A brief review of the 2d/4d correspondences

Understand

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.