[Paper Review] Quantum Langlands duality and conformal field theory

A. V. Stoyanovsky|arXiv (Cornell University)|Oct 31, 2006

Advanced Algebra and Geometry3 references12 citations

TL;DR

This paper proposes a quantum deformation of the geometric Langlands correspondence via derived category equivalences between twisted $\mathcal{D}$-modules on moduli stacks of $G$-bundles and $\mathcal{O}$-modules on the twisted cotangent bundle of $G^\vee$-bundles, parameterized by a complex parameter $\kappa$. The key result is a conjectural kernel object $\mathcal{L}_\kappa$ that interpolates between the classical geometric Langlands correspondence ($\kappa=0$) and its Langlands dual ($\kappa=\infty$), with singular support conditions tied to the Hitchin integrable system.

ABSTRACT

V. Drinfeld proposed conjectures on geometric Langlands correspondence and its quantum deformation. We refine these conjectures and propose their relationship with algebraic conformal field theory.

Motivation & Objective

• To refine and extend V. Drinfeld's conjectures on geometric Langlands duality by introducing a quantum deformation parameterized by $\kappa \in \mathbb{C}^\times$.
• To establish a derived category equivalence between twisted $\mathcal{D}$-modules on $\mathcal{B}un_G$ and $\mathcal{O}$-modules on $\widetilde{T}^*\mathcal{B}un_{G^\vee}$, generalizing the classical correspondence.
• To relate this quantum Langlands correspondence to algebraic conformal field theory, unifying geometric Langlands with representation-theoretic structures.
• To define a kernel object $\mathcal{L}_\kappa$ that interpolates between the classical geometric Langlands kernel ($\kappa=0$) and its Langlands dual ($\kappa=\infty$), with flatness and singular support constraints.

Proposed method

• Propose a derived category equivalence between twisted $\mathcal{D}_{\mathcal{B}un_G}(\xi^{\otimes\kappa})$-modules and twisted $\mathcal{D}_{\mathcal{B}un_{G^\vee}}(\xi^{\vee\otimes\kappa^\vee})$-modules, with $\kappa^\vee = 1/(r\kappa)$ and $r$ the maximal edge multiplicity in the Dynkin diagram.
• Define the kernel $\mathcal{L}_\kappa$ as a $\mathcal{D}_{\mathcal{B}un_G}(\xi^{\otimes\kappa}) \boxtimes \mathcal{D}_{\mathcal{B}un_{G^\vee}}(\xi^{\vee\otimes\kappa^\vee})$-module on $\mathbb{P}^1 \times \mathcal{B}un_G \times \mathcal{B}un_{G^\vee}$, flat over $\mathbb{P}^1$.
• Require that the fiber at $t=\kappa$ is $\mathcal{L}_\kappa$, at $t=0$ is the classical kernel $\mathcal{L}_0$, and at $t=\infty$ is the Langlands dual kernel $\mathcal{L}_\infty$, ensuring interpolation between dual theories.
• Impose a singular support condition: the singular support of $\mathcal{L}_\kappa$ must be the preimage of the diagonal $\Delta_\kappa = \{v_i^\vee = \kappa^{d_i} v_i\}$ under the product of Hitchin maps $\chi_G \times \chi_{G^\vee}$.
• Use the Hitchin integrable system to constrain the singular support of $\mathcal{L}_\kappa$, with the classical limit $\kappa \to 0$ yielding containment in the global nilpotent cone.
• Relate the construction to conformal field theory via the space $\mathcal{C}onf_{G^\vee}$, identified with the space of colored divisors, and use the $\mathcal{B}un_{B/[N,N]}^{>0}$ diagram to define the relevant geometric structures.

Experimental results

Research questions

• RQ1 How can the geometric Langlands correspondence be deformed into a quantum version parameterized by a complex coupling $\kappa$?
• RQ2 What is the precise form of the kernel $\mathcal{L}_\kappa$ that realizes the derived equivalence between twisted $\mathcal{D}$-modules on $\mathcal{B}un_G$ and $\mathcal{O}$-modules on $\widetilde{T}^*\mathcal{B}un_{G^\vee}$?
• RQ3 How does the singular support of $\mathcal{L}_\kappa$ relate to the Hitchin integrable system and the diagonal condition $v_i^\vee = \kappa^{d_i} v_i$?
• RQ4 What is the role of conformal field theory in realizing and justifying this quantum Langlands duality?
• RQ5 How does the classical limit $\kappa \to 0$ recover the original geometric Langlands correspondence, and what is the singular support of the corresponding $\mathcal{D}$-modules?

Key findings

• The derived category of twisted $\mathcal{D}_{\mathcal{B}un_G}(\xi^{\otimes\kappa})$-modules is conjectured to be equivalent to the derived category of twisted $\mathcal{D}_{\mathcal{B}un_{G^\vee}}(\xi^{\vee\otimes\kappa^\vee})$-modules for any $\kappa \in \mathbb{C}^\times$, with $\kappa^\vee = 1/(r\kappa)$.
• The kernel $\mathcal{L}_\kappa$ is a flat $\mathcal{O}_{\mathbb{P}^1}$-module on $\mathbb{P}^1 \times \mathcal{B}un_G \times \mathcal{B}un_{G^\vee}$, with fibers at $t=\kappa$, $t=0$, and $t=\infty$ corresponding to the quantum, classical, and dual classical Langlands kernels, respectively.
• The singular support of $\mathcal{L}_\kappa$ is required to coincide with the preimage of the diagonal $\Delta_\kappa = \{v_i^\vee = \kappa^{d_i} v_i\}$ under the product of Hitchin maps $\chi_G \times \chi_{G^\vee}$, linking the construction to integrable systems.
• The classical limit $\kappa \to 0$ implies that the singular support of the $\mathcal{D}_{\mathcal{B}un_G}$-modules $\mathcal{F}_{\mathcal{P}^\vee}$ is contained in the global nilpotent cone, the preimage of zero under $\chi_G$.
• The space $\mathcal{C}onf_{G^\vee}$ is identified with the space of colored divisors $\{D_i : \deg D_i \geq 2g-2\}$, and is isomorphic to a disjoint union of symmetric products of the curve $C$, providing a geometric realization of conformal blocks.
• The construction is compatible with conformal field theory: the kernel $\mathcal{L}_\kappa$ arises from a flat family over $\mathbb{P}^1$ that interpolates between the classical and dual classical Langlands theories, with the conformal field theory structure encoded in the $\mathcal{B}un_{B/[N,N]}^{>0}$ diagram and the vector bundle $\mathcal{B}un_{\omega,H}$.