Dirac Operators, APS Boundary Conditions, and Spectral Flow on a Finite Warped Cylinder
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915887371517952 |
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| author | Kimura, Taro Sharma, Sanchita |
| author_facet | Kimura, Taro Sharma, Sanchita |
| contents | We study the Dirac operator on a finite warped cylinder coupled to a background $U(1)$ gauge field. We identify the intrinsic endpoint operators defining the Atiyah-Patodi-Singer (APS) boundary condition and derive a determinant characterization of the modewise APS spectrum. In the constant-gauge, invertible setting, the endpoint reduced $η$ contributions cancel, so the APS index vanishes. For smooth gauge families, the APS projector becomes discontinuous when a boundary mode crosses zero. We therefore introduce a regularized APS-type family of self-adjoint endpoint conditions that remains continuous across such crossings. This regularized family admits a real-symplectic boundary formulation within the standard spectral-flow/Maslov framework: for nondegenerate regularization, the zero-mode set coincides with the boundary-zero set, and transverse boundary zeros give isolated regular crossings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_23275 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dirac Operators, APS Boundary Conditions, and Spectral Flow on a Finite Warped Cylinder Kimura, Taro Sharma, Sanchita Mathematical Physics High Energy Physics - Theory Differential Geometry We study the Dirac operator on a finite warped cylinder coupled to a background $U(1)$ gauge field. We identify the intrinsic endpoint operators defining the Atiyah-Patodi-Singer (APS) boundary condition and derive a determinant characterization of the modewise APS spectrum. In the constant-gauge, invertible setting, the endpoint reduced $η$ contributions cancel, so the APS index vanishes. For smooth gauge families, the APS projector becomes discontinuous when a boundary mode crosses zero. We therefore introduce a regularized APS-type family of self-adjoint endpoint conditions that remains continuous across such crossings. This regularized family admits a real-symplectic boundary formulation within the standard spectral-flow/Maslov framework: for nondegenerate regularization, the zero-mode set coincides with the boundary-zero set, and transverse boundary zeros give isolated regular crossings. |
| title | Dirac Operators, APS Boundary Conditions, and Spectral Flow on a Finite Warped Cylinder |
| topic | Mathematical Physics High Energy Physics - Theory Differential Geometry |
| url | https://arxiv.org/abs/2603.23275 |