Collective field theory of gauged multi-matrix models: Integrating out off-diagonal strings

TL;DR

This paper develops a collective field theory for a gauged two-matrix model, revealing new non-local features and the necessity of a mass term for locality, thereby extending understanding of matrix models in higher dimensions.

Contribution

It introduces a novel collective field approach for gauged multi-matrix models, including off-diagonal elements, with a focus on gauge fixing and non-locality.

Findings

The $(2+1)$-dimensional collective field action exhibits non-local features.

Adding a mass term restores time-locality in the potential.

Recovers single matrix quantum mechanics in the appropriate limit.

Abstract

We study a two-matrix toy model with a BFSS-like interaction term using the collective field formalism. The main technical simplification is obtained by gauge-fixing first, and integrating out the off-diagonal elements, before changing to the collective field variable. We show that the resulting $(2+1)$-dimensional collective field action has novel features with respect to non-locality, and that we need to add a mass term to get a time-local potential. As is expected, one recovers the single matrix quantum mechanical collective field Hamiltonian in the proper limit.