Blobbed Topological Recursion of the Quartic Kontsevich Model I: Loop Equations and Conjectures

Abstract

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.

Complete solution of the LSZ Model via Topological Recursion

“…By exact solvability we mean the expressibility, at least in principle, of the model's correlation functions by "known" functions. With new methods, it was recently possible to replace the principle expressibility for the hermitian model with concrete expressions that satisfy a remarkable algebraic structure whose investigation is still an ongoing process [GHW19,BHW22,HW21]. The key to this is the universal structure of topological recursion (TR) of Chekhov, Eynard and Orantin [CEO06,EO07].…”

“…Here, we integrate over the space H N of hermitian N ×N matrices where E ∈ H N has positive eigenvalues (E 1 , ..., E N ). The exact and concrete solution of the 2point function of (1.2), obtained after complexification and a crucial variable transform [SW19, GHW19, PW20], gave the possibility to read off the spectral curve for topological recursion [BHW22]. For the complex LSZ model, now with two external fields, we will follow a very similar strategy.…”

Exact solution of the $Φ_{2}^{3}$ finite matrix model

“…In this paper, we found the exact solutions of the Φ 3 2 finite matrix model (Grosse-Wulkenhaar model). In the Φ 3 2 finite matrix model, multipoint correlation functions were expressed as G In Section 3, the integration of the off-diagonal elements of the Hermitian matrix was calculated using the Harish-Chandra-Itzykson-Zuber integral [23], [28], [20], [31] in calculating the partition function Z[J]. Next the integral the diagonal elements of the Hermitian matrix was calculated using the Airy functions as similar to [24].…”

“…Afterward, the planar 2-point function of the Grosse-Wulkenhaar type Φ 4 model in large N, V limit was solved exactly by Grosse, Wulkenhaar, and Hock [18], n-point functions were solved by Wulkenhaar, and Hock [22]. Wulkenhaar, Branahl, and Hock found blobbed topological recursion of the Φ 4 model [3], [2].…”

A spectral curve for the generation of bipartite maps in topological recursion

“…A final question is dedicated only to quadrangulations. In [BHW20] the quartic Kontsevich model (QKM) was shown to be solvable in terms of correlators ω g,n that follow an extension of TR. In this so-called blobbed topological recursion (BTR; general framework developed in [BS17]), the ω g,n split into parts with poles at the ramification points (polar part) and with poles somewhere else (holomorphic part).…”