Higgs Bundles, Gauge Theories and Quantum Groups

Abstract

The appearance of the Bethe Ansatz equation for the Nonlinea r Schr ?odinger equa- tion in the equivariant integration over the moduli space of Higgs bundles is revisited. We argue that the wave functions of the corresponding two-di mensional topological U ( N ) gauge theory reproduce quantum wave functions of the Nonlin ear Schr ?odinger equation in the N -particle sector. This implies the full equivalence betwee n the above gauge theory and the N -particle sub-sector of the quantum theory of Nonlinear Sch r ?odinger equation. This also implies the explicit correspondence between the g auge theory and the represen- tation theory of degenerate double affine Hecke algebra. We pr opose 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