Schwinger-Dyson operators as invariant vector fields on a matrix-model analogue of the group of loops

TL;DR

This paper introduces a group-theoretic framework for large-N multi-matrix models, where Schwinger-Dyson operators are shown to be invariant vector fields on an algebraic structure analogous to the loop group in Yang-Mills theory.

Contribution

It identifies a Hopf algebra-based group G for matrix models, linking Schwinger-Dyson operators to invariant vector fields, and extends this to continuum Yang-Mills actions.

Findings

Schwinger-Dyson operators are right-invariant vector fields on G.

The group G is the spectrum of a shuffle-deconcatenation Hopf algebra.

Includes models like Gaussian, Chern-Simons, and Yang-Mills as special cases.

Abstract

For a class of large-N multi-matrix models, we identify a group G that plays the same role as the group of loops on space-time does for Yang-Mills theory. G is the spectrum of a commutative shuffle-deconcatenation Hopf algebra that we associate to correlations. G is the exponential of the free Lie algebra. The generating series of correlations is a function on G and satisfies quadratic equations in convolution. These factorized Schwinger-Dyson or loop equations involve a collection of Schwinger-Dyson operators, which are shown to be right-invariant vector fields on G, one for each linearly independent primitive of the Hopf algebra. A large class of formal matrix models satisfying these properties are identified, including as special cases, the zero momentum limits of the Gaussian, Chern-Simons and Yang-Mills field theories. Moreover, the Schwinger-Dyson operators of the continuum Yang-Mills action are shown to be right-invariant derivations of the shuffle-deconcatenation Hopf algebra generated by sources labeled by position and polarization.