Tate’s thesis in the de Rham setting

Hilburn, Justin; Raskin, Sam

doi:10.1090/jams/1010

Tate’s thesis in the de Rham setting
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by Justin Hilburn and Sam Raskin

J. Amer. Math. Soc. 36 (2023), 917-1001

DOI: https://doi.org/10.1090/jams/1010

Published electronically: June 27, 2022

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Abstract:

We calculate the category of $D$-modules on the loop space of the affine line in coherent terms. Specifically, we find that this category is derived equivalent to the category of ind-coherent sheaves on the moduli space of rank one de Rham local systems with a flat section. Our result establishes a conjecture coming out of the $3d$ mirror symmetry program, which obtains new compatibilities for the geometric Langlands program from rich dualities of QFTs that are themselves obtained from string theory conjectures.

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