Abstract
Solving the exact renormalisation group equation à la Wilson-Polchinski perturbatively, we derive a power-counting theorem for general matrix models with arbitrarily non-local propagators. The power-counting degree is determined by two scaling dimensions of the cut-off propagator and various topological data of ribbon graphs. As a necessary condition for the renormalisability of a model, the two scaling dimensions have to be large enough relative to the dimension of the underlying space. In order to have a renormalisable model one needs additional locality properties—typically arising from orthogonal polynomials—which relate the relevant and marginal interaction coefficients to a finite number of base couplings. The main application of our power-counting theorem is the renormalisation of field theories on noncommutative ℝD in matrix formulation.
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Communicated by M.R. Douglas
Acknowledgement We are grateful to José Gracia-Bondía and Edwin Langmann for discussions concerning the integral representation of the ⋆-product and its matrix base. We would like to thank Thomas Krajewski for advertising the Polchinski equation to us and Volkmar Putz for the accompanying study of Polchinski’s original proof. We are grateful to Christoph Kopper for indicating to us a way to reduce in our original power-counting estimation the polynomial in 
to a polynomial in 
thus permitting immediately the limit Λ0→∞. We would also like to thank Manfred Schweda and his group for enjoyable collaboration. We are indebted to the Erwin Schrödinger Institute in Vienna, the Max-Planck-Institute for Mathematics in the Sciences in Leipzig and the Institute for Theoretical Physics of the University of Vienna for the generous support of our collaboration.
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Grosse, H., Wulkenhaar, R. Power-Counting Theorem for Non-Local Matrix Models and Renormalisation. Commun. Math. Phys. 254, 91–127 (2005). https://doi.org/10.1007/s00220-004-1238-9
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DOI: https://doi.org/10.1007/s00220-004-1238-9