Witten deformation for non-Morse functions and gluing formula for analytic torsions
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866908329913090048 |
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| author | Yan, Junrong |
| author_facet | Yan, Junrong |
| contents | This paper concentrates on analyzing Witten deformation for a family of non-Morse functions parameterized by $T\in \mathbb{R}_+$, resulting in a novel, purely analytic proof of the gluing formula for analytic torsions in complete generality due to Brünning-Ma. Intriguingly, the gluing formula in this article could be reformulated as the Bismut-Zhang theorem for non-Morse functions, and from the perspective of Vishik's theory of moving boundary problems, the deformation parameter $T$ parameterize a family of boundary conditions. Our proof also makes use of a connection between small eigenvalues of Witten Laplacians and Mayer-Vietoris sequences. Finally, these new techniques could be extended to analytic torsion forms and play key roles in the study of the higher Cheeger-Müller/Bismut-Zhang theorem for nontrivial flat bundles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_01990 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Witten deformation for non-Morse functions and gluing formula for analytic torsions Yan, Junrong Differential Geometry Analysis of PDEs Spectral Theory This paper concentrates on analyzing Witten deformation for a family of non-Morse functions parameterized by $T\in \mathbb{R}_+$, resulting in a novel, purely analytic proof of the gluing formula for analytic torsions in complete generality due to Brünning-Ma. Intriguingly, the gluing formula in this article could be reformulated as the Bismut-Zhang theorem for non-Morse functions, and from the perspective of Vishik's theory of moving boundary problems, the deformation parameter $T$ parameterize a family of boundary conditions. Our proof also makes use of a connection between small eigenvalues of Witten Laplacians and Mayer-Vietoris sequences. Finally, these new techniques could be extended to analytic torsion forms and play key roles in the study of the higher Cheeger-Müller/Bismut-Zhang theorem for nontrivial flat bundles. |
| title | Witten deformation for non-Morse functions and gluing formula for analytic torsions |
| topic | Differential Geometry Analysis of PDEs Spectral Theory |
| url | https://arxiv.org/abs/2301.01990 |