A brief review of the 2d/4d correspondences

TL;DR

This paper reviews the 2d/4d correspondences linking 2d q-deformed Yang-Mills and Liouville theories to 4d supersymmetric gauge theories, highlighting their derivation from 6d theories and future research directions.

Contribution

It provides an elementary overview of the 2d/4d correspondences, connecting supersymmetric partition functions to 2d theories and discussing their origin from 6d N=(2,0) theory.

Findings

Partition functions on S^3 x S^1 and S^4 match 2d q-Yang-Mills and Liouville theories

Connection between 4d gauge theories and 2d conformal field theories clarified

Discussion of the 6d origin of the 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.