Spectral action and heat kernel trace for Ricci flat manifolds from stochastic flow over second quantized $L^2$-differential forms
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913182108352512 |
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| author | Gakkhar, Sita Marcolli, Matilde |
| author_facet | Gakkhar, Sita Marcolli, Matilde |
| contents | A quantum stochastic differential equation (qsde) on Fock space over $L^2$ differential 1-forms is given from the small "time" flow of which the trace of the connection Laplacian heat kernel for the spinor endomorphism bundle can be computed over any compact Ricci-flat Riemannian manifold. The existence of the stochastic flow is established by adapting the construction from [14]. When the manifold supports a parallel spinor - Ricci-flatness is a required integrability condition for parallel spinors, the trace of Dirac Laplacian heat kernel of the spinor bundle can be recovered. For 4-manifolds, this corresponds to the spectral action, and realizes Einstein-Hilbert action as a stochastic flow. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00643 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral action and heat kernel trace for Ricci flat manifolds from stochastic flow over second quantized $L^2$-differential forms Gakkhar, Sita Marcolli, Matilde Mathematical Physics Differential Geometry Probability A quantum stochastic differential equation (qsde) on Fock space over $L^2$ differential 1-forms is given from the small "time" flow of which the trace of the connection Laplacian heat kernel for the spinor endomorphism bundle can be computed over any compact Ricci-flat Riemannian manifold. The existence of the stochastic flow is established by adapting the construction from [14]. When the manifold supports a parallel spinor - Ricci-flatness is a required integrability condition for parallel spinors, the trace of Dirac Laplacian heat kernel of the spinor bundle can be recovered. For 4-manifolds, this corresponds to the spectral action, and realizes Einstein-Hilbert action as a stochastic flow. |
| title | Spectral action and heat kernel trace for Ricci flat manifolds from stochastic flow over second quantized $L^2$-differential forms |
| topic | Mathematical Physics Differential Geometry Probability |
| url | https://arxiv.org/abs/2401.00643 |