Abstract
We define a new topological polynomial extending the Bollobás–Riordan one, which obeys a four-term reduction relation of the deletion/contraction type and has a natural behaviour under partial duality. This allows to write down a completely explicit combinatorial evaluation of the polynomials, occurring in the parametric representation of the non-commutative Grosse–Wulkenhaar quantum field theory. An explicit solution of the parametric representation for commutative field theories based on the Mehler kernel is also provided.
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Abdesselam A.: The Grassmann-Berezin calculus and theorems of the matrix-tree type. Adv. Appl. Math. 33, 51–70 (2004) arXiv:math.CO/0306396
Aluffi, P., Marcolli, M.: Feynman motives of banana graphs. arXiv:0807.1690. July 2008
Bollobás, B.: Modern graph theory. In: Graduate Texts in Mathematics, vol. 184. Springer, New York (1998)
Bollobás B., Riordan O.: A polynomial of graphs on surfaces. Math. Ann. 323, 81–96 (2002)
Brown, F.: On the periods of some Feynman integrals. arXiv:0910.0114. October 2009
Brown, F., Yeats, K.: Spanning forest polynomials and the transcendental weight of Feynman graphs. arXiv:0910.5429. October 2009
Chmutov S.: Generalized duality for graphs on surfaces and the signed Bollobás-Riordan polynomial. J. Comb. Theory Ser. B 99(3), 617–638 (2009). doi:10.1016/j.jctb.2008.09.007 arXiv:0711.3490
Grosse H., Wulkenhaar R.: Power-counting theorem for non-local matrix models and renormalisation. Commun. Math. Phys. 254(1), 91–127 (2005) arXiv:hep-th/0305066
Grosse H., Wulkenhaar R.: Renormalisation of \({\phi^{4}}\) -theory on noncommutative \({{\mathbb R}^4}\) in the matrix base. Commun. Math. Phys. 256(2), 305–374 (2005) arXiv:hep-th/0401128
Gurău R., Rivasseau V.: Parametric representation of non-commutative field theory. Commun. Math. Phys. 272, 811 (2007) arXiv:math-ph/0606030
Gurău, R.: Topological graph polynomials in colored group field theory. arXiv: 0911.1945. November 2009
Huggett, S., Moffatt, I.: Expansions for the Bollobás-Riordan polynomial of separable ribbon graphs. Ann. Comb. (2011, in press). arXiv:0710.4266
Itzykson C., Zuber J.-B.: Quantum Field Theory. McGraw-Hill, New York (1980)
Kenyon, R.W.: Lectures on dimers. Lecture notes for lectures at the Park City Math Institute, summer 2007. arXiv:0910:3129. October 2009
Krajewski T., Rivasseau V., Tanasă A., Wang Z.: Topological graph polynomials and quantum field theory. Part I: heat kernel theories. J. Noncommut. Geom. 4(1), 29–82 (2010) arXiv:0811.0186
Moffatt I.: Knot invariants and the Bollobás-Riordan of embedded graphs. Eur. J. Comb. 29, 95–107 (2008)
Moffatt I.: Partial duality and Bollobás and Riordan’s ribbon graph polynomial. Discrete Math. 310, 174–183 (2010) arXiv:0809.3014
Nakanishi N.: Graph Theory and Feynman Integrals. Gordon and Breach, New York (1971)
Rivasseau V., Tanasa A.: Parametric representation of “critical” noncommutative QFT models. Commun. Math. Phys. 279, 355 (2008) arXiv:hep-th/0701034
Rivasseau V., Vignes-Tourneret F., Wulkenhaar R.: Renormalization of noncommutative \({\phi^4}\) -theory by multi-scale analysis. Commun. Math. Phys. 262, 565–594 (2006) arXiv:hep-th/0501036
Sokal, A.: The multivariate Tutte polynomial (alias Potts model) for graphs and matroids. In: Survey in Combinatorics. London Mathematical Society Lecture Notes, vol. 327 (2005). arXiv:math/0503607
Tutte, W.T.: Graph theory. In: Encyclopedia of Mathematics and its Applications, vol. 21. Addison-Wesley Publishing Company (1984)
Vignes-Tourneret F.: The multivariate signed Bollobás-Riordan polynomial. Discrete Math. 309, 5968–5981 (2009). doi:10.1016/j.disc.2009.04.026 arXiv: 0811.1584
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Communicated by Raimar Wulkenhaar.
T. Krajewski on leave from Centre de Physique Théorique, CNRS UMR 6207, CNRS Luminy, Case 907, 13288 Marseille Cedex 9, France.
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Krajewski, T., Rivasseau, V. & Vignes-Tourneret, F. Topological Graph Polynomial and Quantum Field Theory Part II: Mehler Kernel Theories. Ann. Henri Poincaré 12, 483–545 (2011). https://doi.org/10.1007/s00023-011-0087-2
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DOI: https://doi.org/10.1007/s00023-011-0087-2