Cohomological Localization of $\mathcal{N}=2$ Gauge Theories with Matter

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 $S^4$ 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.

37 pages including appendices. Partial text overlap with arXiv:1812.06473 in appendices. Minor typo fixes, references added

References in corpus (3)

• Gluing Nekrasov partition functions
• 7D supersymmetric Yang-Mills on curved manifolds
• 7D supersymmetric Yang-Mills on hypertoric 3-Sasakian manifolds

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• Index of the Transversally Elliptic Complex in Pestunization
• Flavor deformations and supersymmetry enhancement in $4d$ $\mathcal{N}=2$ theories