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Asymptotic spectral flow

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In this paper we study the asymptotic behavior of the spectral flow of a one-parameter family \(\{D_s\}\) of Dirac operators acting on the spinor bundle S twisted by a vector bundle E of rank k, with the parameter \(s\in [0,r]\) when r gets sufficiently large. Our method uses the variation of eta invariant and local index theory technique. The key is a uniform estimate of the eta invariant \(\bar{\eta }(D_r)\) which is established via local index theory technique and heat kernel estimate.

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Acknowledgements

The authors would like to thank the anonymous referee for careful reading and valuable suggestions. The second author would like to thank professor Weiping Zhang and Huitao Feng for their constant encouragement and helpful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Santa Barbara, CA, 93106, USA

    Xianzhe Dai

  2. Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, 300071, China

    Yihan Li

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  1. Xianzhe Dai
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Correspondence to Yihan Li.

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Xianzhe Dai was partially supported by the Simons Foundation. Yihan Li was partially supported by Nankai Zhide Foundation.

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Dai, X., Li, Y. Asymptotic spectral flow. Math. Z. 303, 78 (2023). https://doi.org/10.1007/s00209-023-03229-2

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