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.

arXiv comment53 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 (22)

• Renormalisation of ϕ^4-theory on noncommutative R^4 in the matrix base
• Remodeling the B-model
• Vanishing of Beta Function of Non Commutative ${\Phi }_4^4$ Theory to all orders
• Free energy topological expansion for the 2-matrix model
• Exact Solution of Quantum Field Theory on Noncommutative Phase Spaces
• Power-counting theorem for non-local matrix models and renormalisation
• A matrix model for simple Hurwitz numbers, and topological recursion
• Combinatorial Solution of the Two-Matrix Model
• Self-dual noncommutative ϕ^4-theory in four dimensions is a non-perturbatively solvable and non-trivial quantum field theory
• Progress in solving a noncommutative quantum field theory in four dimensions
• Renormalization of the noncommutative phi^3 model through the Kontsevich model
• Blobbed topological recursion: properties and applications
• Topological expansion of mixed correlations in the hermitian 2 Matrix Model and x-y symmetry of the F_g invariants
• Enumeration of maps with self avoiding loops and the O(n) model on random lattices of all topologies
• Solution of all quartic matrix models
• Lambert-W solves the noncommutative ${\Phi }^4$-model
• Perturbative and Geometric Analysis of the Quartic Kontsevich Model
• An algebraic approach to a quartic analogue of the Kontsevich model
• Topological expansion and boundary conditions
• Nested Catalan tables and a recurrence relation in noncommutative quantum field theory
• Matrix Field Theory
• Blobbed topological recursion for the quartic melonic tensor model

Cited by in corpus (18)

• Topological recursion for Kadomtsev-Petviashvili tau functions of hypergeometric type
• Solution of all quartic matrix models
• From scalar fields on quantum spaces to blobbed topological recursion
• Spectral Statistics of Dirac Ensembles
• On the $x$-$y$ Symmetry of Correlators in Topological Recursion via Loop Insertion Operator
• Perturbative and Geometric Analysis of the Quartic Kontsevich Model
• 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
• Approximate treatment of noncommutative curvature in quartic matrix model
• $x-y$ duality in Topological Recursion for exponential variables via Quantum Dilogarithm
• Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme
• Blobbed topological recursion from extended loop equations
• A Note on BKP for the Kontsevich Matrix Model with Arbitrary Potential
• Blobbed topological recursion and KP integrability
• Exact Solutions v.s. Perturbative Calculations of Finite ${\Phi }^3$-${\Phi }^4$ Hybrid-Matrix-Model
• Genus one free energy contribution to the quartic Kontsevich model