Random Finite Noncommutative Geometries and Topological Recursion

Abstract

In this paper we investigate a model for quantum gravity on finite noncommutative spaces using the theory of blobbed topological recursion. The model is based on a particular class of random finite real spectral triples $(\mathcal{A},\mathcal{H},D,{\hat{I}}^3,J)$, called random matrix geometries of type $(1,0)$, with a fixed fermion space $(\mathcal{A},\mathcal{H},{\hat{I}}^3,J)$, and a distribution of the form $e^{-\mathcal{S}(D)}\mathrm{d}D$ over the moduli space of Dirac operators. The action functional $\mathcal{S}(D)$ is considered to be a sum of terms of the form ${\prod }_{i=1}^s\mathrm{Tr}\left(D^{n_i}\right)$ for arbitrary $s⩾1$. The Schwinger-Dyson equations satisfied by the connected correlators $W_n$ of the corresponding multi-trace formal 1-Hermitian matrix model are derived by a differential geometric approach. It is shown that the coefficients $W_{g,n}$ of the large $N$ expansion of $W_n$'s enumerate discrete surfaces, called stuffed maps, whose building blocks are of particular topologies. The spectral curve $\left(\hat{I}\pounds ,\ddot{I}{‰}_{0,1},\ddot{I}{‰}_{0,2}\right)$ of the model is investigated in detail. In particular, we derive an explicit expression for the fundamental symmetric bidifferential $\ddot{I}{‰}_{0,2}$ in terms of the formal parameters of the model.

44 pages, two references added to the new version, published (online first) Ann. Inst. Henri Poincaré Comb. Phys. Interact. (2024)