The ⊗*-monoidal structure on the category of sheaves on the Ran space is not pro-nilpotent in the sense of . However, under some connectivity assumptions, we prove that Koszul duality induces an equivalence of categories and that this equivalence behaves nicely with respect to Verdier duality on the Ran space and integrating along the Ran space, i.e. taking factorization homology. Based on ideas sketched in , we show that these results also offer a simpler alternative to one of the two main steps in the proof of the Atiyah-Bott formula given in  and .
Advances in Mathematics
The author would like to express his gratitude to D. Gaitsgory, without whose tireless guidance and encouragement in pursuing this problem, this work would not have been possible. The author is grateful to his advisor B.C. Ngô for many years of patient guidance and support. This paper is revised while the author is a postdoc in Hausel group at IST Austria. We thank him and the group for providing a wonderful research environment. The author also gratefully acknowledges the support of the Lise Meitner fellowship “Algebro-Geometric Applications of Factorization Homology,” Austrian Science Fund (FWF): M 2751.
Ho QP. The Atiyah-Bott formula and connectivity in chiral Koszul duality. Advances in Mathematics. 2021;392. doi:10.1016/j.aim.2021.107992
Ho, Q. P. (2021). The Atiyah-Bott formula and connectivity in chiral Koszul duality. Advances in Mathematics. Elsevier. https://doi.org/10.1016/j.aim.2021.107992
Ho, Quoc P. “The Atiyah-Bott Formula and Connectivity in Chiral Koszul Duality.” Advances in Mathematics. Elsevier, 2021. https://doi.org/10.1016/j.aim.2021.107992.
Q. P. Ho, “The Atiyah-Bott formula and connectivity in chiral Koszul duality,” Advances in Mathematics, vol. 392. Elsevier, 2021.
Ho QP. 2021. The Atiyah-Bott formula and connectivity in chiral Koszul duality. Advances in Mathematics. 392, 107992.
Ho, Quoc P. “The Atiyah-Bott Formula and Connectivity in Chiral Koszul Duality.” Advances in Mathematics, vol. 392, 107992, Elsevier, 2021, doi:10.1016/j.aim.2021.107992.
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