Random Finite Noncommutative Geometries and Topological Recursion
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arXiv
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| Natura: | Preprint |
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2019
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| _version_ | 1866913346684452864 |
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| author | Azarfar, Shahab Khalkhali, Masoud |
| author_facet | Azarfar, Shahab Khalkhali, Masoud |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1906_09362 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Random Finite Noncommutative Geometries and Topological Recursion Azarfar, Shahab Khalkhali, Masoud Mathematical Physics High Energy Physics - Theory Quantum Algebra 05A16 (primary), 81S05, 53Z05 (secondary) 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. |
| title | Random Finite Noncommutative Geometries and Topological Recursion |
| topic | Mathematical Physics High Energy Physics - Theory Quantum Algebra 05A16 (primary), 81S05, 53Z05 (secondary) |
| url | https://arxiv.org/abs/1906.09362 |