Solution of all quartic matrix models

Grosse, Harald; Hock, Alexander; Wulkenhaar, Raimar

Abstract

We consider the quartic analogue of the Kontsevich model, which is defined by a measure exp(−N Tr(EΦ^2+(λ/4)Φ^4))dΦ on Hermitian N×N-matrices, where E is any positive matrix and λ 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 ≤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 ω_{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 λΦ^4_4-model on noncommutative Moyal space does not have a triviality problem. arXiv:1906.04600 [math]