Gauge Theory and the Analytic Form of the Geometric Langlands Program

TL;DR

This paper offers a gauge-theoretic perspective on the analytic geometric Langlands program, connecting Hitchin Hamiltonians and Hecke operators with quantum operators in a novel framework.

Contribution

It introduces a new gauge-theoretic interpretation of the analytic geometric Langlands program, emphasizing the role of electric-magnetic duality and operator organization.

Findings

Hitchin Hamiltonians and Hecke operators are realized as quantum operators.

Electric-magnetic duality underpins the gauge-theoretic interpretation.

A novel organization of gauge theory ingredients for the analytic Langlands.

Abstract

We present a gauge-theoretic interpretation of the "analytic" version of the geometric Langlands program, in which Hitchin Hamiltonians and Hecke operators are viewed as concrete operators acting on a Hilbert space of quantum states. The gauge theory ingredients required to understand this construction -- such as electric-magnetic duality between Wilson and 't Hooft line operators in four-dimensional gauge theory -- are the same ones that enter in understanding via gauge theory the more familiar formulation of geometric Langlands, but now these ingredients are organized and applied in a novel fashion.