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 $\mathrm{Tr}({\Phi }^3)$ replaced by a quartic term $\mathrm{Tr}({\Phi }^4)$, 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 ${\omega }_{g,n}$ 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 ${\omega }_{g,n}$ 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.

53 pages, 13 figures. v2: proofs in appendix E considerably simplified, typos corrected. v3: added reference arXiv:2103.13271, where we have proved the main conjecture for $g=0$. v4: rearrangements to improve readability, graphical interpretation of loop equations added, analogies to the 2-matrix model emphasised

References in corpus (7)

• Remodeling the B-model
• Vanishing of Beta Function of Non Commutative ${\Phi }_4^4$ Theory to all orders
• Progress in solving a noncommutative quantum field theory in four dimensions
• Topological expansion of mixed correlations in the hermitian 2 Matrix Model and x-y symmetry of the F_g invariants
• Perturbative and Geometric Analysis of the Quartic Kontsevich Model
• Topological expansion and boundary conditions
• Matrix Field Theory

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• Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme
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• Blobbed topological recursion and KP integrability
• Genus one free energy contribution to the quartic Kontsevich model
• Exact Solutions v.s. Perturbative Calculations of Finite ${\Phi }^3$-${\Phi }^4$ Hybrid-Matrix-Model