Complex Langevin analysis of the spontaneous symmetry breaking in dimensionally reduced super Yang-Mills models

TL;DR

This paper demonstrates that the complex Langevin method can effectively study spontaneous symmetry breaking in the 6D type IIB matrix model, providing evidence for SO(6) symmetry breaking to SO(3), which relates to the dimensionality of spacetime.

Contribution

The paper applies the complex Langevin method to a 6D super Yang-Mills model, showing its effectiveness in analyzing symmetry breaking in models with sign problems.

Findings

SO(6) symmetry is broken down to SO(3) in the 6D model

Results agree with previous Gaussian expansion studies

CLM can handle sign problems in superstring related models

Abstract

In recent years the complex Langevin method (CLM) has proven a powerful method in studying statistical systems which suffer from the sign problem. Here we show that it can also be applied to an important problem concerning why we live in four-dimensional spacetime. Our target system is the type IIB matrix model, which is conjectured to be a nonperturbative definition of type IIB superstring theory in ten dimensions. The fermion determinant of the model becomes complex upon Euclideanization, which causes a severe sign problem in its Monte Carlo studies. It is speculated that the phase of the fermion determinant actually induces the spontaneous breaking of the SO(10) rotational symmetry, which has direct consequences on the aforementioned question. In this paper, we apply the CLM to the 6D version of the type IIB matrix model and show clear evidence that the SO(6) symmetry is broken down to SO(3). Our results are consistent with those obtained previously by the Gaussian expansion method.