We study a triangulated category of graded matrix factorizations for a polynomial of type ADE. We show that it is equivalent to the derived category of finitely generated modules over the path algebra of the corresponding Dynkin quiver. Also, we discuss a special stability condition for the triangulated category in the sense of T. Bridgeland, which is naturally defined by the grading.
We discuss how matrix factorizations offer a practical method of computing the quiver and associated superpotential for a hypersurface singularity. This method also yields explicit geometrical interpretations of D-branes (i.e., quiver representations) on a resolution given in terms of Grassmannians. As an example we analyze some non-toric singularities which are resolved by a single CP1 but have "length" greater than one. These examples have a much richer structure than conifolds. A picture is proposed that relates matrix factorizations in Landau-Ginzburg theories to the way that matrix factor…
A factorization formula for certain automorphisms of a Poisson algebra associated to a quiver is proved, which involves framed versions of moduli spaces of quiver representations. This factorization formula is related to wall-crossing formulas for Donaldson-Thomas type invariants of M. Kontsevich and Y. Soibelman.
Inspired by Cachazo, Katz and Vafa (``Geometric transitions and $\mathcal {N}=1$ quiver theories'' (hep-th/0108120)), we examine representations of ``${N}=1$ quivers'' arising from string theory. We derive some mathematical consequences of the physics, and show that these results are a natural extension of Gabriel's ADE theorem. Extending the usual ADE case that relates quiver representations to curves on surfaces, we relate these new quiver representations to curves on threefolds.
We prove a family of factorization formulas for the combinatorial Donaldson--Thomas invariant for an acyclic quiver. A quantum dilogarithm identity due to Reineke, later interpreted by Rimanyi by counting codimensions of quiver loci, gives two extremal cases of our formulation in the Dynkin case. We establish our interpolating factorizations explicitly with a dimension counting argument by defining certain stratifications of the space of representations for the quiver and calculating Betti numbers in the corresponding equivariant cohomology algebras.
This paper introduces a mathematical definition of the category of D-branes in Landau-Ginzburg orbifolds in terms of $A_\infty$-categories. Our categories coincide with the categories of (graded) matrix factorizations for quasi-homogeneous polynomials. After setting up the necessary definitions, we prove that our category for the polynomial $x^{n+1}$ is equivalent to the derived category of representations of the Dynkin quiver of type $A_{n}$. We also construct a special stability condition for the triangulated category in the sense of T. Bridgeland, which should be the "origin" of the space o…
This paper classifies a class of holomorphic D-branes, closely related to framed torsion-free sheaves, on threefolds fibered in resolved ADE surfaces over a general curve C, in terms of representations with relations of a twisted Kronheimer--Nakajima-type quiver in the category Coh(C) of coherent sheaves on C. For the local Calabi--Yau case C\cong\A^1 and special choice of framing, one recovers the N=1 ADE quiver studied by Cachazo--Katz--Vafa.
We consider the moduli space of the McKay quiver representations associated to the binary polyhedral groups G < SU(2) < SU(3). The derived category of such representations is equivalent to the derived category of coherent sheaves on the corresponding ADE resolution Y=G-Hilb(C^3). Following the ideas of Nagao and Nakajima, by making particular choices of parameters in the space of stability conditions on the equivalent derived categories above, we recover Donaldson-Thomas (DT), Pandharipande-Thomas (PT) and Szendroi (NCDT) moduli spaces. We also compute the Gromov-Witten (GW) partition function…
We study preprojective algebras of graphs and their relationship to module categories over representations of quantum SL(2). As an application, ADE quiver varieties of Nakajima are shown to be subvarieties of the variety of representations of a certain associative algebra introduced by Lusztig.
