Unitary matrix models and random partitions: Universality and multi-criticality

Taro Kimura,Ali Zahabi

doi:10.1007/jhep07(2021)100

Abstract

The generating functions for the gauge theory observables are often represented in terms of the unitary matrix integrals. In this work, the perturbative and non-perturbative aspects of the generic multi-critical unitary matrix models are studied by adopting the integrable operator formalism, and the multi-critical generalization of the Tracy-Widom distribution in the context of random partitions. We obtain the universal results for the multi-critical model in the weak and strong coupling phases. The free energy of the instanton sector in the weak coupling regime, and the genus expansion of the free energy in the strong coupling regime are explicitly computed and the universal multi-critical phase structure of the model is explored. Finally, we apply our results in concrete examples of supersymmetric indices of gauge theories in the large N limit.

Highlights

Introduction and summaryThe construction and counting of the gauge invariant observables, such as BPS operators in supersymmetric gauge theories, have been central problems in gauge theory and string theory, for decades
The perturbative and nonperturbative aspects of the generic multi-critical unitary matrix models are studied by adopting the integrable operator formalism, and the multi-critical generalization of the Tracy-Widom distribution in the context of random partitions
We introduce an analytic approach, based on the machinery of the integrable operator formalism in random matrices and random partitions to study the universal features in the phase structure of gauge theories

Summary

Introduction

Introduction and summaryThe construction and counting of the gauge invariant observables, such as BPS operators in supersymmetric gauge theories, have been central problems in gauge theory and string theory, for decades. An essential step in this framework is applying the Weyl integration formula in the gauge theory [3,4,5], to obtain the full generating function of the multi-trace operators as a group/matrix integral of the plethystic exponentiation of the single-trace operator generating function. The potential of this matrix model is turned up to be the double-trace potential, and in the weak coupling limit, one can approximate the pairwise interaction potentials between the eigenvalues of the matrix model with a single-trace potential, i.e. the Gross-Witten-Wadia (GWW) model [6, 7] and its generalization to higher order polynomial potentials

Objectives

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Discussion

Conclusion