  This work concerns single-trace correlations of Euclidean multi-matrix
models. In the large-N limit we show that Schwinger-Dyson equations imply loop
equations and non-anomalous Ward identities. Loop equations are associated to
generic infinitesimal changes of matrix variables (vector fields). Ward
identities correspond to vector fields preserving measure and action. The
former are analogous to Makeenko-Migdal equations and the latter to
Slavnov-Taylor identities. Loop equations correspond to leading large-N
Schwinger-Dyson equations. Ward identities correspond to 1/N^2 suppressed
Schwinger-Dyson equations. But they become leading equations since loop
equations for non-anomalous vector fields are vacuous. We show that symmetries
at infinite N persist at finite N, preventing mixing with multi-trace
correlations. For one matrix, there are no non-anomalous infinitesimal
symmetries. For two 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. Ward identities are homogeneous linear equations. We use them with
the loop equations to determine some correlations of these models. Ward
identities alleviate the underdeterminacy of loop equations. Non-anomalous
symmetries give a naturalness-type explanation for why several linear
combinations of correlations in these models vanish.
