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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.16197 |
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| _version_ | 1866909872342171648 |
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| author | Sanchez Jr, Jesus Valiente, Andres Franco |
| author_facet | Sanchez Jr, Jesus Valiente, Andres Franco |
| contents | Given a contact sub-Riemannian manifold one obtains a non-integrable splitting of the tangent bundle into the directions along the contact distribution and the Reeb field. We generalize the construction of the Bismut superconnection to this non-integrable setting and show that although singularities appear within the superconnection, if one extracts the finite part then the resulting operator is hypoelliptic. We find that the hypoellipticity also holds in the setting of two-step subRiemannian manifolds and produce a modification for arbitrary subRiemannian manifolds which always gives a hypoelliptic operator. A discussion of the explicit form of the operator on principal $\bS^1$-bundles is provided. The index theory is worked out on contact manifolds and a matrix twisting of the Clifford relations produces operators with non-trivial Fredholm index. We conclude with a possible relationship between our hypoelliptic operator and a constrained supersymmetric sigma model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_16197 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hypoellipticity of the Asymptotic Bismut Superconnection on Contact Manifolds Sanchez Jr, Jesus Valiente, Andres Franco Mathematical Physics Given a contact sub-Riemannian manifold one obtains a non-integrable splitting of the tangent bundle into the directions along the contact distribution and the Reeb field. We generalize the construction of the Bismut superconnection to this non-integrable setting and show that although singularities appear within the superconnection, if one extracts the finite part then the resulting operator is hypoelliptic. We find that the hypoellipticity also holds in the setting of two-step subRiemannian manifolds and produce a modification for arbitrary subRiemannian manifolds which always gives a hypoelliptic operator. A discussion of the explicit form of the operator on principal $\bS^1$-bundles is provided. The index theory is worked out on contact manifolds and a matrix twisting of the Clifford relations produces operators with non-trivial Fredholm index. We conclude with a possible relationship between our hypoelliptic operator and a constrained supersymmetric sigma model. |
| title | Hypoellipticity of the Asymptotic Bismut Superconnection on Contact Manifolds |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2505.16197 |