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

TL;DR

This paper connects D-brane effective theories to two-dimensional gauge theories, showing how localization reduces complex instanton calculations in four-dimensional N=2 gauge theories to solvable 2D models, revealing new insights into their structure.

Contribution

It demonstrates the reduction of D-brane effective theories to 2D Yang-Mills models and proposes new partition functions for instanton effects in N=2 gauge theories with various matter content.

Findings

Partition function reduces to generalized 2D Yang-Mills theory via localization.

Provides 2D gauge theories for instanton counting in N=2 theories with different matter representations.

Links Nekrasov's instanton partition function with Dijkgraaf-Vafa theory through large N phase transitions.

Abstract

We consider the effective topological field theory on Euclidean D-strings wrapping on a 2-cycle in the internal space. We evaluate the vev 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 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 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 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.