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

Krishnaswami, G. S. (2006) 'Multi-matrix loop equations : algebraic & differential structures and an approximation based on deformation quantization.', Journal of high energy physics., 08 . 035.

Abstract

Large-N multi-matrix loop equations are formulated as quadratic difference equations in concatenation of gluon correlations. Though non-linear, they involve highest rank correlations linearly. They are underdetermined in many cases. Additional linear equations for gluon correlations, associated to symmetries of action and measure are found. Loop equations aren't differential equations as they involve left annihilation, which doesn't satisfy the Leibnitz rule with concatenation. But left annihilation is a derivation of the commutative shuffle product. Moreover shuffle and concatenation combine to define a bialgebra. Motivated by deformation quantization, we expand concatenation around shuffle in powers of q, whose physical value is 1. At zeroth order the loop equations become quadratic PDEs in the shuffle algebra. If the variation of the action is linear in iterated commutators of left annihilations, these quadratic PDEs linearize by passage to shuffle reciprocal of correlations. Remarkably, this is true for regularized versions of the Yang-Mills, Chern-Simons and Gaussian actions. But the linear equations are underdetermined just as the loop equations were. For any particular solution, the shuffle reciprocal is explicitly inverted to get the zeroth order gluon correlations. To go beyond zeroth order, we find a Poisson bracket on the shuffle algebra and associative q-products interpolating between shuffle and concatenation. This method, and a complementary one of deforming annihilation rather than product are shown to give over and underestimates for correlations of a gaussian matrix model.

Item Type:Article
Keywords:Field theories in lower dimensions, 1/N expansion, Quantum groups, Matrix models.
Full text:Full text not available from this repository.
Publisher Web site:http://dx.doi.org/10.1088/1126-6708/2006/08/035
Record Created:16 Jan 2008
Last Modified:03 Dec 2009 16:00

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