A Study of the Complex Action Problem in a Simple Model for Dynamical Compactification in Superstring Theory Using the Factorization Method

TL;DR

This paper investigates the spontaneous symmetry breaking in a simplified matrix model related to superstring theory, employing the factorization method to overcome complex action issues and confirming lower-dimensional dominance.

Contribution

It introduces the use of the factorization method on the density of states to study symmetry breaking in a simple SO(4) invariant matrix model, supporting the dynamical compactification hypothesis.

Findings

Symmetry breaking confirmed in the SO(4) matrix model.

Scaling properties of the density of states are consistent with theoretical expectations.

Results align with previous Gaussian expansion method calculations.

Abstract

The IIB matrix model proposes a mechanism for dynamically generating four dimensional space--time in string theory by spontaneous breaking of the ten dimensional rotational symmetry $\textrm{SO}(10)$. Calculations using the Gaussian expansion method (GEM) lend support to this conjecture. We study a simple $\textrm{SO}(4)$ invariant matrix model using Monte Carlo simulations and we confirm that its rotational symmetry breaks down, showing that lower dimensional configurations dominate the path integral. The model has a strong complex action problem and the calculations were made possible by the use of the factorization method on the density of states $\rho_n(x)$ of properly normalized eigenvalues $\tilde\lambda_n$ of the space--time moment of inertia tensor. We study scaling properties of the factorized terms of $\rho_n(x)$ and we find them in agreement with simple scaling arguments. These can be used in the finite size scaling extrapolation and in the study of the region of configuration space obscured by the large fluctuations of the phase. The computed values of $\tilde\lambda_n$ are in reasonable agreement with GEM calculations and a numerical method for comparing the free energy of the corresponding ansatze is proposed and tested.