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arXiv · 2003.09411

Five-dimensional cohomological localization and squashed $q$-deformations of two-dimensional Yang-Mills theory

Leonardo Santilli·Richard J. Szabo·Miguel Tierz
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Abstract

We revisit the duality between five-dimensional supersymmetric gauge theories and deformations of two-dimensional Yang-Mills theory from a new perspective. We give a unified treatment of supersymmetric gauge theories in three and five dimensions using cohomological localization techniques and the Atiyah-Singer index theorem. We survey various known results in a unified framework and provide simplified derivations of localization formulas, as well as various extensions including the case of irregular Seifert fibrations. We describe the reductions to four-dimensional gauge theories, and give an extensive description of the dual two-dimensional Yang-Mills theory when the three-dimensional part of the geometry is a squashed three-sphere, including its extension to non-zero area, and a detailed analysis of the resulting matrix model. The squashing parameter $b$ yields a further deformation of the usual $q$-deformation of two-dimensional Yang-Mills theory, which for rational values $b^2=p/s$ yields a new correspondence with Chern-Simons theory on lens spaces $L(p,s)$.

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Leonardo Santilli, Richard J. Szabo, Miguel Tierz. 2020-03-20. Five-dimensional cohomological localization and squashed $q$-deformations of two-dimensional Yang-Mills theory. https://doi.org/10.1007/jhep06(2020)036

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