We study modules and comodules for cohomological Hall algebras equipped with their vertex coproducts arising as objects with classical type stabilizer groups. Specifically we consider how classical type parabolic induction gives rise to actions of CoHAs of quivers with potential, of preprojective algberas, and of dimension zero sheaves on a smooth proper surface. In all cases the CoHA action is compatible with a localised (and vertex) coaction making the module a twisted Yetter-Drinfeld module over the CoHA with its localised braided bialgebra structure. In the case of dimension zero sheaves on a surface the action is related to an approach to the AGT conjecture in classical type using moduli stacks of orthosymplectic perverse coherent sheaves, a compactification of the stack of classical type bundles on a surface.
We derive a combinatorial formula for quantized Donaldson-Thomas invariants of the m-loop quiver. Our main tools are the combinatorics of noncommutative Hilbert schemes and a degenerate version of the Cohomological Hall algebra of this quiver.
The moduli stack of representations of a quiver, or coherent sheaves on a proper curve, carries two structures on its cohomology: a Hall algebra and braided vertex coalgebra. We show that they are compatible, by developing a formulation of abelian localisation which works in the derived/singular setting. An application of these techniques gives an explicit formula for the cohomological Hall algebra products.
This paper aims at a geometric realization of the Yangian of non-simply laced type in terms of quiver with potentials. For every quiver with symmetrizer, there is an extended quiver with superpotential, whose Jacobian algebra is the generalized preprojective algebra of Geiß, Leclerc, and Schröer arXiv:1410.1403. We study the cohomological Hall algebra of Kontsevich and Soibelman associated to this quiver with potential. In particular, we prove a dimensional reduction result, and provide a shuffle formula of this cohomological Hall algebra. In the case when the quiver with symmetrizer comes fro…
We give a short introduction to Cohomological Hall algebras of quivers and describe the semistable Cohomological Hall algebra of central slope of the Kronecker quiver in terms of generators and relations.
We define the cohomological Hall algebra ${AHA}_{Higgs(X)}$ of the ($2$-dimensional) Calabi-Yau category of Higgs sheaves on a smooth projective curve $X$, as well as its nilpotent and semistable variants, in the context of an arbitrary oriented Borel-Moore homology theory. In the case of usual Borel-Moore homology, ${AHA}_{Higgs(X)}$ is a module over the (universal) cohomology ring $\mathbb{H}$ of the stacks of coherent sheaves on $X$ . We show that it is a torsion-free $\mathbb{H}$-module, and we provide an explicit collection of generators (the collection of fundamental classes $[Coh_{r,d}]…
In this paper we investigate the relationship between twisted and untwisted character varieties via a specific instance of the Cohomological Hall algebra for moduli of objects in 3-Calabi-Yau categories introduced by Kontsevich and Soibelman. In terms of Donaldson--Thomas theory, this relationship is completely understood via the calculations of Hausel and Villegas of the E polynomials of twisted character varieties and untwisted character stacks. We present a conjectural lift of this relationship to the cohomological Hall algebra setting.
For any acyclic quiver, we establish a family of structure isomorphisms for its cohomological Hall algebra (CoHA). The family is parameterized by partitions of the quiver into Dynkin subquivers. For each such partition, we write the domain of our isomorphism as a tensor product of subalgebras in two ways. In the first, each tensor factor is isomorphic to the CoHA of the quiver with a single vertex and no arrows. In the second, the tensor factors are each isomorphic to the CoHAs of the corresponding Dynkin subquivers. When the quiver is already an orientation of a simply-laced Dynkin diagram, o…
The equivariant Chern-Schwartz-MacPherson (CSM) class and the equivariant Motivic Chern (MC) class are important characteristic classes of singular varieties in cohomology and K theory---and their theory overlaps with the theory of Okounkov's stable envelopes. We study CSM and MC classes for the orbits of Dynkin quiver representations. We show that the problem of computing the CSM and MC classes of all these orbits can be reduced to some basic classes $c^o_β$, $C^o_β$ parameterized by positive roots $β$. We prove an identity in a deformed version of Kontsevich-Soibelman's Cohomological (and K-…
A multiplication on the 2D cohomological Hall algebra (CoHA) of the variety of commuting matrices was described by Schiffman and Vasserot. This construction can be generalised to other varieties that exist as the zero-locus of a function on a smooth ambient variety. On the 2D CoHA of the character variety of the fundamental group of a genus $g$ Riemann surface, we compare the multiplication induced by the standard presentation and that of a brane tiling presentation.
