Geometric Langlands Conjecture

• Geometric Langlands Conjecture is a deep correspondence connecting automorphic sheaves on moduli spaces of G-bundles with spectral ind‐coherent sheaves on Langlands dual local systems.
• It verifies categorical equivalences across de Rham, Betti, and Dolbeault settings by employing Hecke operators, singular support conditions, and Fourier–Mukai-type transforms.
• Its recent proof in the de Rham setting confirms a multiplicity one property and underpins connections to mirror symmetry, quantum field theory, and advanced representation theory.

The Geometric Langlands Conjecture is a far-reaching correspondence between categories of sheaves on moduli spaces associated to a reductive group $G$ and its Langlands dual group ${}^L G$, formulated in various settings—de Rham, Betti, Dolbeault—over curves defined over arbitrary fields. It governs the spectral decomposition of automorphic sheaves, the microlocal and quantum structure of moduli spaces, and the functoriality underlying the Langlands program. With the recent complete proof in the de Rham setting, the conjecture is now established as a categorical equivalence, incorporating deep geometric and representation-theoretic input, singular support, and mirror symmetry.

1. Formulations and Foundations

Given a smooth projective curve $X$ over a field $k$ of characteristic zero and a connected reductive group $G$, the conjecture posits an equivalence of DG categories:

$\Dmod_{1}(\Bun_G) \simeq \IndCoh_{\Nilp}(\LocSys_{\,{}^L G}),$

where $\Dmod_{1}(\Bun_G)$ is the category of (half-twisted) D-modules on the moduli stack $\Bun_G$ of $G$-bundles on $X$, and ${}^L G$0 is the DG category of ind-coherent sheaves on the (derived) stack ${}^L G$1 of de Rham ${}^L G$2-local systems with singular support in the global nilpotent cone ${}^L G$3. The conjecture is compatible with the spectral action of ${}^L G$4, Hecke eigensheaf structures, and Eisenstein series functoriality (Gaitsgory et al., 2024, Arinkin et al., 2012, Arinkin et al., 2024, Gaitsgory et al., 2024).

Alternate incarnations exist:

• Betti Setting: With sheaves having nilpotent singular support on ${}^L G$5 and ind-coherent sheaves on the character stack of local systems (Ben-Zvi et al., 2016).
• Dolbeault Setting: Relates categories of coherent sheaves (or limit categories) on Higgs moduli spaces for ${}^L G$6 and ${}^L G$7, arising as the classical (${}^L G$8) limit of the de Rham equivalence (Pădurariu et al., 27 Aug 2025).

The conjecture specializes, for tori, to the abelian Fourier--Mukai transform, and—in the non-abelian case—imposes singular support and spectral conditions reflecting stability and automorphic-torsor properties (Arinkin et al., 2012, Gaitsgory, 2013).

2. Automorphic and Spectral Categories

• Automorphic (A-side): Categories of D-modules (or constructible sheaves in Betti) on ${}^L G$9 are acted upon by the symmetric monoidal category of spherical Hecke operators, encapsulating the modification of bundles at points of $X$0. The global nilpotent cone in $X$1 controls singular support, restricting to "nilpotent" or "tempered" automorphic objects (Ben-Zvi, 22 May 2026, Ben-Zvi et al., 2016).
• Spectral (B-side): The spectral counterpart involves ind-coherent sheaves on $X$2 with singular support in the stack of Arthur parameters $X$3 with nilpotent $X$4. Compact generation, functoriality, and perverse $X$5-structures play a crucial role in matching stratifications and extension classes on both sides (Arinkin et al., 2012).

A fundamental role is played by the vacuum Poincaré sheaf ("white-light" generator), which determines a $X$6-linear functor and, via adjunction, the spectral (Hecke-commuting) functor (Gaitsgory et al., 2024, Arinkin et al., 2024).

3. Hecke Operators, Eigenvalues, and Multiplicity

Hecke functors $X$7 indexed by finite-dimensional representations $X$8 of $X$9 and points $k$0 act on $k$1. A Hecke eigensheaf $k$2 with eigenvalue $k$3 satisfies

$k$4

for all $k$5, with the spectral side corresponding to the point $k$6.

The proof of the conjecture establishes a "multiplicity one" theorem: for an irreducible local system $k$7, the eigensheaf is unique up to tensoring by a vector space (with Hom space of dimension one between any two such) (Gaitsgory et al., 2024). For each irreducible $k$8 (with $k$9 automorphism), the cuspidal subcategory is identified fully faithfully with $G$0, with spectral decomposition realized via Fourier–Mukai-type transforms.

