Topological strings and largeNphase transitions I: Nonchiral expansion ofq-deformed Yang-Mills theory

Abstract

Abstract:We examine the problem of counting bound states of BPS black holes on local Calabi-Yau threefolds which are fibrations over a Riemann surface by computing the partition function of q-deformed Yang-Mills theory on the Riemann surface. We study in detail the genus zero case and obtain, at finite N , the instanton expansion of the gauge theory. It can be written exactly as the partition function for U (N ) Chern-Simons gauge theory on a Lens space, summed over all non-trivial vacua, plus a tower of non-perturbative instanton contributions. The correspondence between two and three dimensional gauge theories is elucidated by an explicit mapping between two-dimensional Yang-Mills instantons and flat connections on the Lens space. In the large N limit we find a peculiar phase structure in the model. At weak string coupling the theory reduces exactly to the trivial flat connection sector with instanton contributions exponentially suppressed, and the topological string partition function on the resolved conifold is reproduced in this regime. At a certain critical point all non-trivial vacua contribute, instantons are enhanced and the theory appears to undergo a phase transition into a strong coupling regime. We rederive these results by performing a saddle-point approximation to the exact partition function. We obtain a q-deformed version of the Douglas-Kazakov equation for two-dimensional YangMills theory on the sphere, whose one-cut solution below the transition point reproduces the resolved conifold geometry. Above the critical point we propose a two-cut solution that should reproduce the chiral-antichiral dynamics found for black holes on the Calabi-Yau threefold and the Gross-Taylor string in the undeformed limit. The transition from the strong coupling phase to the weak coupling phase appears to be of third order.

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

“…When the lines reach and go below t = t + c (p), the distribution function ρ becomes larger than 1 and a phase transition occurs. Rather remarkably, the value t + c (p) is very close to the value of the Kähler modulus that triggers the phase transition in the full coupled q-deformed gauge theory [34,38,39]. At this point no physical solution exists until we reach a second critical point t − c (p).…”

“…The saddle point equation governing the distribution of Young tableaux variables in chiral qdeformed Yang-Mills theory coincides with that of the full coupled gauge theory [34,38,39]. The new information is completely encoded in boundary conditions on the solutions to this equation.…”

“…By parameterizing the free energy as 25) we find that the first six contributions are given by Note that N 0 n,k (p) is a polynomial of degree 2n − 2 in p. As a first application of these results, let us check the consistency of our computations with ordinary large N Yang-Mills theory. It is possible to find a limit in which the undeformed theory is recovered [34,38,39], namely p → ∞ with A = 2 pt = p N g s kept fixed. The free energies (2.26) in this limit should reproduce the analogous quantities obtained in ordinary chiral Yang-Mills theory from the Gross-Taylor string expansion.…”

“…In the matrix model approach we therefore have to solve the saddle-point equation [34,38,39] A z 2 =t…”

“…Many of the features of two-dimensional Yang-Mills theory on a sphere, such as its nontrivial instanton driven phase transition in the large N limit [36,37], are preserved by the qdeformation and were thoroughly addressed in [34,38,39]. 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.…”