Index invariants and Eta invariants determine Differential KO theory in degrees that are multiples of 8
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917102291517440 |
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| author | Su, Tan |
| author_facet | Su, Tan |
| contents | Sullivan--Simons developed a Cheeger--Simons differential character analogue for degree (0 mod 2) differential K-theory, giving a complete set of numerical invariants that determine a complex vector bundle with unitary connection on a base manifold X, up to Chern--Simons equivalence of the connection. In this paper we develop a degree (0 mod 8) differential KO-analogue. Namely, given a real vector bundle with orthogonal connection, we construct R/Z -valued eta-invariants in the context of Atiyah--Patodi--Singer and Z2 Atiyah--Singer index invariants that completely determine differential KO-theory in degree (0 mod 8); we call this the differential KO-character. In the second part, for a Riemannian submersion X to B with closed 8k-dimensional spin fibers, we develop two family index theorems in differential KO -theory: one in the differential KO -character model and one in the structured-bundle model. The fact that these two pushforwards agree follows from the Bismut--Cheeger adiabatic limit theorem, providing a new interpretation of that result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_19904 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Index invariants and Eta invariants determine Differential KO theory in degrees that are multiples of 8 Su, Tan K-Theory and Homology Algebraic Topology Differential Geometry Sullivan--Simons developed a Cheeger--Simons differential character analogue for degree (0 mod 2) differential K-theory, giving a complete set of numerical invariants that determine a complex vector bundle with unitary connection on a base manifold X, up to Chern--Simons equivalence of the connection. In this paper we develop a degree (0 mod 8) differential KO-analogue. Namely, given a real vector bundle with orthogonal connection, we construct R/Z -valued eta-invariants in the context of Atiyah--Patodi--Singer and Z2 Atiyah--Singer index invariants that completely determine differential KO-theory in degree (0 mod 8); we call this the differential KO-character. In the second part, for a Riemannian submersion X to B with closed 8k-dimensional spin fibers, we develop two family index theorems in differential KO -theory: one in the differential KO -character model and one in the structured-bundle model. The fact that these two pushforwards agree follows from the Bismut--Cheeger adiabatic limit theorem, providing a new interpretation of that result. |
| title | Index invariants and Eta invariants determine Differential KO theory in degrees that are multiples of 8 |
| topic | K-Theory and Homology Algebraic Topology Differential Geometry |
| url | https://arxiv.org/abs/2511.19904 |