Geometric Langlands Program Research Articles

The Dirac–Higgs Complex and Categorification of (BBB)-Branes

Abstract Let ${\mathcal{M}}_{\operatorname{Dol}}(X,G)$ denote the hyperkähler moduli space of $G$-Higgs bundles over a smooth projective curve $X$. In the context of four dimensional supersymmetric Yang–Mills theory, Kapustin and Witten introduced the notion of (BBB)-brane: boundary conditions that are compatible with the B-model twist in every complex structure of ${\mathcal{M}}_{\operatorname{Dol}}(X,G)$. The geometry of such branes was initially proposed to be hyperkähler submanifolds that support a hyperholomorphic bundle. Gaiotto has suggested a more general type of (BBB)-brane defined by perfect analytic complexes on the Deligne–Hitchin twistor space $\operatorname{Tw}({\mathcal{M}}_{\operatorname{Dol}}(X,G))$. Following Gaiotto’s suggestion, this paper proposes a framework for the categorification of (BBB)-branes, both on the moduli spaces and on the corresponding derived moduli stacks. We do so by introducing the Deligne stack, a derived analytic stack with corresponding moduli space $\operatorname{Tw}({\mathcal{M}}_{\operatorname{Dol}}(X,G))$, defined as a gluing between two analytic Hodge stacks along the Riemann–Hilbert correspondence. We then construct a class of (BBB)-branes using integral functors that arise from higher non-abelian Hodge theory, before discussing their relation to the Wilson functors from the Dolbeault geometric Langlands correspondence.

Opers on the projective line, Wronskian relations, and the Bethe Ansatz

It is well-known that the spectra of the Gaudin model may be described in terms of solutions of the Bethe Ansatz equations. A conceptual explanation for the appearance of the Bethe Ansatz equations is provided by appropriate G-opers: G-connections on the projective line with extra structure. In fact, solutions of the Bethe Ansatz equations are parameterized by an enhanced version of opers called Miura opers; here, the opers appearing have only regular singularities. Moreover, this geometric approach to the spectra of the Gaudin model provides a well-known example of the geometric Langlands correspondence. Feigin, Frenkel, Rybnikov, and Toledano Laredo have introduced an inhomogeneous version of the Gaudin model; this model incorporates an additional twist factor, which is an element of the Lie algebra of G. They exhibited the Bethe Ansatz equations for this model and gave an interpretation of the spectra in terms of opers with an irregular singularity. In this paper, we consider a new geometric approach to the study of the spectra of the inhomogeneous Gaudin model in terms of a further enhancement of opers called twisted Miura-Plücker opers. This approach involves a certain system of nonlinear differential equations called the qq-system, which were previously studied in [20] in the context of the Bethe Ansatz. We show that there is a close relationship between solutions of the inhomogeneous Bethe Ansatz equations and polynomial solutions of the qq-system and use this fact to construct a bijection between the set of solutions of the inhomogeneous Bethe Ansatz equations and the set of nondegenerate twisted Miura-Plücker opers. We further prove that as long as certain combinatorial conditions are satisfied, nondegenerate twisted Miura-Plücker opers are in fact Miura opers.

Tamely Ramified Geometric Langlands Correspondence in Positive Characteristic

Abstract We prove a version of the tamely ramified geometric Langlands correspondence in positive characteristic for $GL_{n}(k)$, where $k$ is an algebraically closed field of characteristic $p> n$. Let $X$ be a smooth projective curve over $k$ with marked points, and fix a parabolic subgroup of $GL_{n}(k)$ at each marked point. We denote by $\operatorname{Bun}_{n,P}$ the moduli stack of (quasi-)parabolic vector bundles on $X$, and by $\mathcal{L}oc_{n,P}$ the moduli stack of parabolic flat connections such that the residue is nilpotent with respect to the parabolic reduction at each marked point. We construct an equivalence between the bounded derived category $D^{b}(\operatorname{QCoh}({\mathcal{L}oc_{n,P}^{0}}))$ of quasi-coherent sheaves on an open substack $\mathcal{L}oc_{n,P}^{0}\subset \mathcal{L}oc_{n,P}$, and the bounded derived category $D^{b}(\mathcal{D}^{0}_{{\operatorname{Bun}}_{n,P}}\operatorname{-mod})$ of $\mathcal{D}^{0}_{{\operatorname{Bun}}_{n,P}}$-modules, where $\mathcal{D}^{0}_{\operatorname{Bun}_{n,P}}$ is a localization of $\mathcal{D}_{\operatorname{Bun}_{n,P}}$ the sheaf of crystalline differential operators on $\operatorname{Bun}_{n,P}$. Thus, we extend the work of Bezrukavnikov–Braverman [ 8] to the tamely ramified case. We also prove a correspondence between flat connections on $X$ with regular singularities and meromorphic Higgs bundles on the Frobenius twist $X^{(1)}$ of $X$ with first-order poles.

