Topological strings and large N phase transitions I: Nonchiral expansion of q-deformed Yang-Mills theory

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 Yang-Mills 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.

arXiv comment68 pages, 4 figures, References added and typos corrected

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• Phase transitions, double-scaling limit, and topological strings
• Duality and Higher Temperature Phases of Large $N$ Chern-Simons Matter Theories on $S^2\times S^1$
• Level-rank duality of the U(N) WZW model, Chern-Simons theory, and 2d qYM theory
• Large N duality beyond the genus expansion
• Quantum Entanglement of Baby Universes
• Large $N$ phase transition in $T\overline{T}$-deformed $2d$ Yang-Mills theory on the sphere
• Branes, Black Holes and Topological Strings on Toric Calabi-Yau Manifolds
• Instanton on toric singularities and black hole countings
• q-deformations of two-dimensional Yang-Mills theory: Classification, categorification and refinement
• Chern-Simons matrix models, two-dimensional Yang-Mills theory and the Sutherland model
• Black Holes in Type IIA String on Calabi-Yau Threefolds with Affine ADE Geometries and q-Deformed 2d Quiver Gauge Theories
• $T\overline{T}$-deformation of $q$-Yang-Mills theory
• Unitary Chern-Simons matrix model and the Villain lattice action
• Universal Character and Large N Factorization in Topological Gauge/String Theory
• Refined Chern-Simons theory and (q,t)-deformed Yang-Mills theory: Semi-classical expansion and planar limit
• $T\bar{T}$-deformed 2D Yang-Mills at large N: collective field theory and phase transitions
• New Phase Transitions in Chern-Simons Matter Theory
• Large N Expansion of q-Deformed Two-Dimensional Yang-Mills Theory and Hecke Algebras
• Five-dimensional cohomological localization and squashed $q$-deformations of two-dimensional Yang-Mills theory
• String Duals of Two-Dimensional Yang-Mills and Symmetric Product Orbifolds
• Chiral expansion and Macdonald deformation of two-dimensional Yang-Mills theory
• A Note on q-Deformed Two-Dimensional Yang-Mills and Open Topological Strings
• Phase diagram of $q$-deformed Yang-Mills theory on $S^2$ at non-zero $\theta$-angle
• Exact Lorentz-violating $q$-deformed O($N$) universality class