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 ${(\mathcal{A}, \mathcal{H}, D , γ, J) \,}$, called random matrix geometries of type ${(1,0) \,}$, with a fixed fermion space ${(\mathcal{A}, \mathcal{H}, γ, J) \,}$, and a distribution of the form ${e^{- \mathcal{S} (D)} {\mathop{}\!\mathrm{d}} D}$ 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 ${s \geqslant 1 \,}$. 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 ${\left( {Σ, ω_{0,1} , ω_{0,2}} \right)}$ 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.

Figures (1)

• Figure 1: Illustration of the Joukowski map in the case where the boundary of $U$ is an ellipse with the foci ${x=\mathfrak{a}}$ and ${x=\mathfrak{b} \,}$. The open neighborhoods $U$, $V^e$, and $V^i$ are colored green, brown and yellow, respectively.

Theorems & Definitions (20)

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