Integrality of the Betti moduli space

If in a given rank r r , there is an irreducible complex local system with torsion determinant and quasi-unipotent monodromies at infinity on a smooth quasi-projective variety, then for every prime number ℓ \ell , there is an absolutely irreducible ℓ \ell -adic local system of the same rank, with the same determinant and monodromies at infinity, up to semi-simplification. A finitely presented group is said to be weakly integral with respect to a torsion character and a rank r r if once there is an irreducible rank r r complex linear representation with determinant isomorphic to this torsion character, then for any ℓ \ell , there is an absolutely irreducible one of rank r r and determinant this given character, which is defined over Z ¯ ℓ \bar { \mathbb {Z}}_\ell . We prove that this property is a new obstruction for a finitely presented group to be the fundamental group of a smooth quasi-projective complex variety. The proofs rely on the arithmetic Langlands program via the existence of Deligne’s companions (L. Lafforgue, Drinfeld) and the geometric Langlands program via de Jong’s conjecture (Gaitsgory for ℓ ≥ 3 \ell \ge 3 ). We also define weakly arithmetic complex local systems and show they are Zariski dense in the Betti moduli. Finally we show that our method gives an arithmetic proof of the Corlette-T. Mochizuki theorem, proved using tame pure imaginary harmonic metrics, which shows the pull-back by a morphism between two smooth complex algebraic varieties of a semi-simple complex local system is semi-simple.

Fourier coefficients and a filtration on Shv(BunG)

We define a filtration by DG-subcategories on the DG-category Shv(BunG) of sheaves on the moduli of G-torsors on a curve, which is stable under the action of Hecke functors. We formulate a conjecture relating this filtration with another filtration on the spectral side of the categorical geometric Langlands conjecture. We also formulate a conjectural compatibility with the parabolic induction.

Hypergeometric sheaves for classical groups via geometric Langlands

In a previous paper, the first and third authors gave an explicit realization of the geometric Langlands correspondence for hypergeometric sheaves, considered as G L n \mathrm {GL}_n -local systems. Certain hypergeometric local systems admit a symplectic or orthogonal structure, which can be viewed as G ˇ \check {G} -local systems, for a classical group G ˇ \check {G} . This article aims to realize the geometric Langlands correspondence for these G ˇ \check {G} -local systems. We study this problem from two aspects. In the first approach, we define the hypergeometric automorphic data for a classical group G G in the framework of Yun, one of whose local components is a new class of euphotic representations in the sense of Jakob–Yun. We prove the rigidity of hypergeometric automorphic data under natural assumptions, which allows us to define G ˇ \check {G} -local systems E G ˇ \mathcal {E}_{\check {G}} on G m \mathbb {G}_m as Hecke eigenvalues (in both ℓ \ell -adic and de Rham settings). In the second approach (which works only in the de Rham setting), we quantize a ramified Hitchin system, following Beilinson–Drinfeld and Zhu, and identify E G ˇ \mathcal {E}_{\check {G}} with certain G ˇ \check {G} -opers on G m \mathbb {G}_m . Finally, we compare these G ˇ \check {G} -opers with hypergeometric local systems.

