Multi-cut solutions in Chern–Simons matrix models

Takeshi Morita,Kento Sugiyama

doi:10.1016/j.nuclphysb.2018.01.028

Abstract

We elaborate the Chern–Simons (CS) matrix models at large N. The saddle point equations of these matrix models have a curious structure which cannot be seen in the ordinary one matrix models. Thanks to this structure, an infinite number of multi-cut solutions exist in the CS matrix models. Particularly we exactly derive the two-cut solutions at finite 't Hooft coupling in the pure CS matrix model. In the ABJM matrix model, we argue that some of multi-cut solutions might be interpreted as a condensation of the D2-brane instantons.

Highlights

Matrix models play crucial roles in various topics in theoretical physics. (See, for example, recent review [1, 2].) in string theory, there are several proposals that matrixes may provide the non-perturbative formulation of non-critical strings [3, 4, 5, 6] and critical strings [7, 8, 9], and, many remarkable results have been obtained in this direction
We argue that some of the multi-cut solutions in the ABJM theory might be interpreted as a condensation of the D2-brane instantons [34]
The organization of this paper is as follows: In section 2, we explain the basic idea of the derivation of the multi-cut solutions in the pure CS matrix model at weak coupling. (We show the derivation at finite coupling in appendix A.) In section 3, we consider the CS

Summary

Introduction

Matrix models play crucial roles in various topics in theoretical physics. (See, for example, recent review [1, 2].) in string theory, there are several proposals that matrixes may provide the non-perturbative formulation of non-critical strings [3, 4, 5, 6] and critical strings [7, 8, 9], and, many remarkable results have been obtained in this direction. Some of the multi-cut solutions in the CS matrix model coupled to adjoint matter case might be related to the poles in the Borel plane [35]. We have not understood the physical interpretations of other multi-cut solutions They might be related to some non-perturbative objects in the CS matrix models and string theory. We find the new multi-cut solutions in the CS matrix models, we have to emphasize that we just evaluate the saddle point equation (2). It means that we cannot answer whether these solutions should be summed up through the path integral (1).

One-cut solution

Two-cut solution

Effective potential and free energy