A key advance is the contractibility of the space of generic oper structures on irreducible local systems, reducing the equivalence to vanishing of higher obstruction classes (Arinkin et al., 2024).

4. Singular Support, Nilpotent Cones, and Temperedness

Singular support, as defined for ind-coherent sheaves, is controlled by the total space $G$1 of $G$2 of the tangent complex, yielding conical substacks corresponding to nilpotent loci in both automorphic and spectral moduli (Arinkin et al., 2012). The equivalence restricts to the nilpotent singular support condition, matching tempered automorphic D-modules with perfect or nilpotent-support coherent sheaves.

The construction of automorphic or spectral objects using singular support ensures compatibility with Hecke, Eisenstein, and Whittaker functor actions, and allows for filtration by Harder–Narasimhan strata or Arthur parameters, underpinning the microlocal and stability-theoretic aspects of the conjecture (Pădurariu et al., 27 Aug 2025).

In characteristic $G$3, the equivalence is established for $G$4-adic sheaves with nilpotent singular support, covering a union of connected components of the stack of Langlands parameters, with essential image determined by existence of nonzero automorphic sheaves (Gaitsgory et al., 4 Aug 2025).

5. Extensions: Dolbeault, Relative, Quantum, and Ramified Variants

The classical limit of the de Rham Geometric Langlands is formulated using limit categories for Higgs moduli stacks, with the Dolbeault conjecture predicting an equivalence

$G$5

for central type $G$6 and topological type $G$7, compatible with Hecke/Wilson actions and the semiorthogonal decomposition into quasi-BPS categories (Pădurariu et al., 27 Aug 2025). Over the regular locus, this reduces to Fourier–Mukai between abelian torsors (Ben-Zvi et al., 2016).

Relative forms, as in the geometric theory of period sheaves, generalize the conjecture to affine homogeneous spherical varieties, with period sheaves and Dirac–Higgs branes forming the brane duals under Fourier–Mukai. The regular quotient of invariant theory provides the correct geometric structure to abelianize the Hitchin fibration for non-group-type settings (Hameister et al., 2024, Feng et al., 2024, Devalapurkar, 2024).

Quantum variants in both characteristic zero and $G$8 (for irrational twist parameters) yield equivalences for derived categories of twisted D-modules, governed by quantum duality conditions and multiplicative gerbes (Travkin, 2011). Ramified, parabolic, and real-group analogs are included via local models and character stacks (Ben-Zvi et al., 2016).

6. Structural and Technical Advances in the Proof

The proof of the de Rham conjecture unfolds via the construction of the spectral functor, local-to-global techniques (via the Fundamental Local Equivalence at critical level), spectral gluing, ambidexterity and Barr–Beck formalism, and the verification of the contractibility of generic opers (Arinkin et al., 2024, Arinkin et al., 2024).

A fundamental role is played by the identification of tautological algebras and coalgebras controlling the monad/comonad structures for the cuspidal versus Eisenstein subcategories, with the contractibility of generic oper spaces forcing the full faithfulness of spectral-eigensheaf correspondences (Arinkin et al., 2024, Ben-Zvi, 22 May 2026).

Multiplicity one, as shown in (Gaitsgory et al., 2024), arises from analysis of the endomorphism algebra of the vacuum Poincaré sheaf and properties of the space of irreducible local systems, coupled with cohomological vanishing on the stack of local systems.

The conjecture admits equivalences between restricted/full, Betti/de Rham, and tempered/non-tempered forms, with Riemann–Hilbert correspondence matching Betti and D-module versions (Gaitsgory et al., 2024, Ben-Zvi et al., 2016).

7. Impact, Mirror Symmetry, and Further Directions

The Geometric Langlands Conjecture underlies deep connections between representation theory, algebraic geometry, quantum field theory, and arithmetic. At its core, it provides a spectral decomposition of automorphic sheaves—manifesting as a categorified, nonabelian harmonic analysis, diagonalizing the commuting family of Hecke operators with spectra labeled by local systems (the "frequencies" of the theory) (Ben-Zvi, 22 May 2026).

The categorical formalism supports extensions to topological quantum field theories, with equivalence of 4d TQFTs, mirror symmetry of Hitchin fibrations, the capstone of brane duality (Kapustin–Witten), and the categorification of $G$9-functions, periods, and quantum group representations (Ben-Zvi et al., 2016, Feng et al., 2024, Pădurariu et al., 27 Aug 2025).

Current research explores ramified, quantum, and real group generalizations, relative dualities, singular support structures, and the interplay with derived algebraic geometry, factorization homology, and global Springer theory, as well as applications to arithmetic Langlands theory, trace formulas, and beyond.