Classification of higher grade $\ell$ graphs for $\mathrm{U}(N)^2\times \mathrm{O}(D)$ multi-matrix models
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arXiv
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| Format: | Preprint |
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2023
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| author | Avohou, Rémi Cocou Toriumi, Reiko Vancraeynest, Matthias |
| author_facet | Avohou, Rémi Cocou Toriumi, Reiko Vancraeynest, Matthias |
| contents | The authors studied in [Ann. Inst. Henri Poincaré D 9, 367-433, (2022)], a complex multi-matrix model with $\mathrm{U}(N)^2 \times \mathrm{O}(D)$ symmetry, and whose double scaling limit where simultaneously the large-$N$ and large-$D$ limits were taken while keeping the ratio $N/\sqrt{D}=M$ finite and fixed. In this double scaling limit, the complete recursive characterization of the Feynman graphs of arbitrary genus for the leading order grade $\ell=0$ was achieved. In this current study, we classify the higher order graphs in $\ell$. More specifically, $\ell=1$ and $\ell=2$ with arbitrary genus, in addition to a specific class of two-particle-irreducible (2PI) graphs for higher $\ell \geqslant 3$ but with genus zero. Furthermore, we demonstrate that each 2PI graph with a single $\mathrm{O}(D)$-loop with an arbitrary $\ell$ corresponds to a reduced alternating knot diagram with $\ell$ crossings as listed in the Rolfsen knot table, or a resulting alternating knot diagram obtained after performing the Tait flyping moves. We generalize to 2PR by considering the connected sum and the Reidemeister move I. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_13789 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Classification of higher grade $\ell$ graphs for $\mathrm{U}(N)^2\times \mathrm{O}(D)$ multi-matrix models Avohou, Rémi Cocou Toriumi, Reiko Vancraeynest, Matthias Mathematical Physics General Relativity and Quantum Cosmology High Energy Physics - Theory Combinatorics The authors studied in [Ann. Inst. Henri Poincaré D 9, 367-433, (2022)], a complex multi-matrix model with $\mathrm{U}(N)^2 \times \mathrm{O}(D)$ symmetry, and whose double scaling limit where simultaneously the large-$N$ and large-$D$ limits were taken while keeping the ratio $N/\sqrt{D}=M$ finite and fixed. In this double scaling limit, the complete recursive characterization of the Feynman graphs of arbitrary genus for the leading order grade $\ell=0$ was achieved. In this current study, we classify the higher order graphs in $\ell$. More specifically, $\ell=1$ and $\ell=2$ with arbitrary genus, in addition to a specific class of two-particle-irreducible (2PI) graphs for higher $\ell \geqslant 3$ but with genus zero. Furthermore, we demonstrate that each 2PI graph with a single $\mathrm{O}(D)$-loop with an arbitrary $\ell$ corresponds to a reduced alternating knot diagram with $\ell$ crossings as listed in the Rolfsen knot table, or a resulting alternating knot diagram obtained after performing the Tait flyping moves. We generalize to 2PR by considering the connected sum and the Reidemeister move I. |
| title | Classification of higher grade $\ell$ graphs for $\mathrm{U}(N)^2\times \mathrm{O}(D)$ multi-matrix models |
| topic | Mathematical Physics General Relativity and Quantum Cosmology High Energy Physics - Theory Combinatorics |
| url | https://arxiv.org/abs/2310.13789 |