A brief review of the 2d/4d correspondences

Abstract

An elementary introduction to the 2d/4d correspondences is given. After quickly reviewing the 2d q-deformed Yang-Mills theory and the Liouville theory, we will introduce 4d theories obtained by coupling trifundamentals to SU(2) gauge fields. We will then see concretely that the supersymmetric partition function of these theories on S^3 x S^1 and on S^4 is given respectively by the q-deformed Yang-Mills theory and the Liouville theory. After giving a short discussion on how this correspondence may be understood from the viewpoint of the 6d N=(2,0) theory, we conclude the review by enumerating future directions. Most of the technical points will be referred to more detailed review articles.

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 v4: a reference corrected

References in corpus (13)

• Localization techniques in quantum field theories
• The 4d Superconformal Index from q-deformed 2d Yang-Mills
• Sphere Partition Functions and the Zamolodchikov Metric
• The supersymmetric index in four dimensions
• Black Holes, Instanton Counting on Toric Singularities and q-Deformed Two-Dimensional Yang-Mills Theory
• Bases in coset conformal field theory from AGT correspondence and Macdonald polynomials at the roots of unity
• Toda Theory From Six Dimensions
• An introduction to supersymmetric field theories in curved space
• Perturbative and nonperturbative aspects of complex Chern-Simons Theory
• N=2 SUSY gauge theories on S^4
• Instanton on toric singularities and black hole countings
• N=2 quiver gauge theories on A-type ALE spaces
• Indices for 6 dimensional superconformal field theories

Cited by in corpus (17)

• Localization techniques in quantum field theories
• IIB flux non-commutativity and the global structure of field theories
• New Modular Invariants in $\mathcal{N}=4$ Super-Yang-Mills Theory
• Taming Defects in $\mathcal{N}=4$ Super-Yang-Mills
• Holomorphic blocks and the 5d AGT correspondence
• A slow review of the AGT correspondence
• Supersymmetric localization in two dimensions
• Introduction to localization in quantum field theory
• An introduction to supersymmetric field theories in curved space
• N=2 SUSY gauge theories on S^4
• A one-dimensional theory for Higgs branch operators
• Trinion Conformal Blocks from Topological strings
• An infrared bootstrap of the Schur index with surface defects
• Super Topological Recursion and Gaiotto Vectors For Superconformal Blocks
• Holomorphic anomaly of 2d Yang-Mills theory on a torus revisited
• Dual infrared limits of 6d $\mathcal{N}$=(2,0) theory
• Matrix models from localization of five-dimensional supersymmetric noncommutative U(1) gauge theory