We introduce a new class of representations of the cohomological Hall algebras of Kontsevich and Soibelman, which we call cohomological Hall modules, or CoHM for short. These representations are constructed from self-dual representations of a quiver with contravariant involution $σ$ and provide a mathematical model for the space of BPS states in orientifold string theory. We use the CoHM to define a generalization of the cohomological Donaldson-Thomas theory of quivers which allows the quiver representations to have orthogonal and symplectic structure groups. The associated invariants are call…
The paper is devoted to several questions related to the notion of Cohomological Hall algebra (COHA for short) introduced few years ago by Maxim Kontsevich and the author. In particular we discuss a class of representations of COHA in the cohomology groups of the moduli spaces of stable framed objects. We speculate about possible sources of interesting representations as well as about some applications to geometry and physics.
This paper concerns the cohomological aspects of Donaldson-Thomas theory for Jacobi algebras and the associated cohomological Hall algebra, introduced by Kontsevich and Soibelman. We prove the Hodge-theoretic categorification of the integrality conjecture and the wall crossing formula, and furthermore realise the isomorphism in both of these theorems as Poincaré-Birkhoff-Witt isomorphisms for the associated cohomological Hall algebra. We do this by defining a perverse filtration on the cohomological Hall algebra, a result of the "hidden properness" of the semisimplification map from the moduli…
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 $\mathfrak{gl}(1)$. Based on that we derive the vertex algebra at the corner $\mathcal{W}_{r_1,r_2,r_3}$ of Gaiotto and Rapcak. We conjecture that our approach works for a big class of Calabi-Yau categories, including those associated with toric Calabi-Yau $3$-folds.
Pursuing the similarity between the Kontsevich--Soibelman construction of the cohomological Hall algebra of BPS states and Lusztig's construction of canonical bases for quantum enveloping algebras, and the similarity between the inetgrality conjecture for motivic Donaldson--Thomas invariants and the PBW theorem for quantum enveloping algebras, we build a coproduct on the cohomological Hall algebra associated to a quiver with potential. We also prove a cohomological dimensional reduction theorem, further linking a special class of cohomological Hall algebras with Yangians, and explaining how to…
Consider the Cohomological Hall Algebra as defined by Kontsevich and Soibelman, associated with a Dynkin quiver. We reinterpret the geometry behind the multiplication map in the COHA, and give an iterated residue formula for it. We show natural subalgebras whose product is the whole COHA (except in the $E_8$ case). The dimension count version of this statement is an identity for quantum dilogarithm series, first proved by Reineke. We also show that natural structure constants of the COHA are universal polynomials representing degeneracy loci, a.k.a. quiver polynomials.
We discuss semicanonical bases from the point of view of Cohomological Hall algebras via the "dimensional reduction" from 3-dimensional Calabi-Yau categories to 2-dimensional ones. Also, we discuss the notion of motivic Donaldson-Thomas invariants (as defined by M. Kontsevich and Y. Soibelman) in the framework of 2-dimensional Calabi-Yau categories. In particular we propose a conjecture which allows one to define Kac polynomials for a 2-dimensional Calabi-Yau category (this is a theorem of S. Mozgovoy in the case of preprojective algebras).