Inspired by recent work of Hernandez-Leclerc and Leclerc-Plamondon we investigate the link between Nakajima's graded affine quiver varieties associated with an acyclic connected quiver Q and the derived category of Q. As Leclerc-Plamondon have shown, the points of these varieties can be interpreted as representations of a category, which we call the (singular) Nakajima category S. We determine the quiver of S and the number of minimal relations between any two given vertices. We construct a delta-functor Phi taking each finite-dimensional representation of S to an object of the derived categor…
We investigate representations of *-algebras associated with posets. Unitarizable representations of the corresponding (bound) quivers (which are polystable representations for some appropriately chosen slope function) give rise to representations of these algebras. Considering posets which correspond to unbound quivers this leads to an ADE-classification which describes the unitarization behaviour of their representations. Considering posets which correspond to bound quivers, it is possible to construct unitarizable representations starting with polystable representations of related unbound q…
We give a moduli-theoretic treatment of the existence and properties of moduli spaces of semistable quiver representations, avoiding methods from geometric invariant theory. Using the existence criteria of Alper--Halpern-Leistner--Heinloth, we show that for many stability functions, the stack of semistable representations admits an adequate moduli space, and prove that this moduli space is proper over the moduli space of semisimple representations. We construct a natural determinantal line bundle that descends to a semiample line bundle on the moduli space and provide new effective bounds for …
We introduce the notion of filtered representations of quivers, which is related to usual quiver representations, but is a systematic generalization of conjugacy classes of $n\times n$ matrices to (block) upper triangular matrices up to conjugation by invertible (block) upper triangular matrices. With this notion in mind, we describe the ring of invariant polynomials for interesting families of quivers, namely, finite $ADE$-Dynkin quivers and affine type $\widetilde{A}$-Dynkin quivers. We then study their relation to an important and fundamental object in representation theory called the Groth…
We initiate the study of (2,0) little string theory of ADE type using its definition in terms of IIB string compactified on an ADE singularity. As one application, we derive a 5d ADE quiver gauge theory that describes the little string compactified on a sphere with three full punctures, at low energies. As a second application, we show the partition function of this theory equals the 3-point conformal block of ADE Toda CFT, q-deformed. To establish this, we generalize the A_n triality of \cite{AHS} to all ADE Lie algebras; IIB string perspective is crucial for this as well.
We study the connection between quadratic Strebel differentials on punctured surfaces and the construction of moduli spaces of matrix factorizations for dimer models using GIT-quotients. We show that for each consistent dimer model and each nondegenerate stability condition $θ$ we can find a Strebel differential for which the horizontal trajectories correspond to the $θ$-stable matrix factorizations and the vertical trajectories correspond to the arrows of the dimer quiver. We give explicit expressions for the $θ$-stable matrix factorizations that can be deduced from these horizontal trajector…
Let r be an orbit of the quiver representation of type A_n (equioriented case). In this paper we study the Poincare dual of the closure of r (a.c.a. Thom polynomial/degeneracy loci formula) in equivariant cohomology. Using general Thom polynomial theory we prove the component formula and its stable version of [Knutson-Miller-Shimozono] as well as the positivity of the quiver coefficients of [Buch-Fulton]. We also show how the component formula follows from Grobner degeneration easily. In relation with the K-theory counterpart of the formula we give a new description of KMS factorizations based…
Kempf [1976] studied proper, G-equivariant maps from equivariant vector bundles over flag manifolds to G-representations V, which he called _collapsings_. We give a simple formula for the G-equivariant cohomology class on V, or_multidegree_, associated to the image of a collapsing: apply a certain sequence of divided difference operators to a certain product of linear polynomials, then divide by the number of components in a general fiber. When that number of components is 1, we construct a desingularization of the image of the collapsing. If in addition the image has rational singularities, w…
We compute the Hilbert-Kunz functions of two-dimensional rings of type ADE by using representations of their indecomposable, maximal Cohen-Macaulay modules in terms of matrix factorizations, and as first syzygy modules of homogeneous ideals.
Coulomb branches of quiver gauge theories with symmetrizers
Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo（东京大学宇宙物理与数学研究所（WPI））；Perimeter Institute for Theoretical Physics（理论物理前沿研究所）
This is a companion paper of arXiv:1601.03586. We study Coulomb branches of unframed and framed quiver gauge theories of type $ADE$. In the unframed case they are isomorphic to the moduli space of based rational maps from ${\mathbb C}P^1$ to the flag variety. In the framed case they are slices in the affine Grassmannian and their generalization. In the appendix, written jointly with Joel Kamnitzer, Ryosuke Kodera, Ben Webster, and Alex Weekes, we identify the quantized Coulomb branch with the truncated shifted Yangian.
