Transmutation of a Trans-series: The Gross-Witten-Wadia Phase Transition

Abstract

We study the change in the resurgent asymptotic properties of a trans-series in two parameters, a coupling $g^2$ and a gauge index $N$, as a system passes through a large $N$ phase transition, using the universal example of the Gross-Witten-Wadia third-order phase transition in the unitary matrix model. This transition is well-studied in the immediate vicinity of the transition point, where it is characterized by a double-scaling limit Painlevé II equation, and also away from the transition point using the pre-string difference equation. Here we present a complementary analysis of the transition at all coupling and all finite N, in terms of a differential equation, using the explicit Tracy-Widom mapping of the Gross-Witten-Wadia partition function to a solution of a Painlevé III equation. This mapping provides a simple method to generate trans-series expansions in all parameter regimes, and to study their transmutation as the parameters are varied. For example, at any finite N the weak coupling expansion is divergent, with a non-perturbative trans-series completion; on the other hand, the strong coupling expansion is convergent, and yet there is still a non-perturbative trans-series completion. We show how the different instanton terms `condense' at the transition point to match with the double-scaling limit trans-series. We also define a uniform large N strong-coupling expansion (a non-linear analogue of uniform WKB), which is much more precise than the conventional large N expansion through the transition region, and apply it to the evaluation of Wilson loops.

arXiv comment46 pages; 20 figures

References in corpus (40)

• Level-Spacing Distributions and the Airy Kernel
• 2D Gravity and Random Matrices
• Two Dimensional Gauge Theories Revisited
• Quantum Field Theory in the Large N Limit: a review
• Random Matrix Theory and Chiral Symmetry in QCD
• Level-Spacing Distributions and the Bessel Kernel
• Nekrasov Functions and Exact Bohr-Sommerfeld Integrals
• Quantization of Integrable Systems and Four Dimensional Gauge Theories
• Lectures on non-perturbative effects in large N gauge theories, matrix models and strings
• Exact scaling functions for one-dimensional stationary KPZ growth
• Large $N$ Phase Transition in Continuum QCD${}_2$
• Non-perturbative effects and the refined topological string
• Multi-Instantons and Exact Results I: Conjectures, WKB Expansions, and Instanton Interactions
• Top eigenvalue of a random matrix: large deviations and third order phase transition
• Non-intersecting Brownian walkers and Yang-Mills theory on the sphere
• Generating Non-perturbative Physics from Perturbation Theory
• Nonperturbative Ambiguities and the Reality of Resurgent Transseries
• Nonperturbative effects and nonperturbative definitions in matrix models and topological strings
• Multi-Instantons and Exact Results II: Specific Cases, Higher-Order Effects, and Numerical Calculations
• Resurgence and the Nekrasov-Shatashvili Limit: Connecting Weak and Strong Coupling in the Mathieu and Lam'e Systems
• Instanton Induced Large $N$ Phase Transitions in Two and Four Dimensional QCD
• Some Properties of Large $N$ Two Dimensional Yang--Mills Theory
• Theta dependence, sign problems and topological interference
• Deconstructing zero: resurgence, supersymmetry and complex saddles
• Instantons in Quantum Mechanics and Resurgent Expansions
• Random Unitary Matrices, Permutations and Painleve
• Complex Saddles in Two-dimensional Gauge Theory
• Borel summability of perturbative series in 4d N=2 and 5d N=1 theories
• The toroidal block and the genus expansion
• Unitary Matrix Models and Painlevé III
• Non-Perturbative Quantum Geometry III
• How to resum perturbative series in 3d N=2 Chern-Simons matter theories
• Supersymmetric $U(N)$ Chern-Simons-matter theory and phase transitions
• Non-Perturbative Quantum Geometry
• Non-Perturbative Quantum Geometry II
• Wilson loops in unitary matrix models at finite $N$
• Universal Randomness
• Two-dimensional Yang-Mills theory, Painleve equations and the six-vertex model
• Tracy-Widom distribution as instanton sum of 2D IIA superstrings
• Unitary Matrix Models and Phase Transition

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