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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Chern-Simons invariant and Deligne-Riemann-Roch isomorphism


Author: Takashi Ichikawa
Journal: Trans. Amer. Math. Soc. 374 (2021), 2987-3005
MSC (2020): Primary 14C40, 58J28; Secondary 14H10, 14H15
DOI: https://doi.org/10.1090/tran/8320
Published electronically: February 8, 2021
MathSciNet review: 4223040
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Abstract: Using the arithmetic Schottky uniformization theory, we show the arithmeticity of $PSL_{2}({\mathbb C})$ Chern-Simons invariant. In terms of this invariant, we give an explicit formula of the Deligne-Riemann-Roch isomorphism as the Zograf-McIntyre-Takhtajan infinite product for families of algebraic curves. Applying this formula to the Liouville theory, we determine the unknown constant which appears in the holomorphic factorization formula of determinants of Laplacians on Riemann surfaces.


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Additional Information

Takashi Ichikawa
Affiliation: Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan
MR Author ID: 253584
Email: ichikawn@cc.saga-u.ac.jp

Keywords: Chern-Simons invariant, Deligne-Riemann-Roch isomorphism, Liouville action, moduli space of curves
Received by editor(s): May 27, 2020
Received by editor(s) in revised form: September 9, 2020
Published electronically: February 8, 2021
Article copyright: © Copyright 2021 American Mathematical Society