Solution of all quartic matrix models

Harald Grosse, Alexander Hock, Raimar Wulkenhaar
June 11, 2019
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 Hermitean $N \times N$-matrices, where $E$ is any positive matrix and $\lambda$ a scalar. We prove that the two-point function admits an explicit solution formula in terms of the roots of a meromorphic function $J$ constructed from the spectrum of $E$. Structures which appear in this solution can be assembled into complex curves. We also solve the large-$N$ limit to unbounded operators $E$. The renormalised two-point function is given by an integral formula involving a regularisation of $J$. We prove triviality of the renormalised four-dimensional quartic matrix model for all positive $\lambda$.
open access link
@article{Grosse:2019jnv, author = "Grosse, Harald and Hock, Alexander and Wulkenhaar, Raimar", title = "{Solution of all quartic matrix models}", year = "2019", eprint = "1906.04600", archivePrefix = "arXiv", primaryClass = "math-ph", SLACcitation = "%%CITATION = ARXIV:1906.04600;%%" }