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