Spontaneous supersymmetry breaking in matrix models from the viewpoints of localization and Nicolai mapping

TL;DR

This paper investigates spontaneous supersymmetry breaking in matrix models using localization and Nicolai mapping, revealing how eigenvalue dynamics and external fields influence SUSY restoration at large N.

Contribution

It extends localization and Nicolai mapping techniques to SUSY matrix models, providing a new matrix-model localization formula and analyzing eigenvalue behavior.

Findings

Eigenvalue dynamics balance localization and Vandermonde forces.

Nicolai mapping computes partition functions with external fields.

SUSY is restored in the large-N limit for models with double-well potentials.

Abstract

In the previous work, it was shown that, in supersymmetric (matrix) discretized quantum mechanics, inclusion of an external field twisting the boundary condition of fermions enables us to discuss spontaneous breaking of supersymmetry (SUSY) in the path-integral formalism in a well-defined way. In the present work, we continue investigating the same systems from the points of view of localization and Nicolai mapping. The localization is studied by changing of integration variables in the path integral, which is applicable whether or not SUSY is explicitly broken. We examine in detail how the integrand of the partition function with respect to the integral over the auxiliary field behaves as the auxiliary field vanishes, which clarifies a mechanism of the localization. In SUSY matrix models, we obtain a matrix-model generalization of the localization formula. In terms of eigenvalues of matrix variables, we observe that eigenvalues' dynamics is governed by balance of attractive force from the localization and repulsive force from the Vandermonde determinant. The approach of the Nicolai mapping works even in the presence of the external field. It enables us to compute the partition function of SUSY matrix models for finite N (N is the rank of matrices) with arbitrary superpotential at least in the leading nontrivial order of an expansion with respect to the small external field. We confirm the restoration of SUSY in the large-N limit of a SUSY matrix model with a double-well scalar potential observed in the previous work.