Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 085, 33 pages      arXiv:2012.02622      https://doi.org/10.3842/SIGMA.2021.085
Contribution to the Special Issue on Algebraic Structures in Perturbative Quantum Field Theory in honor of Dirk Kreimer for his 60th birthday

Perturbative and Geometric Analysis of the Quartic Kontsevich Model

Johannes Branahl a, Alexander Hock b and Raimar Wulkenhaar a
a) Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, 48149 Münster, Germany
b) Mathematical Institute, University of Oxford, Andrew Wiles Building, Woodstock Road, OX2 6GG, Oxford, UK

Received February 26, 2021, in final form September 10, 2021; Published online September 16, 2021

Abstract
The analogue of Kontsevich's matrix Airy function, with the cubic potential $\operatorname{Tr}\big(\Phi^3\big)$ replaced by a quartic term $\operatorname{Tr}\big(\Phi^4\big)$ with the same covariance, provides a toy model for quantum field theory in which all correlation functions can be computed exactly and explicitly. In this paper we show that distinguished polynomials of correlation functions, themselves given by quickly growing series of Feynman ribbon graphs, sum up to much simpler and highly structured expressions. These expressions are deeply connected with meromorphic forms conjectured to obey blobbed topological recursion. Moreover, we show how the exact solutions permit to explore critical phenomena in the quartic Kontsevich model.

Key words: Dyson-Schwinger equations; perturbation theory; exact solutions; topological recursion.

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