@article{14244, abstract = {In this paper, we determine the motivic class — in particular, the weight polynomial and conjecturally the Poincaré polynomial — of the open de Rham space, defined and studied by Boalch, of certain moduli spaces of irregular meromorphic connections on the trivial rank bundle on P1. The computation is by motivic Fourier transform. We show that the result satisfies the purity conjecture, that is, it agrees with the pure part of the conjectured mixed Hodge polynomial of the corresponding wild character variety. We also identify the open de Rham spaces with quiver varieties with multiplicities of Yamakawa and Geiss–Leclerc–Schröer. We finish with constructing natural complete hyperkähler metrics on them, which in the four-dimensional cases are expected to be of type ALF.}, author = {Hausel, Tamás and Wong, Michael Lennox and Wyss, Dimitri}, issn = {1460-244X}, journal = {Proceedings of the London Mathematical Society}, number = {4}, pages = {958--1027}, publisher = {Wiley}, title = {{Arithmetic and metric aspects of open de Rham spaces}}, doi = {10.1112/plms.12555}, volume = {127}, year = {2023}, } @inbook{12303, abstract = {We construct for each choice of a quiver Q, a cohomology theory A, and a poset P a “loop Grassmannian” GP(Q,A). This generalizes loop Grassmannians of semisimple groups and the loop Grassmannians of based quadratic forms. The addition of a “dilation” torus D⊆G2m gives a quantization GPD(Q,A). This construction is motivated by the program of introducing an inner cohomology theory in algebraic geometry adequate for the Geometric Langlands program (Mirković, Some extensions of the notion of loop Grassmannians. Rad Hrvat. Akad. Znan. Umjet. Mat. Znan., the Mardešić issue. No. 532, 53–74, 2017) and on the construction of affine quantum groups from generalized cohomology theories (Yang and Zhao, Quiver varieties and elliptic quantum groups, preprint. arxiv1708.01418).}, author = {Mirković, Ivan and Yang, Yaping and Zhao, Gufang}, booktitle = {Representation Theory and Algebraic Geometry}, editor = {Baranovskky, Vladimir and Guay, Nicolas and Schedler, Travis}, isbn = {9783030820060}, issn = {2297-024X}, pages = {347--392}, publisher = {Springer Nature; Birkhäuser}, title = {{Loop Grassmannians of Quivers and Affine Quantum Groups}}, doi = {10.1007/978-3-030-82007-7_8}, year = {2022}, } @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{7940, abstract = {We prove that the Yangian associated to an untwisted symmetric affine Kac–Moody Lie algebra is isomorphic to the Drinfeld double of a shuffle algebra. The latter is constructed in [YZ14] as an algebraic formalism of cohomological Hall algebras. As a consequence, we obtain the Poincare–Birkhoff–Witt (PBW) theorem for this class of affine Yangians. Another independent proof of the PBW theorem is given recently by Guay, Regelskis, and Wendlandt [GRW18].}, author = {Yang, Yaping and Zhao, Gufang}, issn = {1531-586X}, journal = {Transformation Groups}, pages = {1371--1385}, publisher = {Springer Nature}, title = {{The PBW theorem for affine Yangians}}, doi = {10.1007/s00031-020-09572-6}, volume = {25}, year = {2020}, } @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{6986, abstract = {Li-Nadler proposed a conjecture about traces of Hecke categories, which implies the semistable part of the Betti geometric Langlands conjecture of Ben-Zvi-Nadler in genus 1. We prove a Weyl group analogue of this conjecture. Our theorem holds in the natural generality of reflection groups in Euclidean or hyperbolic space. As a corollary, we give an expression of the centralizer of a finite order element in a reflection group using homotopy theory. }, author = {Li, Penghui}, issn = {1088-6826}, journal = {Proceedings of the American Mathematical Society}, number = {11}, pages = {4597--4604}, publisher = {AMS}, title = {{A colimit of traces of reflection groups}}, doi = {10.1090/proc/14586}, volume = {147}, year = {2019}, } @article{439, abstract = {We count points over a finite field on wild character varieties,of Riemann surfaces for singularities with regular semisimple leading term. The new feature in our counting formulas is the appearance of characters of Yokonuma–Hecke algebras. Our result leads to the conjecture that the mixed Hodge polynomials of these character varieties agree with previously conjectured perverse Hodge polynomials of certain twisted parabolic Higgs moduli spaces, indicating the possibility of a P = W conjecture for a suitable wild Hitchin system.}, author = {Hausel, Tamas and Mereb, Martin and Wong, Michael}, issn = {1435-9855}, journal = {Journal of the European Mathematical Society}, number = {10}, pages = {2995--3052}, publisher = {European Mathematical Society}, title = {{Arithmetic and representation theory of wild character varieties}}, doi = {10.4171/JEMS/896}, volume = {21}, year = {2019}, } @article{322, abstract = {We construct quantizations of multiplicative hypertoric varieties using an algebra of q-difference operators on affine space, where q is a root of unity in C. The quantization defines a matrix bundle (i.e. Azumaya algebra) over the multiplicative hypertoric variety and admits an explicit finite étale splitting. The global sections of this Azumaya algebra is a hypertoric quantum group, and we prove a localization theorem. We introduce a general framework of Frobenius quantum moment maps and their Hamiltonian reductions; our results shed light on an instance of this framework.}, author = {Ganev, Iordan V}, journal = {Journal of Algebra}, pages = {92 -- 128}, publisher = {World Scientific Publishing}, title = {{Quantizations of multiplicative hypertoric varieties at a root of unity}}, doi = {10.1016/j.jalgebra.2018.03.015}, volume = {506}, year = {2018}, } @article{687, abstract = {Pursuing the similarity between the Kontsevich-Soibelman construction of the cohomological Hall algebra (CoHA) of BPS states and Lusztig's construction of canonical bases for quantum enveloping algebras, and the similarity between the integrality conjecture for motivic Donaldson-Thomas invariants and the PBW theorem for quantum enveloping algebras, we build a coproduct on the CoHA associated to a quiver with potential. We also prove a cohomological dimensional reduction theorem, further linking a special class of CoHAs with Yangians, and explaining how to connect the study of character varieties with the study of CoHAs.}, author = {Davison, Ben}, issn = {0033-5606}, journal = {Quarterly Journal of Mathematics}, number = {2}, pages = {635 -- 703}, publisher = {Oxford University Press}, title = {{The critical CoHA of a quiver with potential}}, doi = {10.1093/qmath/haw053}, volume = {68}, year = {2017}, }