Topological Recursion and Random Finite Noncommutative Geometries

Abstract

In this thesis, we investigate a model for quantum gravity on finite noncommutative spaces using the topological recursion method originated from random matrix theory. More precisely, we consider a particular type of finite noncommutative geometries, in the sense of Connes, called spectral triples of type ${(1,0)} \,$, introduced by Barrett. A random spectral triple of type ${(1,0)}$ has a fixed fermion space, and the moduli space of its Dirac operator ${D=\{ H , \cdot \} \, ,}$ ${H \in {\mathcal{H}_N}}$, encoding all the possible geometries over the fermion space, is the space of Hermitian matrices ${\mathcal{H}_N}$. A distribution of the form ${e^{- \mathcal{S} (D)} \mathrm{d} D }$ is considered over the moduli space of Dirac operators. We specify the form of the action functional ${\mathcal{S} (D)}$ such that the topological recursion for a repulsive particles system, introduced by Borot, Eynard and Orantin, holds for the large $N$ topological expansion of the $n$-point correlators ${W_n (x_1 , \cdots , x_n)}$ of our model. In addition, we get the large $N$ topological expansion of the free energy ${F= \log Z_N}$ and the $n$-point correlators of the model in terms of the enumerative combinatorics of the stuffed maps, introduced by Borot, whose elementary 2-cells may have the topology of a disk or of a cylinder. One can compute all the stable coefficients ${W_n^g (x_1 , \cdots , x_n) \, ,}$ ${2g-2+n \u3e0 \, ,}$ ${n \geq 1 \, ,}$ ${g \geq 0}$ of the large $N$ topological expansion of the $n$-point correlators ${W_n (x_1 , \cdots , x_n)}$ of the model using the topological recursion formula, provided the leading order terms ${W_1^0 (x)}$ and ${W_2^0 (x_1 , x_2)}$ are known. We show that, for our model, the leading order term ${W_1^0 (x)}$ satisfies a quadratic algebraic equation ${y^2 + Q(x) \, y - P(x) = 0 \,}$. The spectral curve $\Sigma$ of the model is a genus zero complex algebraic curve, given by the pre-mentioned quadratic equation. We find explicit linear (resp. quadratic) expressions for the coefficients of the polynomial ${Q(x)}$ (resp. ${P(x)}$) in terms of the moments of the jump discontinuity of ${W_1^0 (x) \,}$. We plane to investigate the spectral curve ${(\Sigma , \omega_1^0 , \omega_2^0)}$ of the model in more detail