Perturbative and Geometric Analysis of the Quartic Kontsevich Model

TL;DR

This paper analyzes the quartic Kontsevich model, revealing simplified structured expressions for correlation functions, exploring critical phenomena, and connecting to blobbed topological recursion, thus advancing understanding of quantum field theory toy models.

Contribution

It introduces explicit solutions for the quartic Kontsevich model's correlation functions and links them to topological recursion and critical phenomena.

Findings

Correlation functions sum to structured, simpler expressions

Connections established with blobbed topological recursion

Exact solutions enable exploration of critical phenomena

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.