Geometric Langlands Program

• The geometric Langlands program is a network of dualities and categorifications connecting algebraic geometry, representation theory, and quantum gauge theory.
• It employs derived categories, DG-categories, and Kac–Moody localization to establish a local-to-global framework via factorization structures and Hecke operators.
• The program bridges mathematics and physics by using mirror symmetry and gauge theory to extend classical Langlands correspondences into a geometric and categorical realm.

The geometric Langlands program is a network of deep dualities, equivalences, and categorifications intertwining algebraic geometry, representation theory, and quantum gauge theory. Originating from the classical (arithmetic) Langlands program—which relates automorphic forms and Galois representations—the geometric version reformulates this correspondence in the setting of algebraic curves, replacing number-theoretic objects with sheaves, categorifying representation-theoretic structures, and providing a geometric and categorical framework compatible with advances in derived algebraic geometry, topological quantum field theory, and infinite-dimensional representation theory.

1. Categorical Geometric Langlands Correspondence

The global geometric Langlands conjecture posits an equivalence between two derived (DG) categories associated with a smooth projective curve $X$ over $\mathbb{C}$ and a complex reductive group $G$:

• The automorphic side is the DG-category of (half-twisted) $D$-modules on the moduli stack of principal $G$-bundles over $X$, denoted $\Bun_G$:

$\mathcal{D}(\Bun_G)$

• The spectral side is the DG-category of ind-coherent sheaves (with singular support in the global nilpotent cone) on the derived stack of de Rham $^LG$-local systems on $X$, where $\mathbb{C}$0 is the Langlands dual group:

$\mathbb{C}$1

The functor, constructed in (Gaitsgory et al., 2024) and proven to be an equivalence in conjunction with Kac–Moody localization methods (Arinkin et al., 2024), is required to intertwine the tensor action of vector bundles (spectral side) with Hecke correspondences (automorphic side) and to respect the $\mathbb{C}$2-structure and various categorical symmetries. For Betti and $\mathbb{C}$3-adic contexts, analogous equivalences are conjectured between sheaves on $\mathbb{C}$4 with nilpotent singular support and coherent sheaves on the moduli of $\mathbb{C}$5-local systems (Ben-Zvi et al., 2016).

2. Local–Global Theory and the Fundamental Local Equivalence

At the local level, the geometric Langlands program seeks a categorified analog of the local Langlands correspondence, replacing irreducible representations of $\mathbb{C}$6 with categories of sheaves and modules:

• The local automorphic side involves the DG-category of representations of the affine Kac–Moody algebra at the critical level, integrable with respect to the arc group,

$\mathbb{C}$7

• The local spectral side is described by ind-coherent sheaves on the ind-scheme of monodromy-free $\mathbb{C}$8-opers over the punctured disc:

$\mathbb{C}$9

The Fundamental Local Equivalence (FLE) at the critical level (Theorem 6.1.4 in (Arinkin et al., 2024)) provides a $G$0-exact equivalence of factorization categories:

$G$1

This local equivalence identifies the vacuum module with the structure sheaf of the unit factorization in $G$2, and matches all key functorialities: Satake, restriction to Levi, Eisenstein series, BRST reduction, and Whittaker categories ((Arinkin et al., 2024), Theorems 6.4.5, 8.1.4, 12.3.6, 12.8.5). This result is a cornerstone for constructing the global equivalence, effectively enabling a local-to-global factorization approach.

3. Kac–Moody Localization and Factorization Structures

The Kac–Moody localization functor $G$3 acts as a bridge between local representation categories and global $G$4-modules:

$G$5

This functor, constructed via the double quotient presentation of $G$6 as an ind-scheme and the adjunctions induced by pull-push along the relevant correspondences, is shown to be right $G$7-exact and (on quasi-compact open substacks) a localization—i.e., its right adjoint is fully faithful ((Arinkin et al., 2024), Theorem 13.4.2). The essential surjectivity onto a cogenerator of the global category follows, establishing the crucial compatibility with compactness and global support.

Factorization structures play a central role, both in the local theory (Ran-factored categories, factorization modules over factorization algebras) and in assembling global objects. Appendices of (Arinkin et al., 2024) formalize the necessary structure for ind-coherent sheaves on infinite-type prestacks ($G$8 and $G$9 for placid or ind-placid prestacks) and articulate the machinery of factorization categories and unital/lax-unital functors needed for local-to-global constructions.

