Quantum Geometric Langlands Program

• Quantum Geometric Langlands Program is a framework that extends classical Langlands by incorporating quantum deformations and physical dualities from gauge and string theories.
• It employs six-dimensional superconformal field theories, twisted N=4 Yang–Mills, and q-deformations to relate categories of modules over a group and its dual.
• The program unifies physical dualities, categorical representation theory, and algebraic geometry, offering insights into quantum groups, spectral data, and automorphic phenomena.

The Quantum Geometric Langlands Program is a far-reaching extension of the geometric Langlands correspondence, integrating ideas from four- and six-dimensional gauge theory, quantum field theory, representation theory, algebraic geometry, and topological field theory. The central objects are equivalences between categories associated to a reductive group $G$ and its Langlands dual group $^LG$, involving both “quantum” deformations and physical dualities (notably $S$-duality). Unlike the classical geometric Langlands picture, the quantum theory incorporates deformations by parameters such as $q$ or a “level,” and naturally accommodates spectral and automorphic data arising from quantum groups, $W$-algebras, and their categorifications. Central to the quantum theory are frameworks derived from six-dimensional $(0,2)$ superconformal field theory, four-dimensional $N=4$ super Yang–Mills, quantum $K$-theory, and brane constructions in string theory. These frameworks provide both physical motivation and mathematical techniques for formulating and in some cases proving quantum geometric Langlands dualities.

1. Physical Origin and Six-Dimensional Framework

The six-dimensional $(0,2)$ supersymmetric conformal field theory, when compactified on a two-torus $T^2$, produces in the infrared a four-dimensional $N=4$ super Yang–Mills theory. Specifically, if the six-manifold is $M_6 = M_4 \times T^2$, compactification yields a maximally supersymmetric gauge theory on $M_4$. The effective 4d action is determined by the coupling

$\tau = \frac{\theta}{2\pi} + \frac{4\pi i}{e^2}$

which is identified with the complex structure of $T^2$. The modular symmetry group $SL(2,\mathbb{Z})$ acting on the torus induces $S$-duality on the gauge coupling in four dimensions, manifesting as electric–magnetic duality: $(G, \tau) \longleftrightarrow ({}^L G, -1/n_g\tau)$ with $n_g$ the lacing number of $G$.

To obtain linkages with geometric representation theory, the $N=4$ theory is “twisted,” producing a topological quantum field theory whose categories of boundary conditions encode the data of $D$-modules or coherent sheaves on moduli spaces of bundles. The duality group $SL(2,\mathbb{Z})$ is then realized not just as a numerical symmetry but as a correspondence between categories associated to $G$ and $^LG$ (0905.2720, 0906.2747).

Defects supported on submanifolds of $M_6$, such as complex surfaces or smaller loci, are mapped under this framework to Hecke modifications, ramification data, or more general “quantum” objects on the moduli of bundles. Chern–Simons type terms arise from the geometry of the fibration, and anomaly cancellation mechanisms directly relate to the appearance of affine Kac–Moody representations and quantum $W$-algebra data.

2. Quantum Deformations: Parameters, Functorialities, and $q$-Langlands

Quantum geometric Langlands introduces “quantum” or “categorical” deformations by promoting the classical correspondence to involve categories of twisted or deformed objects. Two principal types of deformations are prominent:

• The Level Deformation: Moving away from the critical level in affine Kac–Moody or $W$-algebras so that the equivalence relates categories of modules over quantum group (or $W$-algebra) at generic noncritical level. For simply-laced $G$, the duality relates $U_\hbar(\widehat{{}^Lg})$ to $\mathcal{W}_{q,t}(g)$ with parameters related by $q = \hbar^{-{}^L(k+h^\vee)}$, $t = q^\beta$ (Aganagic et al., 2017).
• The $q$-Deformation: Replacing differential equations with $q$-difference equations, so that conformal blocks of quantum affine algebras (electric side) are functorially related to those of deformed $W$-algebras (magnetic side). In the limit where the deformation parameter (the string mass in the 6d little string theory) becomes infinite, one recovers the quantum (non-$q$) geometric Langlands correspondence.

A concrete realization of this two-parameter deformation is achieved via the quantum K-theory of Nakajima quiver varieties, where partition functions of 3d or 5d gauge theories compute both quantum affine and deformed $W$ conformal blocks. The resulting equivalence takes the form

$$U_\hbar(\widehat{{}^Lg}) \longleftrightarrow \mathcal{W}_{q,t}(g)$$

with conformal blocks or “vertex functions” from both sides identified as solutions to the same $q$-difference systems. This approach is physically motivated by the 6d little string theory and is proven in the simply-laced case (Aganagic et al., 2017, Haouzi, 2023).

3. Categorical and Vertex-Algebraic Structures

Categorical formulations center on equivalences of derived categories: $$D^b(\mathcal{D}\text{-mod}(\operatorname{Bun}_G)) \simeq D^b(\mathcal{O}\text{-mod}(\operatorname{Loc}_{^LG}))$$ and, in the quantum setting,

$$D^b(\mathcal{D}_\kappa\text{-mod}(\operatorname{Bun}_G)) \simeq D^b(\mathcal{O}_q\text{-mod}(\operatorname{Loc}_{{}^LG}))$$

The geometric Satake correspondence is deformed to the quantum case by relating categories of equivariant perverse sheaves on the affine Grassmannian to categories of representations of quantum groups $U_q \mathfrak{g}$ (Cautis et al., 2015). In the derived and $K$-theoretic setting, convolution categories built from equivariant $K$-theory of fibre products of affine Grassmannians are proved equivalent to categories of $U_q \mathfrak{g}$-equivariant modules, for example via the annular spider category in $SL_n$ (Cautis et al., 2015).

