N=2 gauge theories, instanton moduli spaces and geometric representation theory

TL;DR

This paper explores the deep connections between N=2 gauge theories, instanton moduli spaces, and geometric representation theory, highlighting new constructions and computations that bridge four- and six-dimensional supersymmetric theories.

Contribution

It introduces new toric noncommutative ALE spaces, generalizes instanton moduli space constructions, and computes equivariant partition functions in six-dimensional supersymmetric Yang-Mills theory.

Findings

Construction of new toric noncommutative ALE spaces

Generalization of instanton moduli space constructions

Calculation of equivariant partition functions in 6D Yang-Mills

Abstract

We survey some of the AGT relations between N=2 gauge theories in four dimensions and geometric representations of symmetry algebras of two-dimensional conformal field theory on the equivariant cohomology of their instanton moduli spaces. We treat the cases of gauge theories on both flat space and ALE spaces in some detail, and with emphasis on the implications arising from embedding them into supersymmetric theories in six dimensions. Along the way we construct new toric noncommutative ALE spaces using the general theory of complex algebraic deformations of toric varieties, and indicate how to generalise the construction of instanton moduli spaces. We also compute the equivariant partition functions of topologically twisted six-dimensional Yang-Mills theory with maximal supersymmetry in a general Omega-background, and use the construction to obtain novel reductions to theories in four dimensions.