Topological reduction of 4D SYM to 2D σ-models

Abstract By considering a (partial) topological twisting of supersymmetric Yang-Mills compactified on a 2d space with 't Hooft magnetic flux tumed on we obtain a supersymmetric σ-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 σ-models on moduli space of solutions to Hitchin equations.