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 $(\mathcal{A},\mathcal{H},D,\gamma ,J)$, called random matrix geometries of type $(1,0)$, with a fixed fermion space $(\mathcal{A},\mathcal{H},\gamma ,J)$ 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 ${\prod }_{i=1}^s\mathrm{Tr}(D^{n_i})$ 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 $(\Sigma ,{\omega }_{0,1},{\omega }_{0,2})$ of the model is investigated in detail. In particular, we derive an explicit expression for the fundamental symmetric bidifferential ${\omega }_{0,2}$ in terms of the formal parameters of the model.