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Geometry of Localized Effective Theories, Exact Semi-classical Approximation and the Algebraic Index

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Abstract

In this paper we propose a general framework to study the quantum geometry of \(\sigma \)-models when they are effectively localized to small quantum fluctuations around constant maps. Such effective theories have surprising exact descriptions at all loops in terms of target geometry and can be rigorously formulated. We illustrate how to turn the physics idea of exact semi-classical approximation into a geometric set-up in this framework, using Gauss–Manin connection. As an application, we carry out this program in details by the example of topological quantum mechanics, and explain how to implement the idea of exact semi-classical approximation into a proof of the algebraic index theorem. The proof resembles much of the physics derivation of Atiyah–Singer index theorem and clarifies the geometric face of many other mathematical constructions.

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Acknowledgements

We would like to thank Vladimir Baranovsky, Owen Gwilliam, Weiqiang He, Qin Li, Yifan Li, Ryszard Nest, Xiang Tang, Michèle Vergne, Brian Williams, Edward Witten, Jie Zhou for helpful discussions. We are specially grateful to Michèle Vergne for providing numerous valuable suggestions on the earlier version of this manuscript. This work was partially supported by National Key Research and Development Program of China (NO. 2020YFA0713000), Beijing Municipal Natural Science Foundation (NO. Z180003), and National Natural Science Foundation of China (NO. 11801300). Part of this work was done in Fall 2019 while S.L. was visiting Institute for Advanced Study at Princeton and Z.G. was visiting Center of Mathematical Sciences and Applications at Harvard. We thank for their hospitality and provision of excellent working enviroment.

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Authors and Affiliations

  1. Department of Mathematical Sciences and Yau Mathematical Sciences Center, Tsinghua University, Beijing, China

    Zhengping Gui & Si Li

  2. Institute for Advanced Study, Princeton, NJ, USA

    Si Li

  3. Department of Mathematics, Harvard University, Cambridge, MA, USA

    Kai Xu

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  1. Zhengping Gui
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  2. Si Li
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Correspondence to Si Li.

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Communicated by H.-T. Yau.

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Gui, Z., Li, S. & Xu, K. Geometry of Localized Effective Theories, Exact Semi-classical Approximation and the Algebraic Index. Commun. Math. Phys. 382, 441–483 (2021). https://doi.org/10.1007/s00220-021-03944-z

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