Five-dimensional cohomological localization and squashed q -deformations of two-dimensional Yang-Mills theory
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Springer
Received: April 14, 2020 Accepted: May 12, 2020 Published: June 4, 2020
Leonardo Santilli,a Richard J. Szabob,c,d and Miguel Tierze,a a
Grupo de F´ısica Matem´ atica, Departamento de Matem´ atica, Faculdade de Ciˆencias, Universidade de Lisboa, Campo Grande, Edif´ıcio C6, 1749-016 Lisboa, Portugal b Department of Mathematics, Heriot-Watt University, Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, U.K. c Maxwell Institute for Mathematical Sciences, Bayes Centre, 47 Potterrow, Edinburgh EH8 9BT, U.K. d Higgs Centre for Theoretical Physics, James Clerk Maxwell Building, Kings Buildings, Edinburgh EH9 3JZ, U.K. e Departamento de Matem´ atica, ISCTE — Instituto Universit´ ario de Lisboa, Avenida das For¸cas Armadas, 1649-026 Lisboa, Portugal
E-mail: [email protected], [email protected], [email protected] 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 b2 = p/s yields a new correspondence with Chern-Simons theory on lens spaces L(p, s). Keywords: Field Theories in Higher Dimensions, Field Theories in Lower Dimensions, Supersymmetric Gauge Theory, Differential and Algebraic Geometry ArXiv ePrint: 2003.09411
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP06(2020)036
JHEP06(2020)036
Five-dimensional cohomological localization and squashed q-deformations of two-dimensional Yang-Mills theory
Contents 1
2 Preliminaries on superconformal field theories and localization 2.1 Squashed geometry and six-dimensional superconformal field theories 2.2 Reductions to two and three dimensions 2.3 Basics of cohomological localization 2.4 Localization of N = 1 gauge theories on Seifert manifolds
3 3 5 7 8
3 N = 2 cohomological gauge theories in three dimensions 3.1 Supersymmetric Yang-Mills theory and its cohomological formulation 3.2 One-loop determinant of the vector multiplet in a regular background 3.3 One-loop determinant of the vector multiplet in an irregular background 3.4 One-loop determinant of a hypermultiplet 3.5 Applications of the cohomological localization formulas 3.6 Extension to S 1 × Cg
12 12 15 17 17 18 21
4 N = 1 cohom
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