Black holes, q-deformed 2d Yang–Mills, and non-perturbative topological strings

Abstract

We count the number of bound states of BPS black holes on local Calabi-Yau threefolds 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 Ω 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.

Topological strings and largeNphase transitions II: chiral expansion ofq-deformed Yang-Mills theory

“…with R i labeling the number of boxes in the i-th row of the Young tableau corresponding to R. In [28] a different but related definition is used whereby the quantum dimension is replaced by the quantity S 00 dim q (R) with…”

“…It can be conveniently expressed as [28] dim q R = e gsN 2 n WR q −1 15) where cR is the number of rows inR and WR(q) = WR 0 (q) is related to the large N limit of the modular S-matrix of the SU (N ) WZW model and SU (N ) Chern-Simons gauge theory by…”

“…This conjecture has been addressed and refined in [15]- [26]. A concrete realization of this proposal has been advanced in [27] and subsequently in [28] for local threefolds X p which are fibred over a compact curve Σ g of genus g. In this case a black hole can be engineered as bound states of N D4-branes wrapping a non-compact four-cycle, and an arbitrary number of D-branes wrapping Σ g along with D0-branes. This configuration realizes a four dimensional BPS black hole whose microstates are described by a mixed ensemble of fixed magnetic D4-brane charge N and electric chemical potentials.…”

“…In this paper we will focus on the chiral-antichiral decomposition of q-deformed Yang-Mills theory, which according to [27,28] is the chiral (antichiral) sector that should be related to the holomorphic (antiholomorphic) topological string amplitudes. Similarly to what has been observed in [40,41] for the undeformed theory, we will show that the chiral sector has a non-trivial phase structure that plays a key role in the comparison to string theory.…”

“…Because an exact microscopic computation [28] shows that Z BH coincides with the partition function Z q YM of q-deformed U (N ) Yang-Mills theory on the base S 2 ∼ = P 1 of the fibration (2.1), a natural check of the conjecture is to verify that Z q YM = Z top Z top in the large N limit. In this section we shall analyse this last relation by considering in detail the large N chiral expansion of the q-deformed gauge theory.…”