Higgs Bundles, Gauge Theories and Quantum Groups

TL;DR

This paper explores the deep connections between Higgs bundles, gauge theories, and quantum groups, demonstrating how gauge theory wave functions relate to solutions of the Nonlinear Schrödinger equation and the representation theory of double affine Hecke algebras.

Contribution

It establishes an explicit correspondence between 2D topological gauge theories and quantum integrable systems, linking gauge theory, representation theory, and geometric Langlands correspondence.

Findings

Wave functions of gauge theory reproduce quantum wave functions of NLS.

Full equivalence between gauge theory and NLS N-particle sector.

Connection between gauge theories and double affine Hecke algebra representations.

Abstract

The appearance of the Bethe Ansatz equation for the Nonlinear Schr\"{o}dinger equation in the equivariant integration over the moduli space of Higgs bundles is revisited. We argue that the wave functions of the corresponding two-dimensional topological U(N) gauge theory reproduce quantum wave functions of the Nonlinear Schr\"{o}dinger equation in the $N$-particle sector. This implies the full equivalence between the above gauge theory and the $N$-particle sub-sector of the quantum theory of Nonlinear Schr\"{o}dinger equation. This also implies the explicit correspondence between the gauge theory and the representation theory of degenerate double affine Hecke algebra. We propose similar construction based on the $G/G$ gauged WZW model leading to the representation theory of the double affine Hecke algebra. The relation with the Nahm transform and the geometric Langlands correspondence is briefly discussed.