Large N optimization for multi-matrix systems

Robert De Mello Koch,Antal Jevicki,Kagiso Mathaba,João P Rodrigues,Xianlong Liu

doi:10.1007/jhep01(2022)168

Abstract

In this work we revisit the problem of solving multi-matrix systems through numerical large N methods. The framework is a collective, loop space representation which provides a constrained optimization problem, addressed through master-field minimization. This scheme applies both to multi-matrix integrals (c = 0 systems) and multi-matrix quantum mechanics (c = 1 systems). The complete fluctuation spectrum is also computable in the above scheme, and is of immediate physical relevance in the later case. The complexity (and the growth of degrees of freedom) at large N have stymied earlier attempts and in the present work we present significant improvements in this regard. The (constrained) minimization and spectrum calculations are easily achieved with close to 104 variables, giving solution to Migdal-Makeenko, and collective field equations. Considering the large number of dynamical (loop) variables and the extreme nonlinearity of the problem, high precision is obtained when confronted with solvable cases. Through numerical results presented, we prove that our scheme solves, by numerical loop space methods, the general two matrix model problem.

Highlights

The degrees of freedom, represented by (Wilson) loop variables, which are physical, gauge invariant collective variables for the description of matrix and non-Abelian gauge theories
Numerical approaches were developed in [37, 38]. In these previous studies the nature of the planar solution was understood as related to a constrained minimization problem, where a significant role is played by a set of inequalities associated with invariants in the collective description
We have applied previously developed numerical, master field methods to solve a variety of coupled two-matrix models

Summary

Introduction

The degrees of freedom, represented by (Wilson) loop variables, which are physical, gauge invariant collective variables for the description of matrix and non-Abelian gauge theories. Numerical approaches were developed in [37, 38] In these previous studies the nature of the (infinite N ) planar solution was understood as related to a constrained minimization problem, where a significant role is played by a set of inequalities associated with invariants (loops) in the collective description. Due to potential high relevance in problems of emergent geometry, thermalization and black hole formation we revisit the earlier collective field constrained minimization and numerical master field methods, with interest in increasing the numbers of degrees of freedom, and the potential for high precision results. These are developed in the present work.

Methods

Results

Conclusion