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

Nicola Caporaso, Michele Cirafici, Luca Griguolo, Sara Pasquetti, Domenico Seminara, Richard J. Szabo

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.

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 , 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 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.

Paper Structure

This paper contains 31 sections, 1 theorem, 278 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Black holes, topological strings and ${\boldsymbol {q} }$-deformed Yang-Mills theory
  3. Counting microstates in ${\cal N}=4$ and $q$-deformed Yang-Mills theories
  4. Large $N$ limit and topological strings
  5. ${\boldsymbol {q} }$-deformed Yang-Mills theory on $S^2$
  6. General properties and the undeformed limit
  7. Instanton expansion
  8. Relation to Chern-Simons theory on Lens spaces
  9. Instanton driven large ${\boldsymbol {N} }$ phase transition
  10. Instanton contributions at large $N$
  11. Resurrecting the resolved conifold
  12. Saddle point equation
  13. Deformed Douglas-Kazakov phase transition
  14. Resolved conifold from the one-cut solution
  15. Two-cut solution and the strong-coupling phase
  16. ...and 16 more sections

Key Result

Theorem 1

Let $f(t)$ be a locally integrable function on $[0,\infty)$ and $\{A_k\}$ a sequence of complex numbers. Suppose that $f(t)$ has a large $t$ expansion for each $n=1,2,\dots$ of the form with $f_n(t)={O}(t^{-n-1})$ as $t\to\infty$. Then for every $z,\rho>0$ and $n=1,2,\dots$ one has where for $k=0,1,2,\dots$ the coefficients $B_k$ are given by and empty sums are understood as zero. The remainder

Figures (4)

  • Figure 1: Plotting $F_0$ for $p=1,2,3,4,5$ as a function of $t$
  • Figure 2: The double cut ansatz for the distribution $\rho(z)$.
  • Figure 3: The cut structure of the distribution $\rho(z)$.
  • Figure 4: The right-hand side of the saddle-point equation times $p$ for $p=3,4,7,20$.

Theorems & Definitions (1)

  • Theorem 1