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Abstract Arinkin and Gaitsgory defined a category oftempered‐modules on that is conjecturally equivalent to the category of quasi‐coherent (not ind‐coherent!) sheaves on . However, their definition depends on the auxiliary data of a point of the curve; they conjectured that their definition is independent of this choice. Beraldo has outlined a proof of this conjecture that depends on some technology that is not currently available. Here we provide a short, unconditional proof of the Arinkin–Gaitsgory conjecture.
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We calculate the category of D 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 3 d 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.more » « less
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In quantum geometric Langlands, the Satake equivalence plays a less prominent role than in the classical theory. Gaitsgory and Lurie proposed a conjectural substitute, later termed the fundamental local equivalence . With a few exceptions, we prove this conjecture and its extension to the affine flag variety by using what amount to Soergel module techniques.more » « less
