The analogue of Kontsevich's matrix Airy function, with the cubic potential Tr Φ 3 replaced by a quartic term Tr Φ 4 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.

We provide strong evidence for the conjecture that the analogue of Kontsevich's matrix Airy function, with the cubic potential Tr(Φ 3 ) replaced by a quartic term Tr(Φ 4 ), obeys the blobbed topological recursion of Borot and Shadrin. We identify in the quartic Kontsevich model three families of correlation functions for which we establish interwoven loop equations. One family consists of symmetric meromorphic differential forms ω g,n labelled by genus and number of marked points of a complex curve. We reduce the solution of all loop equations to a straightforward but lengthy evaluation of residues. In all evaluated cases, the ω g,n consist of a part with poles at ramification points which satisfies the universal formula of topological recursion, and of a part holomorphic at ramification points for which we provide an explicit residue formula.

We provide strong evidence for the conjecture that the analogue of Kontsevich’s matrix Airy function, with the cubic potential $$\mathrm {Tr}(\Phi ^3)$$ Tr ( Φ 3 ) replaced by a quartic term $$\mathrm {Tr}(\Phi ^4)$$ Tr ( Φ 4 ) , obeys the blobbed topological recursion of Borot and Shadrin. We identify in the quartic Kontsevich model three families of correlation functions for which we establish interwoven loop equations. One family consists of symmetric meromorphic differential forms $$\omega _{g,n}$$ ω g , n labelled by genus and number of marked points of a complex curve. We reduce the solution of all loop equations to a straightforward but lengthy evaluation of residues. In all evaluated cases, the $$\omega _{g,n}$$ ω g , n consist of a part with poles at ramification points which satisfies the universal formula of topological recursion, and of a part holomorphic at ramification points for which we provide an explicit residue formula.

Previously the exact solution of the planar sector of the self-dual Φ 4 -model on 4dimensional Moyal space was established up to the solution of a Fredholm integral equation. This paper solves, for any coupling constant λ > − 1 π , the Fredholm equation in terms of a hypergeometric function and thus completes the construction of the planar sector of the model. We prove that the interacting model has spectral dimension 4 − 2 arcsin(λπ) π for |λ| < 1 π . It is this dimension drop which for λ > 0 avoids the triviality problem of the matricial Φ 4 4 -model. We also establish the power series approximation of the Fredholm solution to all orders in λ. The appearing functions are hyperlogarithms defined by iterated integrals, here of alternating letters 0 and −1. We identify the renormalisation parameter which gives the same normalisation as the ribbon graph expansion.MSC 2010: 33C05, 45B05, 81Q80, 81Q30

We review the construction of the λφ 4 -model on noncommutative geometries via exact solutions of Dyson-Schwinger equations and explain how this construction relates via (blobbed) topological recursion to problems in algebraic and enumerative geometry.