Geometric Langlands Twists of N = 4 Gauge Theory from Derived Algebraic Geometry

TL;DR

This paper develops derived algebraic geometry techniques to describe moduli spaces of solutions in twisted N=4 supersymmetric gauge theories, revealing their connections to the geometric Langlands program.

Contribution

It introduces a holomorphic twist of N=4 gauge theory and computes the associated derived moduli spaces, linking them to de Rham stacks and algebraic structures relevant for geometric Langlands.

Findings

Derived moduli spaces relate to de Rham stacks of algebraic and flat bundles.

Topological twists can preserve subtle algebraic structures.

New insights into the geometric Langlands correspondence from gauge theory.

Abstract

We develop techniques for describing the derived moduli spaces of solutions to the equations of motion in twists of supersymmetric gauge theories as derived algebraic stacks. We introduce a holomorphic twist of N=4 supersymmetric gauge theory and compute the derived moduli space. We then compute the moduli spaces for the Kapustin-Witten topological twists as its further twists. The resulting spaces for the A- and B-twist are closely related to the de Rham stack of the moduli space of algebraic bundles and the de Rham moduli space of flat bundles, respectively. In particular, we find the unexpected result that the moduli spaces following a topological twist need not be entirely topological, but can continue to capture subtle algebraic structures of interest for the geometric Langlands program.