Let $H_{\mathbf{k}}$ be a symplectic reflection algebra corresponding to a cyclic subgroup $Γ\subseteq SL_2 \C$ of order $n$ and $U_{\mathbf{k}} = eH_{\mathbf{k}} e$ the spherical subalgebra of $H_{\mathbf{k}}$.
We show that for suitable ${\mathbf{k}}$ there is a filtered $\Z^{n-1}$-algebra $R$ such that (1) there is an equivalence of categories $R-\mathrm{qgr} \simeq U_{\bf k}$-mod, (2) there is an equivalence of categories $gr R-\mathrm{qgr} \simeq \ttt{Coh}(Hilb_Γ\mathbb{C}^2)$.
Here $ \ttt{Coh}(Hilb_Γ\mathbb{C}^2)$ is the category of coherent sheaves on the $Γ$-Hilbert scheme. and for a graded algebra $\mathcal{R},$ we write $ \mathcal{R}-\mathrm{qgr}$ for the quotient category of finitely generated graded $\mathcal{R}$-modules modulo torsion.
Kleinian singularities, i.e., the varieties corresponding to the algebras of invariants of Kleinian groups are of fundamental importance for Algebraic geometry, Representation theory and Singularity theory. The filtered deformations of these algebras of invariants were classified by Slodowy (the commutative case) and Losev (the general case). To an inclusion of Kleinian groups, there is the corresponding inclusion of algebras of invariants. We classify deformations of these inclusions when a smaller subgroup is normal in the larger.
The notion of a Harish-Chandra bimodule, i.e. finitely generated $U(\mathfrak{g})$-bimodule with locally finite adjoint action, was generalized to any filtered algebra in a work of Losev [Ivan Losev, Dimensions of irreducible modules over W-algebras and Goldie ranks. arXiv:1209.1083]. Similarly to the classical case we can define the notion of a unitarizable bimodule. We investigate a question when the regular bimodule, i.e. the algebra itself, for a deformation of Kleinian singularity of type $A$ is unitarizable. We obtain a partial classification of unitarizable regular bimodules.
In this paper we classify all Morita equivalent pairs of (classical) generalized Weyl algebras for generic values of the parameters, thus positively settling a 30 year old question posed by T.Hodges. We also prove a similar result for noncommutative deformations of arbitrary Kleinian singularities provided the corresponding parameters are very generic.
For a certain class of (nonunital) subalgebras of deformed preprojective algebra of affine type we describe their centers as certain deformation of Kleinian singularity and find their PI-degree. These results can be applied to algebras generated by elements with given spectrum and scalar sum.
We describe the spaces of stability conditions on certain triangulated categories associated to Dynkin diagrams. These categories can be defined either algebraically via module categories of preprojective algebras, or geometrically via coherent sheaves on resolutions of Kleinian singularities. The resulting spaces are related to regular subsets of the corresponding complexified Cartan algebras.
For any $(m,n)$-periodic higher spin six-vertex configuration $\mathscr{L}$, we construct a one-parameter family $Δ_ξ$ of pseudo-unitarizable representations of the corresponding noncommutative fiber product $\mathcal{A}(\mathscr{L})$ by difference operators acting on the space of sections of a complex line bundle $L_ξ$ over the face lattice $F$. The indefinite inner product is given explicitly in terms of a combinatorial sign function defined on $F$. We prove that each simple integral weight $\mathcal{A}(\mathscr{L})$-module (previously classified by the author, see arXiv:1612.08125) occurs a…
We establish faithfulness of braid group actions generated by twists along an ADE configuration of $2$-spherical objects in a derived category. Our major tool is the Garside structure on braid groups of type ADE. This faithfulness result provides the missing ingredient in Bridgeland's description of a space of stability conditions associated to a Kleinian singularity.