Airy sheaves for reductive groups

We construct a class of ℓ $\ell$ -adic local systems on A 1 $\mathbb {A}^1$ that generalizes the Airy sheaves defined by N. Katz to reductive groups. These sheaves are finite field analogues of generalizations of the classical Airy equation y ′ ′ ( z ) = z y ( z ) $y^{\prime \prime }(z)=zy(z)$ . We employ the geometric Langlands correspondence to construct the sought-after local systems as eigenvalues of certain rigid Hecke eigensheaves, following the methods developed by Heinloth, Ngô, and Yun. The construction is motivated by a special case of Adler and Yu's construction of tame supercuspidal representations. The representations that we consider can be viewed as deeper analogues of simple supercuspidals. For GL n $\mathrm{GL}_n$ , we compute the Frobenius trace of the local systems in question and show that they agree with Katz's Airy sheaves. We make precise conjectures about the ramification behavior of the local systems at ∞ $\infty$ . These conjectures, in particular, imply cohomological rigidity of Airy sheaves.

Gauge Theory and the Analytic Form of the Geometric Langlands Program

We present a gauge-theoretic interpretation of the “analytic” version of the geometric Langlands program, in which Hitchin Hamiltonians and Hecke operators are viewed as concrete operators acting on a Hilbert space of quantum states. The gauge theory ingredients required to understand this construction—such as electric–magnetic duality between Wilson and ’t Hooft line operators in four-dimensional gauge theory—are the same ones that enter in understanding via gauge theory the more familiar formulation of geometric Langlands, but now these ingredients are organized and applied in a novel fashion.

The Deligne–Simpson problem for connections on [formula omitted] with a maximally ramified singularity

The classical additive Deligne–Simpson problem is the existence problem for Fuchsian connections with residues at the singular points in specified adjoint orbits. Crawley-Boevey found the solution in 2003 by reinterpreting the problem in terms of quiver varieties. A more general version of this problem, solved by Hiroe, allows additional unramified irregular singularities. We apply the theory of fundamental and regular strata due to Bremer and Sage to formulate a version of the Deligne–Simpson problem in which certain ramified singularities are allowed. These allowed singular points are called toral singularities; they are singularities whose leading term with respect to a lattice chain filtration is regular semisimple. We solve this problem in the special case of connections on Gm with a maximally ramified singularity at 0 and possibly an additional regular singular point at infinity. Examples of such connections arise from Airy, Bessel, and Kloosterman differential equations. They play an important role in recent work in the geometric Langlands program. We also give a complete characterization of all such connections which are rigid, under the additional hypothesis of unipotent monodromy at infinity.

An Informal Introduction to Categorical Representation Theory and the Local Geometric Langlands Program

We provide a motivated introduction to the theory of categorical actions of groups and the local geometric Langlands program. Along the way, we emphasize applications, old and new, to the usual representation theory of reductive and affine Lie algebras.

Tate’s thesis in the de Rham setting

We calculate the category of D D -modules on the loop space of the affine line in coherent terms. Specifically, we find that this category is derived equivalent to the category of ind-coherent sheaves on the moduli space of rank one de Rham local systems with a flat section. Our result establishes a conjecture coming out of the 3 d 3d mirror symmetry program, which obtains new compatibilities for the geometric Langlands program from rich dualities of QFTs that are themselves obtained from string theory conjectures.

Geometric Langlands for hypergeometric sheaves

Generalised hypergeometric sheaves are rigid local systems on the punctured projective line with remarkable properties. Their study originated in the seminal work of Riemann on the Euler–Gauss hypergeometric function and has blossomed into an active field with connections to many areas of mathematics. In this paper, we construct the Hecke eigensheaves whose eigenvalues are the irreducible hypergeometric local systems, thus confirming a central conjecture of the geometric Langlands program for hypergeometrics. The key new concept is the notion of hypergeometric automorphic data. We prove that this automorphic data is generically rigid (in the sense of Zhiwei Yun) and identify the resulting Hecke eigenvalue with hypergeometric sheaves. The definition of hypergeometric automorphic data in the tame case involves the mirabolic subgroup, while in the wild case, semistable (but not necessarily stable) vectors coming from principal gradings intervene.

