Solution of all quartic matrix models

Abstract

We consider the quartic analogue of the Kontsevich model, which is defined by a measure $\exp (-N\mathrm{Tr}(E{\Phi }^2+(\lambda /4){\Phi }^4))d\Phi$ on Hermitian $N\times N$-matrices, where $E$ is any positive matrix and $\lambda$ a scalar. It was previously established that the large-$N$ limit of the second moment (the planar two-point function) satisfies a non-linear integral equation. By employing tools from complex analysis, in particular the Lagrange-Bürmann inversion formula, we identify the exact solution of this non-linear problem, both for finite $N$ and for a large-$N$ limit to unbounded operators $E$ of spectral dimension $\le 4$. For finite $N$, the two-point function is a rational function evaluated at the preimages of another rational function $R$ constructed from the spectrum of $E$. Subsequent work has constructed from this formula a family ${\omega }_{g,n}$ of meromorphic differentials which obey blobbed topological recursion. For unbounded operators $E$, the renormalised two-point function is given by an integral formula involving a regularisation of $R$. This allowed a proof, in subsequent work, that the $\lambda {\Phi }_4^4$-model on noncommutative Moyal space does not have a triviality problem.

31 pages, LaTeX. v2: A problem in v1 specific to dimension 4 is now solved. v3: representation of 2-point function as rational function is added. v4: completely rewritten

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Cited by in corpus (14)

• Blobbed topological recursion of the quartic Kontsevich model I: Loop equations and conjectures
• Solution of the self-dual ${\Phi }^4$ QFT-model on four-dimensional Moyal space
• From scalar fields on quantum spaces to blobbed topological recursion
• Second moment fuzzy-field-theory-like matrix models
• Spectral Statistics of Dirac Ensembles
• Perturbative and Geometric Analysis of the Quartic Kontsevich Model
• An algebraic approach to a quartic analogue of the Kontsevich model
• Nested Catalan tables and a recurrence relation in noncommutative quantum field theory
• Large-$d$ behavior of the Feynman amplitudes for a just-renormalizable tensorial group field theory
• Matrix Field Theory
• Computing the spectral action for fuzzy geometries: from random noncommutative geometry to bi-tracial multimatrix models
• Complex (super)-matrix models with external sources and $q$-ensembles of Chern-Simons and ABJ(M) type
• A Note on BKP for the Kontsevich Matrix Model with Arbitrary Potential
• Genus one free energy contribution to the quartic Kontsevich model