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

Abstract

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 i G = Gξ i G 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.

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

“…The conditions χχ −1 = χ −1 χ = ǫ are precisely the same as the compatibility conditions (7) of the antipode S with product sh and coproduct ∆. Indeed, using the homomorphism property of χ and the second equality above,…”

“…We have not found any nice proof of this, though we verified it explicitly for n ≤ 3 and observed a pattern of cancelations for higher n which leads us to conjecture that it is an identity. Cartier [19] mentions that sh-deconc must form a Hopf algebra on general grounds, though we would still like an explicit proof of (7). The minus signs in the definition of the antipode are crucial for this compatibility condition to hold.…”

“…The fSDE obtained here must be supplemented by gauge-fixing and ghost contributions for a proper treatment of gauge invariance, before we can look for physical solutions. In [7] we have indicated how to incorporate gauge fixing and ghost terms in the context of matrix models. Mathematically, this means the shuffle-deconcatenation Hopf algebra generated by ξ µ (x) needs to be modified.…”

“…al. [14,8], and builds on our previous papers [9,7]. There are of course other approaches to multi-matrix models such as those related to integrable models and algebraic geometry, see for instance [15,16].…”

“…Finally, the SD operators of the Yang-Mills action are obtained and shown to be right-invariant derivations of the shuffle-deconcatenation Hopf algebra on a continuously infinite number of generators labeled by space-time position and polarization. However, in the case of Yang-Mills theory, we still need to pass to a quotient of this Hopf algebra to account for gauge invariance and recover the group of loops [5,6] (or need to gauge-fix and introduce additional generators for ghosts [7]), before we can look for physical solutions to the fSDE.…”