Multi-matrix loop equations: algebraic & differential structures and an approximation based on deformation quantization

Large-N multi-matrix loop equations are formulated as quadratic differenceequations in concatenation of gluon correlations. Though non-linear, theyinvolve highest rank correlations linearly. They are underdetermined in manycases. Additional linear equations for gluon correlations, associated tosymmetries of action and measure are found. Loop equations aren't differentialequations as they involve left annihilation, which doesn't satisfy the Leibnitzrule with concatenation. But left annihilation is a derivation of thecommutative shuffle product. Moreover shuffle and concatenation combine todefine a bialgebra. Motivated by deformation quantization, we expandconcatenation around shuffle in powers of q, whose physical value is 1. Atzeroth order the loop equations become quadratic PDEs in the shuffle algebra.If the variation of the action is linear in iterated commutators of leftannihilations, these quadratic PDEs linearize by passage to shuffle reciprocalof correlations. Remarkably, this is true for regularized versions of theYang-Mills, Chern-Simons and Gaussian actions. But the linear equations areunderdetermined just as the loop equations were. For any particular solution,the shuffle reciprocal is explicitly inverted to get the zeroth order gluoncorrelations. To go beyond zeroth order, we find a Poisson bracket on theshuffle algebra and associative q-products interpolating between shuffle andconcatenation. This method, and a complementary one of deforming annihilationrather than product are shown to give over and underestimates for correlationsof a gaussian matrix model.Comment: 40 page