Large N limit of 2D Yang-Mills Theory and Instanton Counting

TL;DR

This paper demonstrates that the large N limit of 2D U(N) Yang-Mills theory reproduces 4D N=2 instanton counting, connecting gauge theory, string theory, and random partitions.

Contribution

It establishes a link between 2D Yang-Mills large N limit and 4D instanton counting, introducing a double scaling limit and a string theory interpretation.

Findings

Large N limit reproduces Nekrasov's instanton counting.

Double scaling limit yields Douglas-Kazakov effective action.

String theory perspective via brane configurations.

Abstract

We examine the two-dimensional U(N) Yang-Mills theory by using the technique of random partitions. We show that the large N limit of the partition function of the 2D Yang-Mills theory on S^2 reproduces the instanton counting of 4D N=2 supersymmetric gauge theories introduced by Nekrasov. We also discuss that we can take the ``double scaling limit'' by fixing the product of the N and cell size in Young diagrams, and the effective action given by Douglas and Kazakov is naturally obtained by taking this limit. We give an interpretation for our result from the view point of the superstring theory by considering a brane configuration that realizes 4D N=2 supersymmetric gauge theories.