Loop soup representation of zeta-regularised determinants and equivariant Symanzik identities
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| Format: | Preprint |
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2024
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| _version_ | 1866913589223227392 |
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| author | Perruchaud, Pierre Sauzedde, Isao |
| author_facet | Perruchaud, Pierre Sauzedde, Isao |
| contents | We derive a stochastic representation for determinants of Laplace-type operators on vectors bundles over manifolds. Namely, inverse powers of those determinants are written as the expectation of a product of holonomies defined over Brownian loop soups. Our results hold over compact manifolds of dimension 2 or 3, in the presence of a mass or a boundary. We derive a few consequences, including some continuity of these determinants as a function of the operator, and the conformal invariance of the determinant on surfaces.
This expression allows us to construct the scalar field minimally coupled to a prescribed random smooth gauge field, which we prove obeys the so-called Symanzik identities. Some of these results are continuous analogues of the work of A. Kassel and T. Lévy in the discrete. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_00767 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Loop soup representation of zeta-regularised determinants and equivariant Symanzik identities Perruchaud, Pierre Sauzedde, Isao Probability Mathematical Physics Differential Geometry 58J65, 58J52, 81T13, 60J57 We derive a stochastic representation for determinants of Laplace-type operators on vectors bundles over manifolds. Namely, inverse powers of those determinants are written as the expectation of a product of holonomies defined over Brownian loop soups. Our results hold over compact manifolds of dimension 2 or 3, in the presence of a mass or a boundary. We derive a few consequences, including some continuity of these determinants as a function of the operator, and the conformal invariance of the determinant on surfaces. This expression allows us to construct the scalar field minimally coupled to a prescribed random smooth gauge field, which we prove obeys the so-called Symanzik identities. Some of these results are continuous analogues of the work of A. Kassel and T. Lévy in the discrete. |
| title | Loop soup representation of zeta-regularised determinants and equivariant Symanzik identities |
| topic | Probability Mathematical Physics Differential Geometry 58J65, 58J52, 81T13, 60J57 |
| url | https://arxiv.org/abs/2402.00767 |