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

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 $L(p,s)$.

arXiv comment57 pages; v2: minor corrections and changes; Final version to be published in JHEP

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• The 5d Superconformal Index at Large $N$ and Black Holes
• Phases of five-dimensional supersymmetric gauge theories
• 3d defects in 5d: RG flows and defect F-maximization
• Partition Functions and Fibering Operators on the Coulomb Branch of 5d SCFTs
• Large $N$ limits of supersymmetric quantum field theories: A pedagogical overview