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A DK Phase Transition in q-Deformed Yang-Mills on S^2 and Topological Strings

Daniel Jafferis, Joseph Marsano

TL;DR

The paper demonstrates a large-$N$ Douglas–Kazakov–type phase transition in q-deformed Yang–Mills theory on $S^2$ for $p>2$, driven by instanton contributions and studied in the ’t Hooft limit with a fixed $\lambda=g_{YM}^2 N$. It analyzes both $\theta=0$ and $\theta\neq 0$ cases, deriving a phase boundary in the $\lambda$–$\theta$ plane and arguing for a rich, multi-instanton structure that shapes the strong-coupling phase, including a strong-coupling saddle described by elliptic integrals. The work then turns to the chiral decomposition and the topological-string interpretation, showing that the trivial chiral block (topological string partition function) itself undergoes a phase transition at a different coupling than the full theory, signaling a breakdown of OSV-type factorization in the weak-coupling regime and revealing nonperturbative corrections to the factorization. Overall, the results reveal a nuanced phase structure in qYM on $S^2$ and its topological-string connections, with implications for understanding nonperturbative effects in OSV-like dualities and the dynamics of chiral blocks at large $N$.

Abstract

We demonstate the existence of a large $N$ phase transition with respect to the 't Hooft coupling in q-deformed Yang-Mills theory on $S^2$. The strong coupling phase is characterized by the formation of a clump of eigenvalues in the associated matrix model of Douglas-Kazakov (DK) type (hep-th/9305047). By understanding this in terms of instanton contributions to the q-deformed Yang-Mills theory, we gain some insight into the strong coupling phase as well as probe the phase diagram at nonzero values of the $θ$ angle. The Ooguri-Strominger-Vafa relation (hep-th/0405146) of this theory to topological strings on the local Calabi-Yau $\mathcal{O}(-p) \oplus \mathcal{O}(p-2) \to \mathbb{P}^1$ via a chiral decompostion at large $N$ hep-th/0411280, motivates us to investigate the phase structure of the trivial chiral block, which corresponds to the topological string partition function, for $p>2$. We find a phase transition at a different value of the coupling than in the full theory, indicating the likely presence of a rich phase structure in the sum over chiral blocks.

A DK Phase Transition in q-Deformed Yang-Mills on S^2 and Topological Strings

TL;DR

The paper demonstrates a large- Douglas–Kazakov–type phase transition in q-deformed Yang–Mills theory on for , driven by instanton contributions and studied in the ’t Hooft limit with a fixed . It analyzes both and cases, deriving a phase boundary in the – plane and arguing for a rich, multi-instanton structure that shapes the strong-coupling phase, including a strong-coupling saddle described by elliptic integrals. The work then turns to the chiral decomposition and the topological-string interpretation, showing that the trivial chiral block (topological string partition function) itself undergoes a phase transition at a different coupling than the full theory, signaling a breakdown of OSV-type factorization in the weak-coupling regime and revealing nonperturbative corrections to the factorization. Overall, the results reveal a nuanced phase structure in qYM on and its topological-string connections, with implications for understanding nonperturbative effects in OSV-like dualities and the dynamics of chiral blocks at large .

Abstract

We demonstate the existence of a large phase transition with respect to the 't Hooft coupling in q-deformed Yang-Mills theory on . The strong coupling phase is characterized by the formation of a clump of eigenvalues in the associated matrix model of Douglas-Kazakov (DK) type (hep-th/9305047). By understanding this in terms of instanton contributions to the q-deformed Yang-Mills theory, we gain some insight into the strong coupling phase as well as probe the phase diagram at nonzero values of the angle. The Ooguri-Strominger-Vafa relation (hep-th/0405146) of this theory to topological strings on the local Calabi-Yau via a chiral decompostion at large hep-th/0411280, motivates us to investigate the phase structure of the trivial chiral block, which corresponds to the topological string partition function, for . We find a phase transition at a different value of the coupling than in the full theory, indicating the likely presence of a rich phase structure in the sum over chiral blocks.

Paper Structure

This paper contains 13 sections, 134 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The Douglas-Kazakov Phase Transition in Pure 2-D Yang-Mills theory
  3. Review of the Exact Solution
  4. The Phase Transition at $\theta=0$
  5. Instanton Sectors and $\theta\ne 0$
  6. The $q$-deformed Theory
  7. Review of the Exact Solution
  8. The Phase Transition at $\theta=0$
  9. Instanton Sectors and $\theta\ne 0$
  10. Chiral decomposition, and phase transition in topological string
  11. Interpretation of the trivial chiral block
  12. Phase Structure of the Trivial Chiral Block
  13. Concluding Remarks

Figures (5)

  • Figure 1: Sketch of the dominant distribution of $n_i$ in the weak and strong coupling phases of 2-dimensional Yang-Mills on $S^2$
  • Figure 2: Transition line \ref{['pymgthc']} in the $\lambda/\theta$ plane
  • Figure 3: Conjecture for the 2D Yang-Mills phase diagram provided we take the curve \ref{['pymgthc']} seriously over the entire range $0\le\theta\le\pi^2$. The dotted lines represent the first of numerous conjectured transition lines that cluster near $\theta=\pi$.
  • Figure 4: Plot of the curve \ref{['qdthcrit']} for $r=1,p=3$.
  • Figure 5: The saddle point density $\rho(u)$ as a function of $\lambda$ in the case $p=3$ for $\lambda>\approx 1.74174$.