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Differential K-theory and localization formula for \(\eta \)-invariants

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In this paper we obtain a localization formula in differential K-theory for \(S^1\)-actions. We establish a localization formula for equivariant \(\eta \)-invariants by combining this result with our extension of Goette’s result on the comparison of two types of equivariant \(\eta \)-invariants. An important step in our approach is to construct a pre-\(\lambda \)-ring structure in differential K-theory.

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Acknowledgements

We would like to thank Professors Jean-Michel Bismut and Weiping Zhang for helpful discussions. We are especially grateful to the referees for their very helpful comments and suggestions. We are also indebted to George Marinescu for his critical comments. BL is partially supported by NNSFC Nos. 11931997, 11971168 and Science and Technology Commission of Shanghai Municipality (STCSM), Grant No. 18dz2271000. XM is partially supported by NNSFC Nos. 11528103, 11829102, ANR-14-CE25-0012-01, and funded through the Institutional Strategy of the University of Cologne within the German Excellence Initiative.

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  1. School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, 200241, People’s Republic of China

    Bo Liu

  2. Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université de Paris, CNRS, 75013, Paris, France

    Xiaonan Ma

  3. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui Province, People’s Republic of China

    Xiaonan Ma

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Liu, B., Ma, X. Differential K-theory and localization formula for \(\eta \)-invariants. Invent. math. 222, 545–613 (2020). https://doi.org/10.1007/s00222-020-00973-8

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