4. Hecke Operators, Satake Equivalence, and Functoriality

Hecke operators in the geometric setting are implemented by correspondences acting on $D$0-modules or sheaves on $D$1; their spectral counterparts are functors on sheaves of categories over stacks of $D$2-local systems. The geometric Satake equivalence, both in the $D$3-adic and derived settings, identifies the category of $D$4-equivariant perverse sheaves on the affine Grassmannian $D$5 with the tensor category of finite-dimensional representations of $D$6 (Frenkel, 2012, Lafforgue, 2018). In the factorized setting, the spherical Hecke category becomes a monoidal DG-category acting compatibly on the relevant categories, providing the link between geometric and spectral symmetries.

All symmetries and compatibilities in the conjecture (e.g., spherical Hecke, restriction along Levi, Eisenstein functors, BRST reduction, coefficient/Whittaker functors, Poincaré series) are controlled under the FLE and Kac–Moody localization by explicit intertwining results, ensuring that the categorical equivalences respect the enriched structure on both sides ((Arinkin et al., 2024), §3 and §6).

5. Gauge Theory, Mirror Symmetry, and Physical Approaches

Physical realizations, notably via $D$7-duality in 4d $D$8 super-Yang–Mills, underlie the homological mirror symmetry perspective on geometric Langlands. The compactification of the gauge theory, with either brane or boundary condition insertions, yields sigma models whose target is the Hitchin moduli space $D$9 (0906.2747, 0911.4586, Gaiotto, 2016, Gaiotto et al., 2021).

• A-branes (coisotropic or Lagrangian) under $G$0-duality correspond to $G$1-modules on $G$2;
• B-branes (skyscraper sheaves) correspond to spectral objects (local systems).

This structure provides a natural explanation for the tensor/Hecke functors correspondence and for mirror symmetry between the two categories. Formulations in terms of spectral decompositions of Hilbert spaces (analytic approaches) and spectral operators also arise in analytic gauge-theoretic frameworks (Gaiotto et al., 2021).

6. Arithmetic, Sheaf–Function Dictionary, and Further Developments

The geometric context illuminates and refines the classical arithmetic Langlands program. In the function field case, V. Lafforgue's work on shtukas and excursion operators realizes the spectral decomposition of automorphic forms via the cohomology of moduli stacks and their relation to spaces of $G$3-local systems (Lafforgue, 2018, Zhu, 10 Apr 2025). These advances rely on the geometric Satake equivalence, the function–sheaf dictionary, and the categorification of both sides of the correspondence.

The program encompasses several further extensions:

• Twisted/metaplectic geometric Langlands for central extensions and tori (Lysenko, 2013).
• Quantum, Betti, and integral variants, where the spectral side utilizes quantum groups, factorization homology, or categories of sheaves on character stacks (Ben-Zvi et al., 2016).
• Ramification and local functoriality via modifications of bundles, parahoric group schemes, and rigid automorphic data (Kamgarpour et al., 2020, Farang-Hariri, 2015).
• Connection to physics: quantum Hall effect and dualities in topological phases (Ikeda, 2017).

The ongoing proof of the geometric Langlands conjecture for general $G$4—notably via the combination of critical-level Kac–Moody localization, factorization categories, and the compatible integration of local and global machinery—represents an overview of these strands (Gaitsgory et al., 2024, Arinkin et al., 2024).

7. Conceptual Summary and Outlook

The geometric Langlands program establishes a structured, categorical equivalence between automorphic data ($G$5-modules or sheaves on moduli of bundles) and spectral data (sheaves on stacks of local systems), controlled by a network of functorialities, compatibilities, and symmetries. At its core, it unifies geometric representation theory, derived algebraic geometry, mathematical physics (especially gauge theory and mirror symmetry), and arithmetic geometry within a common categorical framework, opening paths to new dualities, constructions, and applications across mathematics and theoretical physics. Recent work (Arinkin et al., 2024, Gaitsgory et al., 2024, Zhu, 10 Apr 2025, Ben-Zvi et al., 2016, Lafforgue, 2018), and related references reflects rapid progress and a convergence of methods achieving the long-sought global and local correspondences.