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arXiv 1906.09362math-phhep-thmath.MPmath.QA
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随机有限非交换几何与拓扑递归

Random Finite Noncommutative Geometries and Topological Recursion

Shahab Azarfar, Masoud Khalkhali

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

本文用斑块化拓扑递归研究类型 (1,0) 随机矩阵几何，推导多迹 1-Hermitian 矩阵模型的 Schwinger-Dyson 方程，并给出谱曲线基本双微分的显式表达。

摘要：

本文利用 blobbed topological recursion（斑块化拓扑递归）理论，研究有限非交换空间上的量子引力模型。该模型基于一类特定的随机有限实谱三元组 ${(\mathcal{A}, \mathcal{H}, D, \gamma, J)}$，称为类型 ${(1,0)}$ 的随机矩阵几何；其中费米子空间 ${(\mathcal{A}, \mathcal{H}, \gamma, J)}$ 固定，并在 Dirac 算子的模空间上赋予形如 ${e^{-\mathcal{S}(D)}\mathrm{d}D}$ 的分布。作用量泛函 ${\mathcal{S}(D)}$ 被视为若干项之和，每项具有 ${\prod_{i=1}^s \mathrm{Tr}(D^{n_i})}$ 的形式，其中任意 ${s\geqslant 1}$。我们通过微分几何方法，推导了相应多迹形式 1-Hermitian 矩阵模型的连通关联函数 ${W_n}$ 所满足的 Schwinger-Dyson 方程。结果表明，${W_n}$ 的大 $N$ 展开系数 ${W_{g,n}}$ 枚举了称为 stuffed maps（填充映射）的离散曲面，其构造块具有特定拓扑。本文还详细研究了模型的谱曲线 ${(\Sigma,\omega_{0,1},\omega_{0,2})}$；特别地，我们依据模型的形式参数推导了基本对称双微分 ${\omega_{0,2}}$ 的显式表达式。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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