higgs bundle $(E,\phi)$, an element $\sigma$ in the Lie algebra of its infinitesimal automorphisms as well as other data. (This will be discussed in more detail in the forthcoming paper [FGu].) Physically, the field $\sigma$ has non-zero “ghost number” $2$. Mathematically, this means that these additional degrees of freedom have cohomological grading $2$, and so $\widetilde{\mathcal{M}}_{H}(G)$ and $\widetilde{\mathcal{M}}_{H}({}{}^{L}\negthinspace G)$ are actually differential graded (DG) stacks. Similar DG stacks have been recently studied in the context of the categorical Langlands correspondence by V. Lafforgue [LafV].

Références