Non-anomalous `Ward' identities to supplement large-N multi-matrix loop equations for correlations

Abstract

This work concerns single-trace correlations of Euclidean multi-matrix models. In the large-N limit we show that Schwinger-Dyson equations (SDE) imply loop equations (LE) and non-anomalous Ward identities (WI). LE are associated to generic infinitesimal changes of matrix variables (vector fields). WI correspond to vector fields preserving measure and action. The former are analogous to Makeenko-Migdal equations and the latter to Slavnov-Taylor identities. LE correspond to leading large-N SDE. WI correspond to 1/N2 suppressed SDE. But they become leading equations since LE for non-anomalous vector fields are vacuous. We show that symmetries at N = ∞ persist at finite N, preventing mixing with multi-trace correlations. For 1 matrix, there are no non-anomalous infinitesimal symmetries. For 2 or more matrices, measure preserving vector fields form an infinite dimensional graded Lie algebra, and non-anomalous action preserving ones a subalgebra. For Gaussian, Chern-Simons and Yang-Mills models we identify up to cubic non-anomalous vector fields, though they can be arbitrarily non-linear. WI are homogeneous linear equations. We use them with the LE to determine some correlations of these models. WI alleviate the underdeterminacy of LE. Non-anomalous symmetries give a naturalness-type explanation for why several linear combinations of correlations in these models vanish.