Classification of higher grade $\ell$ graphs for $\mathrm{U}(N{)}^2\times \mathrm{O}(D)$ multi-matrix models

Abstract

The authors studied in [Ann. Inst. Henri Poincaré D 9 (2022), 367–433], 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 $\ell ⩾3$ 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.