Topological Reduction of 4D SYM to 2D $\sigma$--Models

TL;DR

This paper demonstrates how topological twisting and compactification of 4D supersymmetric Yang-Mills theories reduce them to 2D sigma-models, revealing deep connections between gauge theories, dualities, and geometric structures.

Contribution

It introduces a method to derive 2D sigma-models from 4D SYM via topological twisting and flux compactification, linking gauge theory observables to geometric invariants.

Findings

Maps Donaldson invariants to quantum cohomology of flat connection moduli space

Relates S-duality in N=4 SYM to T-duality in sigma-models

Establishes a geometric framework connecting supersymmetric gauge theories and moduli spaces

Abstract

By considering a (partial) topological twisting of supersymmetric Yang-Mills compactified on a 2d space with `t Hooft magnetic flux turned on we obtain a supersymmetric $\sigma$-model in 2 dimensions. For N=2 SYM this maps Donaldson observables on products of two Riemann surfaces to quantum cohomology ring of moduli space of flat connections on a Riemann surface. For N=4 SYM it maps $S$-duality to $T$-duality for $\sigma$-models on moduli space of solutions to Hitchin equations.