Localization on the D-brane, two-dimensional gauge theory and matrix models

Abstract

We consider the effective topological field theory on Euclidean D-strings wrapping on a 2-cycle in the internal space. We evaluate the vacuum expectation value of a suitable operator corresponding to the chemical potential of vortices bounded to the D-strings, and find that it reduces to the partition function of generalized two-dimensional Yang-Mills theory as a result of localization. We argue that the partition function gives a grand canonical ensemble of multi-instanton corrections for four-dimensional $\mathcal{N}=2$ gauge theory in a suitable large $N$ limit. We find two-dimensional gauge theories that provide the instanton partition function for four-dimensional $\mathcal{N}=2$ theories with the hypermultiplets in the adjoint and fundamental representations. We also propose a partition function that gives the instanton contributions to four-dimensional $\mathcal{N}=2$ quiver gauge theory. We discuss the relation between Nekrasov's instanton partition function and the Dijkgraaf-Vafa theory in terms of large $N$ phase transitions of the generalized two-dimensional Yang-Mills theory.