Perturbative and Geometric Analysis of the Quartic Kontsevich Model

Abstract

The analogue of Kontsevich's matrix Airy function, with the cubic potential $\mathrm{Tr}\left({\Phi }^3\right)$ replaced by a quartic term $\mathrm{Tr}\left({\Phi }^4\right)$ 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.

arXiv comment33 pages, 13 figures. v2: distinction between our model and GKM, exact formula for genus zero closed graphs, example higher-order ramification: correct limit. Small corrections. v3: some material added to sec 5.1+5.3. v4: published version

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Cited by in corpus (10)

• Blobbed topological recursion of the quartic Kontsevich model I: Loop equations and conjectures
• From scalar fields on quantum spaces to blobbed topological recursion
• From Noncommutative Geometry to Random Matrix Theory
• An algebraic approach to a quartic analogue of the Kontsevich model
• A Laplacian to compute intersection numbers on ${\bar{\mathcal{M}}}_{g,n}$ and correlation functions in NCQFT
• Nested Catalan tables and a recurrence relation in noncommutative quantum field theory
• Complete solution of the LSZ Model via Topological Recursion
• Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme
• Blobbed topological recursion from extended loop equations
• Genus one free energy contribution to the quartic Kontsevich model