We define a new type of Hall algebras associated e.g. with quivers with polynomial potentials. The main difference with the conventional definition is that we use cohomology of the stack of representations instead of constructible sheaves or functions. In order to take into account the potential we introduce a generalization of theory of mixed Hodge structures, related to exponential integrals. Generating series of our Cohomological Hall algebra is a generalization of the motivic Donaldson-Thomas invariants introduced in arXiv:0811.2435. Also we prove a new integrality property of motivic Dona…
Categories of Line Defects and Cohomological Hall Algebras
Perimeter Institute for Theoretical Physics（Perimeter理论物理研究所）；University of Waterloo（滑铁卢大学）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）
Commutativity of nilpotent cohomological Hall algebras of $\mathbf{A}^2$
Université de Picardie Jules Verne（皮卡第朱尔·凡尔纳大学）；Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, CNRS UMR 7352（阿米安基础与应用数学实验室，法国国家科学研究中心联合研究单位7352）
For a smooth quasi-projective surface S over complex numbers we consider the Borel-Moore homology of the stack of coherent sheaves on S with compact support and make this space into an associative algebra by a version of the Hall multiplication. This multiplication involves data (virtual pullbacks) governing the derived moduli stack, i.e., the perfect obstruction theory naturally existing on the non-derived stack. By restricting to sheaves with support of given dimension, we obtain several types of Hecke operators. In particular, we study R(S), the Hecke algebra of 0-dimensional sheaves. For t…
In the paper \cite{KS}, Kontsevich and Soibelman in particular associate to each finite quiver $Q$ with a set of vertices $I$ the so-called Cohomological Hall algebra $\cH,$ which is $\Z_{\geq 0}^I$-graded. Its graded component $\cH_γ$ is defined as cohomology of Artin moduli stack of representations with dimension vector $γ.$ The product comes from natural correspondences which parameterize extensions of representations. In the case of symmetric quiver, one can refine the grading to $\Z_{\geq 0}^I\times\Z,$ and modify the product by a sign to get a super-commutative algebra $(\cH,\star)$ (wit…
This review gives an introduction to cohomological Donaldson-Thomas theory: the study of a cohomology theory on moduli spaces of sheaves on Calabi-Yau threefolds, and of complexes in 3-Calabi-Yau categories, categorifying their numerical DT invariant. Local and global aspects of the theory are both covered, including representations of quivers with potential. We will discuss the construction of the DT sheaf, a nontrivial topological coefficient system on such a moduli space, along with some cohomology computations. The Cohomological Hall Algebra, an algebra structure on cohomological DT spaces…
Rutgers, The State University Of New Jersey（新泽西州立罗格斯大学）；The University of Tokyo（东京大学）；Simion Stoilow Institute of Mathematics（罗马尼亚科学院西米恩·斯托伊洛数学研究所）；Université de Paris（巴黎大学）
Cohomological Hall algebras of one-dimensional sheaves on surfaces and Yangians
Rutgers, The State University Of New Jersey（新泽西州立罗格斯大学）；The University of Tokyo（东京大学）；Simion Stoilow Institute of Mathematics（西蒙·斯托伊洛数学研究所）；Université de Paris（巴黎大学）
In the present paper, we introduce two-dimensional categorified Hall algebras of smooth curves and smooth surfaces. A categorified Hall algebra is an associative monoidal structure on the stable $\infty$-category $\mathsf{Coh}^{\mathsf{b}}(\mathbb{R}\mathsf{M})$ of complexes of sheaves with bounded coherent cohomology on a derived moduli stack $\mathbb{R}\mathsf{M}$. In the surface case, $\mathbb{R}\mathsf{M}$ is a suitable derived enhancement of the moduli stack $\mathsf{M}$ of coherent sheaves on the surface. This construction categorifies the K-theoretical and cohomological Hall algebras of…
We introduce and study a fermionization procedure for the cohomological Hall algebra $\mathcal{H}_{Π_Q}$ of representations of a preprojective algebra, that selectively switches the cohomological parity of the BPS Lie algebra from even to odd. We do so by determining the cohomological Donaldson--Thomas invariants of central extensions of preprojective algebras studied in the work of Etingof and Rains, via deformed dimensional reduction. Via the same techniques, we determine the Borel-Moore homology of the stack of representations of the $μ$-deformed preprojective algebra introduced by Crawley-…
The semi-stable ChowHa of a quiver with stability is defined as an analog of the Cohomological Hall algebra of Kontsevich and Soibelman via convolution in equivariant Chow groups of semi-stable loci in representation varieties of quivers. We prove several structural results on the semi-stable ChowHa, namely isomorphism of the cycle map, a tensor product decomposition, and a tautological presentation. For symmetric quivers, this leads to an identification of their quantized Donaldson-Thomas invariants with the Chow-Betti numbers of moduli spaces.
