Perturbative and nonperturbative aspects of complex Chern-Simons Theory

Abstract

We present an elementary review of some aspects of Chern-Simons theory with complex gauge group SL(N,C). We discuss some of the challenges in defining the theory as a full-fledged TQFT, as well as some successes inspired by the 3d-3d correspondence. The 3d-3d correspondence relates partition functions (and other aspects) of complex Chern-Simons theory on a 3-manifold M to supersymmetric partition functions (and other observables) in an associated 3d theory T[M]. Many of these observables may be computed by supersymmetric localization. We present several prominent applications to 3-manifold topology and number theory in light of the 3d-3d correspondence.

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 (18)

• Localization techniques in quantum field theories
• Localization on three-dimensional manifolds
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• Three-dimensional de Sitter horizon thermodynamics
• A slow review of the AGT correspondence
• Holomorphic blocks and the 5d AGT correspondence
• Precision Microstate Counting for the Entropy of Wrapped M5-branes
• Teichmuller TQFT vs Chern-Simons Theory
• Introduction to localization in quantum field theory
• A brief review of the 2d/4d correspondences
• A one-dimensional theory for Higgs branch operators
• $q$-nonabelianization for line defects
• Knots and Their Related $q$-Series
• On the Quantum K-Theory of the Quintic
• Non-semisimple Levin-Wen Models and Hermitian TQFTs from quantum (super)groups
• 3D-3D Correspondence from Seifert Fibering Operators
• On geometric bases for A-polynomials II: ${\mathfrak{su}}_3$ and Kuberberg bracket
• Matrix models from localization of five-dimensional supersymmetric noncommutative U(1) gauge theory