Abstract
We construct a large class of gauge theories with extended supersymmetry on four-dimensional manifolds with a Killing vector field and isolated fixed points. We extend previous results limited to super Yang-Mills theory to general mathcal{N} = 2 gauge theories including hypermultiplets. We present a general framework encompassing equivariant Donaldson-Witten theory and Pestun’s theory on S4 as two particular cases. This is achieved by expressing fields in cohomological variables, whose features are dictated by supersymmetry and require a generalized notion of self-duality for two-forms and of chirality for spinors. Finally, we implement localization techniques to compute the exact partition function of the cohomological theories we built up and write the explicit result for manifolds with diverse topologies.
Highlights
Topological twisting and Pestun’s supersymmetric localization on the four-sphere, were unified in a single framework in [1]
We present a general framework encompassing equivariant Donaldson-Witten theory and Pestun’s theory on S4 as two particular cases. This is achieved by expressing fields in cohomological variables, whose features are dictated by supersymmetry and require a generalized notion of self-duality for two-forms and of chirality for spinors
The inclusion of hypermultiplets generically requires the manifold to admit a spin structure. This substantially enlarges the set of supersymmetric field theories which can be analyzed using localization techniques
Summary
We assume that the Killing spinor equations introduced in the previous subsection are satisfied in some supergravity background. We present the structure of the supersymmetry variations for vector multiplets and hypermultiplets coupled to this background
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