Deligne–Lusztig duality on the stack of local systems

Abstract In the setting of the geometric Langlands conjecture, we argue that the phenomenon of divergence at infinity on Bun G {\operatorname{Bun}_{G}} (that is, the difference between ! {!} -extensions and * {*} -extensions) is controlled, Langlands-dually, by the locus of semisimple G ˇ {{\check{G}}} -local systems. To see this, we first rephrase the question in terms of Deligne–Lusztig duality and then study the Deligne–Lusztig functor 𝖣𝖫 G spec {\mathsf{DL}_{G}^{\mathrm{spec}}} acting on the spectral Langlands DG category IndCoh 𝒩 ⁢ ( LS G ) {{\mathrm{IndCoh}}_{\mathcal{N}}({\mathrm{LS}}_{G})} . We prove that 𝖣𝖫 G spec {\mathsf{DL}_{G}^{\mathrm{spec}}} is the projection IndCoh 𝒩 ⁢ ( LS G ) ↠ QCoh ⁢ ( LS G ) {{\mathrm{IndCoh}}_{\mathcal{N}}({\mathrm{LS}}_{G})\twoheadrightarrow{\mathrm{% QCoh}}({\mathrm{LS}}_{G})} , followed by the action of a coherent D-module St G ∈ 𝔇 ⁢ ( LS G ) {{\mathrm{St}}_{G}\in\mathfrak{D}({\mathrm{LS}}_{G})} , which we call the Steinberg D-module. We argue that St G {{\mathrm{St}}_{G}} might be regarded as the dualizing sheaf of the locus of semisimple G-local systems. We also show that 𝖣𝖫 G spec {\mathsf{DL}_{G}^{\mathrm{spec}}} , while far from being conservative, is fully faithful on the subcategory of compact objects.

Linear Periods of Automorphic Sheaves for GL2n

Abstract We calculate, in the framework of the geometric Langlands program, the periods of cuspidal automorphic sheaves for $\operatorname{GL}_{2n}$ along the Levi subgroup $\operatorname{GL}_n\times \operatorname{GL}_n$. We also solve the corresponding local problem.

The center of the categorified ring of differential operators

Let $\mathcal Y$ be a derived algebraic stack satisfying some mild conditions. The purpose of this paper is three-fold. First, we introduce and study $\mathbb H(\mathcal Y)$, a monoidal DG category that might be regarded as a categorification of the ring of differential operators on $\mathcal Y$. When $\mathcal Y = \mathrm {LS}\_G$ is the derived stack of $G$-local systems on a smooth projective curve, we expect $\mathbb H (\mathrm {LS}g)$ to act on both sides of the geometric Langlands correspondence, compatibly with the conjectural Langlands functor. Second, we construct a novel theory of D-modules on derived algebraic stacks. In contrast to usual D-modules, this new theory, to be denoted by $\mathcal D^{\mathrm{der}}$, is sensitive to the derived structure. Third, we identify the Drinfeld center of $\mathbb H(\mathcal Y)$ with $\mathcal D^{\mathrm{der}}(L\mathcal Y)$, the DG category of $\mathcal D^{\mathrm{der}}$-modules on the loop stack $L\mathcal Y$: = $\mathcal Y \times{\mathcal Y \times \mathcal Y}\mathcal Y$.

Uniformization of semistable bundles on elliptic curves

Let G be a connected reductive complex algebraic group, and E a complex elliptic curve. Let GE denote the connected component of the trivial bundle in the stack of semistable G-bundles on E. We introduce a complex analytic uniformization of GE by adjoint quotients of reductive subgroups of the loop group of G. This can be viewed as a nonabelian version of the classical complex analytic uniformization E≃C⁎/qZ. We similarly construct a complex analytic uniformization of G itself via the exponential map, providing a nonabelian version of the standard isomorphism C⁎≃C/Z, and a complex analytic uniformization of GE generalizing the standard presentation E≃C/(Z⊕Zτ). Finally, we apply these results to the study of sheaves with nilpotent singular support. As an application to Betti geometric Langlands conjecture in genus 1, we define a functor from ShN(GE) (the semistable part of the automorphic category) to IndCohNˇ(LocsysGˇ(E)) (the spectral category).

