An algebraic approach to a quartic analogue of the Kontsevich model

Abstract

We consider an analogue of Kontsevich's matrix Airy function where the cubic potential $\mathrm{Tr}({\Phi }^3)$ is replaced by a quartic term $\mathrm{Tr}({\Phi }^4)$. Cumulants of the resulting measure are known to decompose into cycle types for which a recursive system of equations can be established. We develop a new, purely algebraic geometrical solution strategy for the two initial equations of the recursion, based on properties of Cauchy matrices. These structures led in subsequent work to the discovery that the quartic analogue of the Kontsevich model obeys blobbed topological recursion.

28 pages. v3: We changed title and outlook in view of subsequent results obtained in arXiv:2008.12201 and arXiv:2103.13271 . v2: symmetry of 2-point functions made manifest, references added, outlook updated

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
• Blobbed topological recursion of the quartic Kontsevich model I: Loop equations and conjectures
• Solution of all quartic matrix models
• Lambert-W solves the noncommutative ${\Phi }^4$-model
• Perturbative and Geometric Analysis of the Quartic Kontsevich Model

Cited by in corpus (8)

• Blobbed topological recursion of the quartic Kontsevich model I: Loop equations and conjectures
• Solution of all quartic matrix models
• From scalar fields on quantum spaces to blobbed topological recursion
• Perturbative and Geometric Analysis of the Quartic Kontsevich Model
• From Noncommutative Geometry to Random Matrix Theory
• Blobbed topological recursion from extended loop equations
• A Note on BKP for the Kontsevich Matrix Model with Arbitrary Potential
• Genus one free energy contribution to the quartic Kontsevich model