Chern-Simons invariant and Deligne-Riemann-Roch isomorphism

Ichikawa, Takashi

doi:10.1090/tran/8320

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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