Journal of Noncommutative Geometry


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Volume 4, Issue 1, 2010, pp. 29–82
DOI: 10.4171/JNCG/49

Published online: 2010-01-01

Topological graph polynomials and quantum field theory
Part I: heat kernel theories

Thomas Krajewski[1], Vincent Rivasseau[2], Adrian Tanasă[3] and Zhituo Wang[4]

(1) Université Paris XI
(2) Université Paris XI
(3) Ecole Polytechnique, Palaiseau
(4) Université Paris XI

We investigate the relationship between the universal topological polynomials for graphs in mathematics and the parametric representation of Feynman amplitudes in quantum field theory. In this first article we consider translation invariant theories with the usual heat-kernel-based propagator. We show how the Symanzik polynomials of quantum field theory are particular multivariate versions of the Tutte polynomial, and how the new polynomials of noncommutative quantum field theory are special versions of the Bollobás–Riordan polynomials.

Keywords: Parametric representation in (non)commutative field theory, Tutte polynomial, Bollobás–Riordan polynomial

Krajewski Thomas, Rivasseau Vincent, Tanasă Adrian, Wang Zhituo: Topological graph polynomials and quantum field theory
Part I: heat kernel theories. J. Noncommut. Geom. 4 (2010), 29-82. doi: 10.4171/JNCG/49