Schwinger–Dyson operator of Yang–Mills matrix models with ghosts and derivations of the graded shuffle algebra

We consider large-N multi-matrix models whose action closely mimics that of Yang–Mills theory, including gauge-fixing and ghost terms. We show that the factorized Schwinger–Dyson loop equations, expressed in terms of the generating series of gluon and ghost correlations G(ξ), are quadratic equations S G = GξG in concatenation of correlations. The Schwinger–Dyson operator S i is built from the left annihilation operator, which does not satisfy the Leibnitz rule with respect to concatenation. So the loop equations are not differential equations. We show that left annihilation is a derivation of the graded shuffle product of gluon and ghost correlations. The shuffle product is the point-wise product of Wilson loops, expressed in terms of correlations. So in the limit where concatenation is approximated by shuffle products, the loop equations become differential equations. Remarkably, the Schwinger–Dyson operator as a whole is also a derivation of the graded shuffle product. This allows us to turn the loop equations into linear equations for the shuffle reciprocal, which might serve as a starting point for an approximation scheme. PACS numbers: 11.15.−q, 11.15.Pg, 02.10.Hh Mathematics Subject Classification: 16W25, 16W50, 81T13