Topological graph polynomials and quantum field theory, Part II: Mehler kernel theories

TL;DR

This paper introduces a new topological polynomial extending existing ones, with explicit combinatorial evaluation methods, applicable to non-commutative and commutative quantum field theories involving Mehler kernels.

Contribution

It defines a novel topological polynomial with a four-term reduction relation and explicit evaluation techniques for quantum field theory applications.

Findings

New polynomial extends Bollobas-Riordan polynomial

Explicit combinatorial evaluation formulas derived

Applicable to non-commutative and commutative QFTs with Mehler kernels

Abstract

We define a new topological polynomial extending the Bollobas-Riordan one, which obeys a four-term reduction relation of the deletion/contraction type and has a natural behavior 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.