The exceptional configuration of the minimal resolution $\hat{S}_G $ of a Kleinian quotient surface $S_G (:= \CZ^2/G)$ is depicted by a $A$-$D$-$E$ Coxeter-Dynkin diagram. In this article, we show that branching indices of the affine $A$-$D$-$E$ diagram is geometrically characterized by a certain special function $F$ of $S_G$ as the multiplicities of its divisor components in $\hat{S}_G$, a version parallel to the elliptic fibration near certain types of simple singular fibers in Kodaira's elliptic surface theory. We further obtain the uniqueness property of the function $F$ (modular local uni…
For a finite subgroup $Γ\subset \mathrm{SL}(2,\mathbb{C})$ and $n\geq 1$, we construct the (reduced scheme underlying the) Hilbert scheme of $n$ points on the Kleinian singularity $\mathbb{C}^2/Γ$ as a Nakajima quiver variety for the framed McKay quiver of $Γ$, taken at a specific non-generic stability parameter. We deduce that this Hilbert scheme is irreducible (a result previously due to Zheng), normal, and admits a unique symplectic resolution. More generally, we introduce a class of algebras obtained from the preprojective algebra of the framed McKay quiver by a process called cornering, a…
In this paper we prove an analogue of a recent result of Gordon and Stafford that relates the representation theory of certain noncommutative deformations of the coordinate ring of the n-th symmetric power of C^2 with the geometry of the Hilbert scheme of n points in C^2 through the formalism of Z-algebras. Our work produces, for every regular noncommutative deformation O^λof a Kleinian singularity X=C^2/Γ, as defined by Crawley-Boevey and Holland, a filtered Z-algebra which is Morita equivalent to O^λ, such that the associated graded Z-algebra is Morita equivalent to the minimal resolution of…
C. Weibel and Thomason-Trobaugh proved (under some assumptions) that algebraic K-theory with coefficients is A1-homotopy invariant. In this article we generalize this result from schemes to the broad setting of dg categories. Along the way, we extend Bass-Quillen's fundamental theorem as well as Stienstra's foundational work on module structures over the big Witt ring to the setting of dg categories. Among other cases, the above A1-homotopy invariance result can now be applied to sheaves of (not necessarily commutative) dg algebras over stacks. As an application, we compute the algebraic K-the…
The minimal degeneration singularities in the affine Grassmannians of simple simply-laced algebraic groups are determined to be either Kleinian singularities of type A, or closures of minimal orbits in nilpotent cones. The singularities for non-simply-laced types are studied by intersection cohomology and equivariant Chow group methods.
A finite subgroup of ${\rm SL}_2(\CC)$ defines a (Kleinian) rational surface singularity. The McKay correspondence yields a relation between the Poincaré series of the algebra of invariants of such a group and the characteristic polynomials of certain Coxeter elements determined by the corresponding singularity. Here we consider some non-abelian finite subgroups $G$ of ${\rm SL}_3(\CC)$. They define non-isolated three-dimensional Gorenstein quotient singularities. We consider suitable hyperplane sections of such singularities which are Kleinian or Fuchsian surface singularities. We show that w…
We present a faster method to determine all singularities of quiver moduli spaces up to smooth equivalence. We show that every quiver controls a large family of noncommutative compact manifolds.
This thesis gives a complete description of the Grothendieck group and divisor class group for large families of two and three dimensional singularities. The main results presented throughout, and summarised in Theorem 8.1.1, give an explicit description of the Grothendieck group and class group of Kleinian singularities, their deformations, and compound Du Val (cDV) singularities in a variety of settings. For such rings R, the main results assert that there exists an isomorphism between $G_0(R)$ and $\mathbb{Z} \oplus \mathrm{Cl}(R)$, and the class group is explicitly presented. More precisel…
We construct Knörrer type equivalences outside of the hypersurface case, namely, between singularity categories of cyclic quotient surface singularities and certain finite dimensional local algebras. This generalises Knörrer's equivalence for singularities of Dynkin type A (between Krull dimensions $2$ and $0$) and yields many new equivalences between singularity categories of finite dimensional algebras. Our construction uses noncommutative resolutions of singularities, relative singularity categories, and an idea of Hille & Ploog yielding strongly quasi-hereditary algebras which we describe …
For a large class of good moduli spaces $X$ of symmetric stacks $\mathcal{X}$, we define noncommutative motives $\mathbb{D}^{\text{nc}}(X)$ which can be regarded as categorifications of the intersection cohomology of $X$. These motives are summands of noncommutative resolutions of singularities $\mathbb{D}(X)\subset D^b(\mathcal{X})$ of $X$. The category $\mathbb{D}(X)$ is a global analogue of the noncommutative resolutions of singularities of $V//G$ for $V$ a symmetric representation of a reductive group $G$ constructed by Špenko-Van den Bergh.
