### Abstract

We study the relationship between instanton counting in N = 4 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. © 2007 Elsevier B.V. All rights reserved.

Original language | English |
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Pages (from-to) | 1-24 |

Number of pages | 24 |

Journal | Nuclear Physics B |

Volume | 772 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 11 Jun 2007 |

### Keywords

- Black holes
- Brane dynamics in gauge theories
- Chern-Simons theories
- Field theories in lower dimensions
- Solitons monopoles and instantons

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## Cite this

*Nuclear Physics B*,

*772*(1-2), 1-24. https://doi.org/10.1016/j.nuclphysb.2007.02.030