Schwinger–Dyson operators as invariant vector fields on a matrix model analog of the group of loops

Abstract

For a class of large-N multimatrix 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 with 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.