Locally, the “fundamental local equivalence” (FLE) conjecture, and in many cases established, realizes categories of modules over equivariant affine $W$-algebras (obtained via principal quantum Hamiltonian reduction of chiral differential operators at generic levels) as equivalent to categories of finite-dimensional representations of $^LG$ (Simon, 8 Oct 2025). The construction of simple modules via spectral flow twisting and Hamiltonian reduction matches precisely the combinatorics of dominant weights of the Langlands dual group.

4. Quantum Categories: Stacks of Parameters and Derived Geometry

Quantum deformations are parametrized functorially by the “algebraic stack of quantum parameters,” $\operatorname{Par}_G$, encoding Lagrangian subspaces and extension data in $\mathfrak{g} \oplus \mathfrak{g}^*$. Families of (quasi-)twistings $J(K, E)$ parametrized by $\operatorname{Par}_G$ yield DG categories interpolating between twisted $\mathcal{D}$-modules on $\operatorname{Bun}_G$ and quasi-coherent sheaves on the DG stack of local systems $\operatorname{LocSys}_G$; at “infinite level” the category specializes to $\mathrm{QCoh}(\operatorname{LocSys}_G)$ (Zhao, 2017). These families provide a universal geometric domain for quantum (or “categorical”) versions of Langlands duality, and enable continuous deformation between classical and quantum settings.

On the Betti side, the program extends to “Betti quantum Langlands,” where automorphic categories of nilpotent sheaves on $\operatorname{Bun}_G(X)$ are matched with quantum deformations of spectral categories of sheaves on the character stack $\operatorname{Loc}_{{}^LG}(X)$ endowed with monodromic $q$-twists (Ben-Zvi et al., 2016). Quantum versions further relate topological field theory structures such as factorization homology and enhanced gluing theorems.

5. Gauge-Theoretic, Topological, and String-Theoretical Realizations

The gauge-theoretic origin of quantum geometric Langlands is grounded in the $S$-duality of 4d topologically twisted $N=4$ super-Yang–Mills with surface operators and line operators. Duality acts at the level of categories, with Wilson line observables interchanged with Hecke operators under $S$-duality (0906.2747, Ashwinkumar et al., 2019, Frenkel et al., 2018). Boundary conditions in these gauge theories are mapped to kernel D-modules, which implement derived equivalences between categories of twisted D-modules or $W$-algebra modules.

Brane constructions in string theory (notably the D4/NS5 system with partial topological–holomorphic twist) realize connections between lattice models (4d Chern–Simons theory), analytic continuation of Chern–Simons link invariants, and quantum geometric Langlands, ultimately providing physical explanations for the Gaitsgory-Lurie conjecture linking quantum groups and Whittaker D-modules (Ashwinkumar et al., 2019).

The analytic form of quantum geometric Langlands is realized in the construction of Hilbert spaces with commuting quantized Hitchin Hamiltonians and Hecke/'t Hooft operators. These operators are manifested as line operators in gauge theory, with endpoint data described by critical Kac–Moody chiral algebras and spectral flow automorphisms (Gaiotto et al., 2021). The construction yields a quantization of the Hitchin integrable system, with eigenfunctions characterized by opers and spectral data intrinsic to the quantum correspondence.

6. Ramification, Moduli Spaces, and Physical Applications

Quantum ramification phenomena—implemented via ramified blocks, Drinfeld polynomials, and the moduli of quasimaps to quiver varieties—are integrated in the program by analyzing monodromy defects and their $S$-duality in 4d gauge theory. In this setting, quantum affine weights and representations of deformed $W$-algebras are related to the supersymmetric indices of vortex theories, and vertex functions become enumerative invariants of quiver moduli spaces (Haouzi, 2023).

Physical phenomena such as the quantum Hall effect and the fractal spectrum of Hofstadter’s butterfly are interpreted within the quantum geometric Langlands framework by realizing plateaus of Hall conductance as Hecke eigensheaves and topological dualities (such as particle–vortex duality) as manifestations of $S$-duality in Chern–Simons theory. The quantum group symmetry underlying the Hofstadter spectrum reflects a quantum Langlands duality between representations of $U_q(\mathfrak{sl}_2)$ and its Langlands dual at dual deformation parameters (Ikeda, 2017, Ikeda, 2017, Ikeda, 2018).

7. Impact, Extensions, and Open Problems

The Quantum Geometric Langlands Program has generated a powerful synthesis linking physical dualities, quantum deformation theory, categorical representation theory, and algebraic geometry. Quantum K-theoretic geometric Satake, vertex-algebraic fundamental local equivalence, and derived modular functorialities have been rigorously developed in key cases (Cautis et al., 2015, Simon, 8 Oct 2025). The extension to Betti, de Rham, and Dolbeault forms allows unification across analytic, topological, and algebraic formulations (Ben-Zvi et al., 2016, Gaitsgory et al., 2024). The introduction of stack-theoretic parameters and derived moduli provides a universal construction for quantum equivalences.

Open directions include the full characterization of quantum ramification phenomena (wild/irregular as well as tame), the extension of duality to all types of groups (including exceptional and supergroups), the relation to categorified knot homology and link invariants, and the understanding of quantum spectral data for higher-dimensional varieties. The systematic deployment of higher-categorical and topological quantum field theory methods (including factorization homology and extended TFTs) offers the prospect of organizing and extending the quantum Langlands web to a wider range of geometric and physical contexts.

In all cases, the Quantum Geometric Langlands Program realizes physical dualities as mathematical equivalences which deform and enhance the traditional Langlands correspondence, intertwining the arithmetic, geometric, and quantum symmetries of both mathematics and theoretical physics.