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Real embedding and equivariant eta forms

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Abstract

Bismut and Zhang (Math Ann 295(4):661–684, 1993) establish a \({\mathrm {mod}}\, \mathbb {Z}\) embedding formula of Atiyah–Patodi–Singer reduced eta invariants. In this paper, we explain the hidden \({\mathrm {mod}}\, \mathbb {Z}\) term as a spectral flow and extend this embedding formula to the equivariant family case. In this case, the spectral flow is generalized to the equivariant Chern character of some equivariant Dai–Zhang higher spectral flow.

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Notes

  1. The author thanks Prof. Xiaonan Ma for pointing out this simplification, which is related to a remark in [8, Section 7.5].

  2. One example is the exterior bundle with the \(\mathbb {Z}_2\)-grading induced by the Hodge star operator (see e.g., [5, pp. 150]).

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Acknowledgements

The author gratefully acknowledges the many helpful discussions with Prof. Xiaonan Ma and Shu Shen during the preparation of this paper. He wishes to thank University of California, Santa Barbara, especially Prof. Xianzhe Dai for financial support and hospitality. Part of the work was done while the author was visiting Institut des Hautes Études Scientifiques (IHES) and Max Planck Institute for Mathematics (MPIM) which he thanks the financial support. The author is indebted to a referee for his careful reading and helpful comments on an earlier version of this paper. This research is partially supported by the China Postdoctoral Science Foundation (2017M621404) and Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice.

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  1. School of Mathematical Sciences, East China Normal University, 500 Dongchuan Road, Shanghai, 200241, People’s Republic of China

    Bo Liu

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Liu, B. Real embedding and equivariant eta forms. Math. Z. 292, 849–878 (2019). https://doi.org/10.1007/s00209-018-2119-9

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