$$({{\,\mathrm{\mathrm {SL}}\,}}(N),q)$$-Opers, the q-Langlands Correspondence, and Quantum/Classical Duality

A special case of the geometric Langlands correspondence is given by the relationship between solutions of the Bethe ansatz equations for the Gaudin model and opers - connections on the projective line with extra structure. In this paper, we describe a deformation of this correspondence for $SL(N)$. We introduce a difference equation version of opers called $q$-opers and prove a $q$-Langlands correspondence between nondegenerate solutions of the Bethe ansatz equations for the XXZ model and nondegenerate twisted $q$-opers with regular singularities on the projective line. We show that the quantum/classical duality between the XXZ spin chain and the trigonometric Ruijsenaars-Schneider model may be viewed as a special case of the $q$-Langlands correspondence. We also describe an application of $q$-opers to the equivariant quantum $K$-theory of the cotangent bundles to partial flag varieties.

Vertex Algebras for S-duality

We define new deformable families of vertex operator algebras $$\mathfrak {A}[\mathfrak {g}, \Psi , \sigma ]$$ associated to a large set of S-duality operations in four-dimensional supersymmetric gauge theory. They are defined as algebras of protected operators for two-dimensional supersymmetric junctions which interpolate between a Dirichlet boundary condition and its S-duality image. The $$\mathfrak {A}[\mathfrak {g}, \Psi , \sigma ]$$ vertex operator algebras are equipped with two $$\mathfrak {g}$$ affine vertex subalgebras whose levels are related by the S-duality operation. They compose accordingly under a natural convolution operation and can be used to define an action of the S-duality operations on a certain space of vertex operator algebras equipped with a $$\mathfrak {g}$$ affine vertex subalgebra. We give a self-contained definition of the S-duality action on that space of vertex operator algebras. The space of conformal blocks (in the derived sense, i.e. chiral homology) for $$\mathfrak {A}[\mathfrak {g}, \Psi , \sigma ]$$ is expected to play an important role in a broad generalization of the quantum Geometric Langlands program. Namely, we expect the S-duality action on vertex operator algebras to extend to an action on the corresponding spaces of conformal blocks. This action should coincide with and generalize the usual quantum Geometric Langlands correspondence. The strategy we use to define the $$\mathfrak {A}[\mathfrak {g}, \Psi , \sigma ]$$ vertex operator algebras is of broader applicability and leads to many new results and conjectures about deformable families of vertex operator algebras.

Exotic t-structures and actions of quantum affine algebras

We explain how quantum affine algebra actions can be used to systematically construct t-structures. The main idea, roughly speaking, is to take advantage of the two different descriptions of quantum affine algebras, the Drinfeld--Jimbo and the Kac--Moody realizations. Our main application is to obtain exotic t-structures on certain convolution varieties defined using the Beilinson--Drinfeld and affine Grassmannians. These varieties play an important role in the geometric Langlands program, knot homology constructions, K-theoretic geometric Satake and the coherent Satake category. As a special case we also recover the exotic t-structures of Bezrukavnikov--Mirkovic on the (Grothendieck--)Springer resolution in type A.

Is There an Analytic Theory of Automorphic Functions for Complex Algebraic Curves?

The geometric Langlands correspondence for complex algebraic curves differs from the original Langlands correspondence for number fields in that it is formulated in terms of sheaves rather than functions (in the intermediate case of curves over finite fields, both formulations are possible). In a recent preprint, Robert Langlands made a proposal for developing an analytic theory of automorphic forms on the moduli space of $G$-bundles on a complex algebraic curve. Langlands envisioned these forms as eigenfunctions of some analogues of Hecke operators. In these notes I show that if $G$ is an abelian group then there are well-defined Hecke operators, and I give a complete description of their eigenfunctions and eigenvalues. For non-abelian $G$, Hecke operators involve integration, which presents some difficulties. However, there is an alternative approach to developing an analytic theory of automorphic forms, based on the existence of a large commutative algebra of global differential operators acting on half-densities on the moduli stack of $G$-bundles. This approach (which implements some ideas of Joerg Teschner) is outlined here, as a preview of a joint work with Pavel Etingof and David Kazhdan.