# Random finite noncommutative geometries and topological recursion

### Shahab Azarfar

University of Western Ontario, London, Canada### Masoud Khalkhali

University of Western Ontario, London, Canada

## 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 $(A,H,D,γ,J)$, called random matrix geometries of type $(1,0)$, with a fixed fermion space $(A,H,γ,J)$ and a distribution of the form $e_{−S(D)}dD$ over the moduli space of Dirac operators. The action functional $S(D)$ is considered to be a sum of terms of the form $∏_{i=1}Tr(D_{n_{i}})$ 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 $(Σ,ω_{0,1},ω_{0,2})$ of the model is investigated in detail. In particular, we derive an explicit expression for the fundamental symmetric bidifferential $ω_{0,2}$ in terms of the formal parameters of the model.

## Cite this article

Shahab Azarfar, Masoud Khalkhali, Random finite noncommutative geometries and topological recursion. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 11 (2024), no. 3, pp. 409–451

DOI 10.4171/AIHPD/188