Quantum Analytic Langlands Correspondence

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 $\mathcal{N}=4$ 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.

88 pages

References in corpus (6)

• Loop operators and S-duality from curves on Riemann surfaces
• A Note on Quantum Geometric Langlands Duality, Gauge Theory, and Quantization of the Moduli Space of Flat Connections
• The complex symplectic geometry of the deformation space of complex projective structures
• Bispectral duality and separation of variables from surface defect transition
• Raviolo vertex algebras
• Classical limit of the geometric Langlands correspondence for $SL(2,\mathbb{C})$

Cited by in corpus (1)

• A microscopic realization of dS${}_3$