Solution of all quartic matrix models
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arXiv
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| Format: | Preprint |
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2019
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| author | Grosse, Harald Hock, Alexander Wulkenhaar, Raimar |
| author_facet | Grosse, Harald Hock, Alexander Wulkenhaar, Raimar |
| contents | We consider the quartic analogue of the Kontsevich model, which is defined by a measure $\exp(-{N}\,\mathrm{Tr}(EΦ^2+(λ/4)Φ^4)) dΦ$ on Hermitian ${N}\times{N}$-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 $\leq 4$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1906_04600 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Solution of all quartic matrix models Grosse, Harald Hock, Alexander Wulkenhaar, Raimar Mathematical Physics High Energy Physics - Theory Algebraic Geometry 30E20, 14H81, 39B32, 81Q80 We consider the quartic analogue of the Kontsevich model, which is defined by a measure $\exp(-{N}\,\mathrm{Tr}(EΦ^2+(λ/4)Φ^4)) dΦ$ on Hermitian ${N}\times{N}$-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 $\leq 4$. 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. |
| title | Solution of all quartic matrix models |
| topic | Mathematical Physics High Energy Physics - Theory Algebraic Geometry 30E20, 14H81, 39B32, 81Q80 |
| url | https://arxiv.org/abs/1906.04600 |