Let us consider a specialization of an untwisted quantum affine algebra of type $ADE$ at a nonzero complex number, which may or may not be a root of unity. The Grothendieck ring of its finite dimensional representations has two bases, simple modules and standard modules. We identify entries of the transition matrix with special values of ``computable'' polynomials, similar to Kazhdan-Lusztig polynomials. At the same time we ``compute'' $q$-characters for all simple modules. The result is based on ``computations'' of Betti numbers of graded/cyclic quiver varieties. (The reason why we put `` '' …
6d superconformal field theories (SCFTs) are the SCFTs in the highest possible dimension. They can be geometrically engineered in F-theory by compactifying on non-compact elliptic Calabi-Yau manifolds. In this paper we focus on the class of SCFTs whose base geometry is determined by $-2$ curves intersecting according to ADE Dynkin diagrams and derive the corresponding mirror Calabi-Yau manifold. The mirror geometry is uniquely determined in terms of the mirror curve which has also an interpretation in terms of the Seiberg-Witten curve of the four-dimensional theory arising from torus compactif…
In this paper, we give a method for relating the generalized category $\mathcal{O}$ defined by the author and collaborators to explicit finitely presented algebras, and apply this to quiver varieties. This allows us to describe combinatorially not just the structure of these category $\mathcal{O}$'s but also how certain interesting families of derived equivalences, the shuffling and twisting functors, act on them. In the case of Nakajima quiver varieties, the algebras that appear are weighted KLR algebras and their steadied quotients, defined by the author in earlier work. In particular, these…
Let g be a simple simply laced Lie algebra. In this paper two families of varieties associated to the Dynkin graph of g are described: ``tensor product'' and ``multiplicity'' varieties. These varieties are closely related to Nakajima's quiver varieties and should play an important role in the geometric constructions of tensor products and intertwining operators. In particular it is shown that the set of irreducible components of a tensor product variety can be equipped with a structure of g-crystal isomorphic to the crystal of the canonical basis of the tensor product of several simple finite …
In this short paper we introduce a new triangulated category for rational surface singularities which in the non-Gorenstein case acts as a substitute for the stable category of matrix factorizations. The category is formed as a Frobenius quotient of the category of special CM modules, and we classify the relatively projective-injective objects and thus describe the AR quiver of the quotient. Connections to the corresponding reconstruction algebras are also discussed.
We use F. Ferrari's methods relating matrix models to Calabi-Yau spaces in order to explain Intriligator and Wecht's ADE classification of $\N=1$ superconformal theories which arise as RG fixed points of $\N = 1$ SQCD theories with adjoints. The connection between matrix models and $\N = 1$ gauge theories can be seen as evidence for the Dijkgraaf--Vafa conjecture. We find that ADE superpotentials in the Intriligator--Wecht classification exactly match matrix model superpotentials obtained from Calabi-Yau's with corresponding ADE singularities. Moreover, in the additional $\Hat{O}, \Hat{A}, \Ha…
We first prove semi-orthogonal decompositions of derived factorization categories arising from sums of potentials of gauged Landau-Ginzburg models, where the sums are not necessarily Thom--Sebastiani type. We then apply the result to the category ${\rm HMF}^{L_f}(f)$ of maximally graded matrix factorizations of an invertible polynomial $f$ of chain type, and explicitly construct a full strong exceptional collection $E_1$,..., $E_μ$ in ${\rm HMF}^{L_f}(f)$ whose length $μ$ is the Milnor number of the Berglund--Hübsch transpose $\widetilde{f}$ of $f$. This proves a conjecture, which postulates t…
We investigate the triangle singularity $f=x^a+y^b+z^c$, or $S=k[x,y,z]/(f)$, attached to a weighted projective line $X$ given by the weight triple $(a,b,c)$. We investigate the stable category of vector bundles on $X$ obtained from the vector bundles by factoring out all line bundles. This category is triangulated and has Serre duality. It is, moreover, naturally equivalent to the stable category of graded maximal Cohen-Macaulay modules over $S$ (or matrix factorizations of $f$), and then by results of Buchweitz and Orlov to the graded singularity category of $f$. We show that this category i…
In this thesis we compute the Hilbert-Kunz functions of two-dimensional rings of type ADE by using representations of their indecomposable, maximal Cohen-Macaulay modules in terms of matrix factorizations, and as first syzygy modules of homogeneous ideals. For these computations we need to compute the Hilbert-series of the involved first syzygy modules as well as the fact that the surface rings of type ADE defined over the complex numbers are the rings of invariants of C[x,y] under actions defined by the finite subgroups of SL2(C). Finally, we will show that the developed tools can be used to …
