Quasimaps to Zero-Dimensional A∞-Quiver Varieties

2020 ◽  
Author(s):  
Hunter Dinkins ◽  
Andrey Smirnov
Keyword(s):  
Euler Characteristic ◽  
Moduli Spaces ◽  
Combinatorial Formula ◽  
Quiver Varieties ◽  
Symplectic Duality

Abstract We consider the moduli spaces of quasimaps to zero-dimensional $A_{\infty }$ Nakajima quiver varieties. An explicit combinatorial formula for the equivariant Euler characteristic of these moduli spaces is obtained and applications to symplectic duality are discussed.

scholarly journals Local BPS Invariants: Enumerative Aspects and Wall-Crossing

2018 ◽  
Vol 2020 (17) ◽  
pp. 5450-5475 ◽  
Author(s):  
Jinwon Choi ◽  
Michel van Garrel ◽  
Sheldon Katz ◽  
Nobuyoshi Takahashi
Keyword(s):  
Projective Space ◽  
Euler Characteristic ◽  
Moduli Spaces ◽  
Del Pezzo Surfaces ◽  
Arithmetic Genus ◽  
One Dimensional ◽  
Wall Crossing ◽  
Bps Invariants ◽  
Del Pezzo ◽  
Dimensional Projective Space

Abstract We study the BPS invariants for local del Pezzo surfaces, which can be obtained as the signed Euler characteristic of the moduli spaces of stable one-dimensional sheaves on the surface $S$. We calculate the Poincaré polynomials of the moduli spaces for the curve classes $\beta $ having arithmetic genus at most 2. We formulate a conjecture that these Poincaré polynomials are divisible by the Poincaré polynomials of $((-K_S).\beta -1)$-dimensional projective space. This conjecture motivates the upcoming work on log BPS numbers [8].

scholarly journals Quivers and the Euclidean algebra (Extended abstract)

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Alistair Savage
Keyword(s):  
Moduli Spaces ◽  
Euclidean Group ◽  
Quiver Varieties ◽  
Nous Permet ◽  
Preprojective Algebras ◽  
Nous Montrons ◽  
Wild Representation Type ◽  
Il Y A ◽  
International Audience ◽  
Euclidean Algebra

International audience We show that the category of representations of the Euclidean group $E(2)$ is equivalent to the category of representations of the preprojective algebra of the quiver of type $A_{\infty}$. Furthermore, we consider the moduli space of $E(2)$-modules along with a set of generators. We show that these moduli spaces are quiver varieties of the type considered by Nakajima. These identifications allow us to draw on known results about preprojective algebras and quiver varieties to prove various statements about representations of $E(2)$. In particular, we show that $E(2)$ has wild representation type but that if we impose certain combinatorial restrictions on the weight decompositions of a representation, we obtain only a finite number of indecomposable representations. Nous montrons que la catégorie des représentations du groupe d'Euclide $E(2)$ est équivalente à la catégorie des représentations de l'algèbre préprojective de type $A_{\infty}$. De plus, nous considérons l'espace classifiant de modules de $E(2)$ avec un ensemble de générateurs. Nous montrons que ces espaces sont de variétés de carquois de Nakajima. Cette identification nous permet d'utiliser des résultats des algèbres préprojectives et des variétés de carquois pour prouver des affirmations sur des représentations de $E(2)$. En particulier, nous montrons que le type de représentations de $E(2)$ est sauvage mais si nous imposons des restrictions aux poids d'une représentation, il y a seulement un nombre fini de représentations qui ne sont pas décomposables.

scholarly journals Perverse bundles and Calogero–Moser spaces

2008 ◽  
Vol 144 (6) ◽  
pp. 1403-1428 ◽  
Author(s):  
David Ben-Zvi ◽  
Thomas Nevins
Keyword(s):  
Moduli Spaces ◽  
Vector Bundles ◽  
Hilbert Schemes ◽  
Simple Description ◽  
Quiver Varieties ◽  
Complex Curves ◽  
Noncommutative Version ◽  
Cotangent Bundles ◽  
Rank One ◽  
Torsion Free

AbstractWe present a simple description of moduli spaces of torsion-free 𝒟-modules (𝒟-bundles) on general smooth complex curves, generalizing the identification of the space of ideals in the Weyl algebra with Calogero–Moser quiver varieties. Namely, we show that the moduli of 𝒟-bundles form twisted cotangent bundles to moduli of torsion sheaves on X, answering a question of Ginzburg. The corresponding (untwisted) cotangent bundles are identified with moduli of perverse vector bundles on T*X, which contain as open subsets the moduli of framed torsion-free sheaves (the Hilbert schemes T*X[n] in the rank-one case). The proof is based on the description of the derived category of 𝒟-modules on X by a noncommutative version of the Beilinson transform on P1.

scholarly journals On vector bundles over reducible curves with a node

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Filippo F. Favale ◽  
Sonia Brivio
Keyword(s):  
Irreducible Component ◽  
Euler Characteristic ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Closed Subset ◽  
Vector Bundles ◽  
Projective Bundle ◽  
Single Node ◽  
Semistable Vector Bundles

AbstractLet C be a curve with two smooth components and a single node, and let 𝓤C(w, r, χ) be the moduli space of w-semistable classes of depth one sheaves on C having rank r on both components and Euler characteristic χ. In this paper, under suitable assumptions, we produce a projective bundle over the product of the moduli spaces of semistable vector bundles of rank r on each component and we show that it is birational to an irreducible component of 𝓤C(w, r, χ). Then we prove the rationality of the closed subset containing vector bundles with given fixed determinant.

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