@article{9359, abstract = {We prove that the factorization homologies of a scheme with coefficients in truncated polynomial algebras compute the cohomologies of its generalized configuration spaces. Using Koszul duality between commutative algebras and Lie algebras, we obtain new expressions for the cohomologies of the latter. As a consequence, we obtain a uniform and conceptual approach for treating homological stability, homological densities, and arithmetic densities of generalized configuration spaces. Our results categorify, generalize, and in fact provide a conceptual understanding of the coincidences appearing in the work of Farb--Wolfson--Wood. Our computation of the stable homological densities also yields rational homotopy types, answering a question posed by Vakil--Wood. Our approach hinges on the study of homological stability of cohomological Chevalley complexes, which is of independent interest. }, author = {Ho, Quoc P}, issn = {1364-0380}, journal = {Geometry & Topology}, keywords = {Generalized configuration spaces, homological stability, homological densities, chiral algebras, chiral homology, factorization algebras, Koszul duality, Ran space}, number = {2}, pages = {813--912}, publisher = {Mathematical Sciences Publishers}, title = {{Homological stability and densities of generalized configuration spaces}}, doi = {10.2140/gt.2021.25.813}, volume = {25}, year = {2021}, } @article{9998, abstract = {We define quantum equivariant K-theory of Nakajima quiver varieties. We discuss type A in detail as well as its connections with quantum XXZ spin chains and trigonometric Ruijsenaars-Schneider models. Finally we study a limit which produces a K-theoretic version of results of Givental and Kim, connecting quantum geometry of flag varieties and Toda lattice.}, author = {Koroteev, Peter and Pushkar, Petr and Smirnov, Andrey V. and Zeitlin, Anton M.}, issn = {1420-9020}, journal = {Selecta Mathematica}, number = {5}, publisher = {Springer Nature}, title = {{Quantum K-theory of quiver varieties and many-body systems}}, doi = {10.1007/s00029-021-00698-3}, volume = {27}, year = {2021}, } @article{10033, abstract = {The ⊗*-monoidal structure on the category of sheaves on the Ran space is not pro-nilpotent in the sense of [3]. 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 [4], 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 [7] and [5].}, author = {Ho, Quoc P}, issn = {1090-2082}, journal = {Advances in Mathematics}, keywords = {Chiral algebras, Chiral homology, Factorization algebras, Koszul duality, Ran space}, publisher = {Elsevier}, title = {{The Atiyah-Bott formula and connectivity in chiral Koszul duality}}, doi = {10.1016/j.aim.2021.107992}, volume = {392}, year = {2021}, } @article{7004, abstract = {We define an action of the (double of) Cohomological Hall algebra of Kontsevich and Soibelman on the cohomology of the moduli space of spiked instantons of Nekrasov. We identify this action with the one of the affine Yangian of gl(1). Based on that we derive the vertex algebra at the corner Wr1,r2,r3 of Gaiotto and Rapčák. We conjecture that our approach works for a big class of Calabi–Yau categories, including those associated with toric Calabi–Yau 3-folds.}, author = {Rapcak, Miroslav and Soibelman, Yan and Yang, Yaping and Zhao, Gufang}, issn = {1432-0916}, journal = {Communications in Mathematical Physics}, pages = {1803--1873}, publisher = {Springer Nature}, title = {{Cohomological Hall algebras, vertex algebras and instantons}}, doi = {10.1007/s00220-019-03575-5}, volume = {376}, year = {2020}, } @article{7683, abstract = {For any free oriented Borel–Moore homology theory A, we construct an associative product on the A-theory of the stack of Higgs torsion sheaves over a projective curve C. We show that the resulting algebra AHa0C admits a natural shuffle presentation, and prove it is faithful when A is replaced with usual Borel–Moore homology groups. We also introduce moduli spaces of stable triples, heavily inspired by Nakajima quiver varieties, whose A-theory admits an AHa0C-action. These triples can be interpreted as certain sheaves on PC(ωC⊕OC). In particular, we obtain an action of AHa0C on the cohomology of Hilbert schemes of points on T∗C.}, author = {Minets, Sasha}, issn = {14209020}, journal = {Selecta Mathematica, New Series}, number = {2}, publisher = {Springer Nature}, title = {{Cohomological Hall algebras for Higgs torsion sheaves, moduli of triples and sheaves on surfaces}}, doi = {10.1007/s00029-020-00553-x}, volume = {26}, year = {2020}, }