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

TL;DR

This work develops a collective-field description of a gauged two-matrix toy model with BFSS-like interactions to study emergent spatial dimensions. By fixing the gauge, diagonalizing one matrix, and integrating out the off-diagonal modes, the authors derive a 2+1D effective theory for the diagonal eigenvalues and their densities, encoded in a two-dimensional collective field $oldsymbol{}(x,y,t)$. A key finding is that the massless off-diagonal sector yields a time-nonlocal effective action, which is regularized into a time-local theory by introducing a small mass term; this mass also induces a cosmological-constant-like piece that can be discarded in the collective-field formulation. The resulting 2+1D collective field Hamiltonian contains a BFSS-inspired nonlocal spatial interaction, a local mass term for transverse directions, and a nontrivial $oldsymbol{}^3$-type self-interaction, reducing to the known single-matrix collective-field theory in the appropriate limit. This framework points toward understanding how emergent 2D spatial geometry and, potentially, an emergent spacetime metric arise from matrix eigenvalue dynamics and offers avenues for studying vacuum structure, fluctuations, and entanglement in multi-matrix models.

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.