Tilting objects play a key role in the study of triangulated categories. A famous result due to Iyama and Takahashi asserts that the stable categories of graded maximal Cohen-Macaulay modules over quotient singularities have tilting objects. This paper proves a noncommutative generalization of Iyama and Takahashi's theorem using noncommutative algebraic geometry. Namely, if $S$ is a noetherian AS-regular Koszul algebra and $G$ is a finite group acting on $S$ such that $S^G$ is a "Gorenstein isolated singularity", then the stable category ${\underline {\operatorname {CM}}}^{\Bbb Z}(S^G)$ of gra…
We study the relationship between singularity categories and relative singularity categories and discuss constructions of differential graded algebras of relative singularity categories. As consequences, we obtain structural results, which are known or generalise known results, on singularity categories of algebras with radical square zero, of non-commutative deformations of Kleinian singularities, of $SL_3(\mathbb{C})$-quotient singularities and of Gorenstein toric threefolds.
In this paper, we introduce a notion of ampleness of a group action $G$ on a right noetherian graded algebra $A$, and show that it is strongly related to the notion of $A^G$ to be a graded isolated singularity introduced by the second author of this paper. Moreover, if $S$ is a noetherian AS-regular algebra and $G$ is a finite ample group acting on $S$, then we will show that ${\mathcal D}^b(\operatorname{tails} S^G)\cong {\cal D}^b(\operatorname{mod} \nabla S*G)$ where $\nabla S$ is the Beilinson algebra of $S$. We will also explicitly calculate a quiver $Q_{S, G}$ such that ${\mathcal D}^b(\…
This is an announcement of conjectures and results concerning the generating series of Euler characteristics of Hilbert schemes of points on surfaces with simple (Kleinian) singularities. For a quotient surface C^2/G with G a finite subgroup of SL(2, C), we conjecture a formula for this generating series in terms of Lie-theoretic data, which is compatible with existing results for type A singularities. We announce a proof of our conjecture for singularities of type D. The generating series in our conjecture can be seen as a specialized character of the basic representation of the corresponding…
We relate completions of infinitesimal Hecke algebras of sl2 to noncommutative deformations of Kleinian singularities of type D of Crawley-Boevey and Holland. As a consequence, we show an analogue of the inequality of Bernstein and simplicity of generic maximal primitive quotients of these algebras. We also establish Skryabin type equivalence for these algebras.
We establish a connection between the representation theory of certain noncommutative singular varieties and two-dimensional lattice models. Specifically, we consider noncommutative biparametric deformations of the fiber product of two Kleinian singularities of type $A$. Special examples are closely related to Lie-Heisenberg algebras, the affine Lie algebra $A_1^{(1)}$, and a finite W-algebra associated to $\mathfrak{sl}_4$. The algebras depend on two scalars and two polynomials that must satisfy the Mazorchuk-Turowska Equation (MTE), which we re-interpret as a quantization of the ice rule (lo…
We show that the algebras describing blocks of the category of cuspidal weight (respectively generalized weight) $\mathfrak{sl}_n$-modules are one-parameter (respectively multi-parameter) deformations of certain Brauer tree algebras. We explicitly determine these deformations both graded and ungraded. The algebras we deform also appear as special centralizer subalgebras of Temperley-Lieb algebras or as generalized Khovanov algebras. They show up in the context of highest weight representations of the Virasoro algebra, in the context of rational representations of the general linear group and S…
We introduce new mathematical aspects of the Bell states using matrix factorizations, nonnoetherian singularities, and noncommutative blowups. A matrix factorization of a polynomial $p$ consists of two matrices $ϕ_1,ϕ_2$ such that $ϕ_1ϕ_2 = ϕ_2ϕ_1 = p \operatorname{id}$. Using this notion, we show how the Bell states emerge from the separable product of two mixtures, by defining pure states over complex matrices rather than just the complex numbers. We then show in an idealized algebraic setting that pure states are supported on nonnoetherian singularities. Moreover, we find that the collapse …
