Localization on three-dimensional manifolds

Abstract

In this review article we describe the localization of three dimensional N=2 supersymmetric theories on compact manifolds, including the squashed sphere, S^3_b, the lens space, S^3_b/Z_p, and S^2 x S^1. We describe how to write supersymmetric actions on these spaces, and then compute the partition functions and other supersymmetric observables by employing the localization argument. We briefly survey some applications of these computations.

This is a contribution to the review volume `Localization techniques in quantum field theories' (eds. V. Pestun and M. Zabzine) which contains 17 Chapters. The complete volume is summarized in arXiv:1608.02952 and it can be downloaded at https://arxiv.org/src/1608.02952/anc/LocQFT.pdf or http://pestun.ihes.fr/pages/LocalizationReview/LocQFT.pdf

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Cited by in corpus (40)

• Localization techniques in quantum field theories
• Entanglement entropy: holography and renormalization group
• The supersymmetric index in four dimensions
• Three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories and partition functions on Seifert manifolds: A review
• Large $N$ Partition Functions of the ABJM Theory
• The F-Theorem and F-Maximization
• Gapped Chiral Fermions
• Holomorphic blocks and the 5d AGT correspondence
• 3d $\mathcal{N}=4$ OPE Coefficients from Fermi Gas
• Squashing, Mass, and Holography for 3d Sphere Free Energy
• Supersymmetric localization in two dimensions
• Introduction to localization in quantum field theory
• Localization at large N in Chern-Simons-matter theories
• An introduction to supersymmetric field theories in curved space
• Perturbative and nonperturbative aspects of complex Chern-Simons Theory
• Supersymmetric dS/CFT
• Chiral 3d SU(3) SQCD and N=2 mirror duality
• Squashing and supersymmetry enhancement in three dimensions
• Star shaped quivers with flux
• Matrix Models and Holography: Mass Deformations of Long Quiver Theories in 5d and 3d
• 3d N=4 Bootstrap and Mirror Symmetry
• Twisted indices, Bethe ideals and 3d $\mathcal{N}=2$ infrared dualities
• Higher spin supermultiplets in three dimensions: (2,0) AdS supersymmetry
• 3d defects in 5d: RG flows and defect F-maximization
• Phase transitions and Wilson loops in antisymmetric representations in Chern-Simons-matter theory
• The $\mathcal{N}=2$ Prepotential and the Sphere Free Energy
• A parafermionic hypergeometric function and supersymmetric 6j-symbols
• Lens partition function, pentagon identity and star-triangle relation
• Hyperbolic and trigonometric hypergeometric solutions to the star-star equation
• Pentagon identities arising in supersymmetric gauge theory computations
• Trinions for the $3d$ compactification of the $5d$ rank 1 $E_{N_f+1}$ SCFTs
• Five-dimensional cohomological localization and squashed $q$-deformations of two-dimensional Yang-Mills theory
• Exact results and Schur expansions in quiver Chern-Simons-matter theories
• Compactifying 5d superconformal field theories to 3d
• The 3d $A$-model and generalised symmetries, Part I: bosonic Chern-Simons theories
• One-form symmetries and the 3d $\mathcal{N}=2$ $A$-model: Topologically twisted indices and CS theories
• Dualities and loops on squashed $S^3$
• The Holographic Contributions to the Sphere Free Energy
• Partition functions for supersymmetric gauge theories on spheres
• Large $N$ limits of supersymmetric quantum field theories: A pedagogical overview