Abstract

We study the relationship between instanton counting in Yang–Mills theory on a generic four-dimensional toric orbifold and the semi-classical expansion of q-deformed Yang–Mills theory on the blowups of the minimal resolution of the orbifold singularity, with an eye to clarifying the recent proposal of using two-dimensional gauge theories to count microstates of black holes in four dimensions. We describe explicitly the instanton contributions to the counting of D-brane bound states which are captured by the two-dimensional gauge theory. We derive an intimate relationship between the two-dimensional Yang–Mills theory and Chern–Simons theory on generic Lens spaces, and use it to show that the known instanton counting is only reproduced when the Chern–Simons contributions are treated as non-dynamical boundary conditions in the D4-brane gauge theory. We also use this correspondence to discuss the counting of instantons on higher genus ruled Riemann surfaces.

Keywords: Black holes; Solitons monopoles and instantons; Brane dynamics in gauge theories; Chern–Simons theories; Field theories in lower dimensions

Article Outline

1. Introduction

2. D-brane partition function on toric orbifolds

3. q-Deformed gauge theory on toric singularities

3.1. Sewing construction of the partition function

3.2. Semi-classical expansion

3.3. Emergence of four-dimensional instantons

3.4. Example: Line bundles over

3.5. Example: ALE spaces

4. Chern–Simons gauge theory on Lens spaces

4.1. Classical solutions

4.2. Semi-classical expansion

5. Instantons on higher genus ruled surfaces

6. Conclusions

Acknowledgements

Appendix A. Continued fractions and the Cartan matrix

References