Global unitary fixing and matrix-valued correlations in matrix models

TL;DR

This paper investigates how matrix models with global unitary invariance behave under different unitary fixings, revealing that trace-based averages are invariant, while matrix-valued correlations depend on the fixing, with implications for Ward identities.

Contribution

It introduces a Faddeev-Popov framework for unitary fixing in matrix models, clarifying the invariance of trace averages and the dependence of matrix-valued correlations on the fixing.

Findings

Trace averages are independent of unitary fixing.

Matrix-valued correlations depend on the choice of fixing.

The formalism enables derivation of Ward identities for matrix correlations.

Abstract

We consider the partition function for a matrix model with a global unitary invariant energy function. We show that the averages over the partition function of global unitary invariant trace polynomials of the matrix variables are the same when calculated with any choice of a global unitary fixing, while averages of such polynomials without a trace define matrix-valued correlation functions, that depend on the choice of unitary fixing. The unitary fixing is formulated within the standard Faddeev-Popov framework, in which the squared Vandermonde determinant emerges as a factor of the complete Faddeev-Popov determinant. We give the ghost representation for the FP determinant, and the corresponding BRST invariance of the unitary-fixed partition function. The formalism is relevant for deriving Ward identities obeyed by matrix-valued correlation functions.