Quantum Analytic Langlands Correspondence

TL;DR

This work develops a one-parameter deformation of the Analytic Langlands Correspondence by embedding the spectral problem for quantised Hitchin Hamiltonians into a quantum-field-theoretic framework based on the $H_3^+$ WZNW model. It constructs a two-copy skein-algebra action via Verlinde line operators on generalized partition functions, and conjectures that these operators generate a basis for the Hilbert space, thereby realizing a deformation of the Analytic Langlands picture. The proposal is substantiated through a triad of quantum-field-theoretic realizations (2d Kac–Moody/CFT, 3d HT-BF, 4d Kapustin–Witten), a detailed treatment of real opers via grafting and Fenchel–Nielsen coordinates, and a sophisticated link between Liouville theory, the H3+ model, and grafting in the critical level limit. The results illuminate a path to a quantum analytic Langlands program where real opers correspond to single-valued, $L^2$-normalisable KZB solutions, with Verlinde operators encoding the grafting data and diagrammatic skein relations that connect to the mapping class group and Fenchel–Nielsen-type coordinates, potentially impacting class ${ m S}$ theories and complex Chern–Simons quantisation.

Abstract

The analytic Langlands correspondence describes the solution to the spectral problem for the quantised Hitchin Hamiltonians. It is related to the S-duality of $\cal{N}=4$ super Yang-Mills theory. We propose a one-parameter deformation of the Analytic Langlands Correspondence, and discuss its relations to quantum field theory. The partition functions of the $H_3^+$ WZNW model are interpreted as the wave-functions of a spherical vector in the quantisation of complex Chern-Simons theory. Verlinde line operators generate a representation of two copies of the quantised skein algebra on generalised partition functions. We conjecture that this action generates a basis for the underlying Hilbert space, and explain in which sense the resulting quantum theory represents a deformation of the Analytic Langlands Correspondence.

Figures (9)

• Figure 1: Left: the 3d complex HT-BF theory compactified on $C$ gives the desired Hilbert space ${\mathcal{H}}$. Dirichlet boundary conditions give distributional boundary states labelled by a bundle and support holomorphic and anti-holomorphic critical Kac-Moody currents. Right: The analytic continuation procedure gives an embedding in a $G \times G$ A-twisted Kapustin-Witten theory compactified on $[0,1] \times C$, with $B_{cc}\times B_{cc}$ boundary conditions at one end of the segment and a diagonal reflection boundary condition at the other end. Middle: this can be unfolded to the 4d theory compactified on $[-1,1]\times C$. In either case, the Kac-Moody currents live at the corners between the $B_{cc}$ and Dirichlet boundary conditions.
• Figure 2: A fundamental domain for a once-punctured torus.
• Figure 3: Left side: Representation of an annular neighbourhood of a geodesic (red) in the upper half plane model.
• Figure 4: Result of a grafting operation producing four singularity lines.
• Figure 5: A fundamental domain for grafted once-punctured tori.

Theorems & Definitions (1)

• Theorem 1