Given a quiver, we consider its cohomological Hall algebra (CoHA) as well as CoHA modules built of cohomology groups of non-commutative Hilbert schemes. We investigate cell decompositions of non-commutative Hilbert schemes and the corresponding (non-canonical) bases of CoHA modules. We construct canonical bases of CoHA modules, which consist of products of Chern classes of tautological vector bundles over non-commutative Hilbert schemes. This result generalizes classical results for Grassmannians and (partial) flag varieties.
We prove that Chow groups of certain non-commutative Hilbert schemes have a basis consisting of monomials in Chern classes of the universal bundle. Furthermore, we realize the cohomology of non-commutative Hilbert schemes as a module over the Cohomological Hall algebra.
We study a variant of the semi-stable Cohomological Hall algebra which we construct using equivariant Chow groups. This algebra, we call it the semi-stable ChowHa, arises as a quotient of the CoHa. Smooth models of quiver moduli give rise to modules over the semi-stable ChowHa. We prove that these modules are cyclic and we compute a presentation using Harder-Narasimhan methods.
Let $Q$ be the doubled of a quiver. According to Efimov, Kontsevich and Soibelman, the cohomological Hall algebra (CoHA) associated to $Q$ is a free super-commutative algebra. In this short note, we confirm a conjecture of Hausel which gives a geometric realisation of the generators of the CoHA.
We show that the categorified cohomological Hall algebra structures on surfaces constructed by Porta-Sala descend to those on Donaldson-Thomas categories on local surfaces introduced in the author's previous paper. A similar argument also shows that Pandharipande-Thomas categories on local surfaces admit actions of categorified COHA for zero dimensional sheaves on surfaces. We also construct annihilator actions of its simple operators, and show that their commutator in the K-theory satisfies the relation similar to the one of Weyl algebras. This result may be regarded as a categorification of …
Critical CoHAs, vertex coalgebras and Deformed Drinfeld coproducts
Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo（东京大学汤川理论物理研究所（WPI））；The University of Tokyo Institutes for Advanced Study, The University of Tokyo（东京大学高等研究院）；Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京国际数学研究中心）
Hecke operators on symplectic surfaces and $χ$-independence
University of Edinburgh（爱丁堡大学）；Université de Picardie Jules Verne（约尔·凡尔纳皮卡第大学）；Kyoto University（京都大学）；Université Paris-Saclay（巴黎萨克雷大学）；Simion Stoilow Institute of Mathematics（西米翁·斯托伊洛瓦数学研究所）；Université Paris-Cité（巴黎城市大学）
We study the Borel-Moore homology of stacks of representations of preprojective algebras $Π_Q$, via the study of the DT theory of the undeformed 3-Calabi-Yau completion $Π_Q[x]$. Via a result on the supports of the BPS sheaves for $Π_Q[x]$-mod, we prove purity of the BPS cohomology for the stack of $Π_Q[x]$-modules, and define BPS sheaves for stacks of $Π_Q$-modules. These are mixed Hodge modules on the coarse moduli space of $Π_Q$-modules that control the Borel-Moore homology and geometric representation theory associated to these stacks. We show that the hypercohomology of these objects is p…
The famous pentagon identity for quantum dilogarithms has a generalization for every Dynkin quiver, due to Reineke. A more advanced generalization is associated with a pair of alternating Dynkin quivers, due to Keller. The description and proof of Keller's identities involves cluster algebras and cluster categories, and the statement of the identity is implicit. In this paper we describe Keller's identities explicitly, and prove them by a dimension counting argument. Namely, we consider quiver representations $\boldsymbol{\mathrm{Rep}}_γ$ together with a superpotential function $W_γ$, and calc…
Coherent sheaves on surfaces, COHAs and deformed $W_{1+\infty}$-algebras
Faculty of Mathematics, University of Vienna（维也纳大学数学系）；Max Planck Institute for Mathematics（马克斯·普朗克数学研究所）；Mathematisches Institut, University of Bonn（波恩大学数学研究所）；Département de Mathématiques, Université de Paris-Sud Paris-Saclay（巴黎萨克雷大学巴黎南校区数学系）；Simion Stoilow Institute of Mathematics（西米翁·斯托伊洛瓦数学研究所）；Université Paris Cité（巴黎西岱大学）