This note discusses recent new approaches to studying flopping curves on 3-folds. In a joint paper, the author and Michael Wemyss introduced a 3-fold invariant, the contraction algebra, which may be associated to such curves. It characterises their geometric and homological properties in a unified manner, using the theory of noncommutative deformations. Toda has now clarified the enumerative significance of the contraction algebra for flopping curves, calculating its dimension in terms of Gopakumar-Vafa invariants. Before reviewing these results, and others, I give a brief introduction to the …
Higgs Branch and VOA of 4d $\mathcal{N}=2$ SCFTs from IIB
School of Physics, Peking University（北京大学物理学院）；Center for High Energy Physics, Peking University（北京大学高能物理中心）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
We prove that the functor of noncommutative deformations of every flipping or flopping irreducible rational curve in a 3-fold is representable, and hence associate to every such curve a noncommutative deformation algebra. This new invariant extends and unifies known invariants for flopping curves in 3-folds, such as the width of Reid, and the bidegree of the normal bundle. It also applies in the settings of flips and singular schemes. We show that the noncommutative deformation algebra is finite dimensional, and give a new way of obtaining the commutative deformations of the curve, allowing us…
This paper is based on a talk at the conference `The McKay correspondence, mutation and related topics' from July 2020. We provide an introduction to joint work of the author with Søren Gammelgaard, Ádám Gyenge and Balázs Szendrői that constructs the reduced scheme underlying the Hilbert scheme of $n$ points on an ADE singularity as a Nakajima quiver variety for a particular stability parameter. After drawing a parallel with two well-known constructions of the Hilbert scheme of $n$ points in $\mathbb{A}^2$, we summarise results of the author and Gwyn Bellamy before describing the main result b…
In this article, we prove that for a finite quiver $Q$ the equivalence class of a potential up to formal change of variables of the complete path algebra $\widehat{\mathbb{C} Q}$, is determined by its Jacobi algebra together with the class in its 0-th Hochschild homology represented by the potential assuming the Jacobi algebra is finite dimensional. This is an noncommutative analogue of the famous theorem of Mather and Yau on isolated hypersurface singularities. We also prove that the right equivalence class of a potential is determined by its sufficiently high jet assuming the Jacobi algebra …
We investigate the structure of certain protected operator algebras that arise in three-dimensional N=4 superconformal field theories. We find that these algebras can be understood as a quantization of (either of) the half-BPS chiral ring(s). An important feature of this quantization is that it has a preferred basis in which the structure constants of the quantum algebra are equal to the OPE coefficients of the underlying superconformal theory. We identify several nontrivial conditions that the quantum algebra must satisfy in this basis. We consider examples of theories for which the moduli sp…
For a finite subgroup $Γ\subset \mathrm{SL}(2,\mathbb{C})$ and for $n\geq 1$, we use variation of GIT quotient for Nakajima quiver varieties to study the birational geometry of the Hilbert scheme of $n$ points on the minimal resolution $S$ of the Kleinian singularity $\mathbb{C}^2/Γ$. It is well known that $X:=\mathrm{Hilb}^{[n]}(S)$ is a projective, crepant resolution of the symplectic singularity $\mathbb{C}^{2n}/Γ_n$, where $Γ_n=Γ\wr\mathfrak{S}_n$ is the wreath product. We prove that every projective, crepant resolution of $\mathbb{C}^{2n}/Γ_n$ can be realised as the fine moduli space of $…
Let $R$ be an isolated Gorenstein singularity with a non-commutative resolution $A=End_R(R\oplus M)$. In this paper, we show that the relative singularity category $Δ_R(A)$ of $A$ has a number of pleasant properties, such as being Hom-finite. Moreover, it determines the classical singularity category $D_{sg}(R)$ of Buchweitz and Orlov as a certain canonical quotient category. If $R$ has finite CM type, which includes for example Kleinian singularities, then we show the much more surprising result that $D_{sg}(R)$ determines $Δ_R(Aus(R))$, where $Aus(R)$ is the corresponding Auslander algebra. …
