Black Holes, q-Deformed 2d Yang-Mills, and Non-perturbative Topological Strings

TL;DR

This paper explores the connection between black hole bound states, q-deformed 2D Yang-Mills theory, and topological strings, revealing a non-perturbative framework that links gauge theory, string amplitudes, and black hole entropy.

Contribution

It demonstrates that black hole partition functions can be expressed as sums of chiral blocks related to topological string amplitudes, providing a non-perturbative definition of topological strings in certain backgrounds.

Findings

Black hole partition functions relate to sums of chiral blocks.

Leading block corresponds to closed topological string amplitudes.

Sub-leading blocks involve D-brane insertions on Riemann surfaces.

Abstract

We count the number of bound states of BPS black holes on local Calabi-Yau three-folds involving a Riemann surface of genus g. We show that the corresponding gauge theory on the brane reduces to a q-deformed Yang-Mills theory on the Riemann surface. Following the recent connection between the black hole entropy and the topological string partition function, we find that for a large black hole charge N, up to corrections of $O(e^{-N})$, $Z_{BH}$ is given as a sum of a square of chiral blocks, each of which corresponds to a specific D-brane amplitude. The leading chiral block, the vacuum block, corresponds to the closed topological string amplitudes. The sub-leading chiral blocks involve topological string amplitudes with D-brane insertions at 2g-2 points on the Riemann surface analogous to the $\Omega$ points in the large N 2d Yang-Mills theory. The finite N amplitude provides a non-perturbative definition of topological strings in these backgrounds. This also leads to a novel non-perturbative formulation of c=1 non-critical string at the self-dual radius.