Torsion-free Sheaves Research Articles

Branes in the moduli space of framed sheaves

In the physicist's language, a brane in a hyperkähler manifold is a submanifold which is either complex or lagrangian with respect to three Kähler structures of the ambient manifold. By considering the fixed loci of certain involutions, we describe branes in Nakajima quiver varieties of all possible types. We then focus on the moduli space of framed torsion free sheaves on the projective plane, showing how the involutions considered act on sheaves, and proving the existence of branes in some cases.

Wall crossing for derived categories of moduli spaces of sheaves on rational surfaces

We remove the global quotient presentation input in the theory of windows in derived categories of smooth Artin stacks of finite type. As an application, we use existing results on flipping of strata for wall-crossing of Gieseker semi-stable torsion-free sheaves of rank two on rational surfaces to produce semi-orthogonal decompositions relating the different moduli stacks. The complementary pieces of these semi-orthogonal decompositions are derived categories of products of Hilbert schemes of points on the surface.

Rank 2 Sheaves on Toric 3-Folds: Classical and Virtual Counts

Let $\mathcal{M}$ be the moduli space of rank 2 stable torsion free sheaves with Chern classes $c_i$ on a smooth 3-fold $X$. When $X$ is toric with torus $T$, we describe the $T$-fixed locus of the moduli space. Connected components of $\mathcal{M}^T$ with constant reflexive hulls are isomorphic to products of $\mathbb{P}^1$. We mainly consider such connected components, which typically arise for any $c_1$, "low values" of $c_2$, and arbitrary $c_3$. In the classical part of the paper, we introduce a new type of combinatorics called double box configurations, which can be used to compute the generating function $\mathsf{Z}(q)$ of topological Euler characteristics of $\mathcal{M}$ (summing over all $c_3$). The combinatorics is solved using the double dimer model in a companion paper. This leads to explicit formulae for $\mathsf{Z}(q)$ involving the MacMahon function. In the virtual part of the paper, we define Donaldson-Thomas type invariants of toric Calabi-Yau 3-folds by virtual localization. The contribution to the invariant of an individual connected component of the $T$-fixed locus is in general not equal to its signed Euler characteristic due to $T$-fixed obstructions. Nevertheless, the generating function of all invariants is given by $\mathsf{Z}(q)$ up to signs.

BPS/CFT correspondence II: Instantons at crossroads, moduli and compactness theorem

Gieseker-Nakajima moduli spaces $M_{k}(n)$ parametrize the charge $k$ noncommutative $U(n)$ instantons on ${\bf R}^{4}$ and framed rank $n$ torsion free sheaves $\mathcal{E}$ on ${\bf C\bf P}^{2}$ with ${\rm ch}_{2}({\mathcal{E}}) = k$. They also serve as local models of the moduli spaces of instantons on general four-manifolds. We study the generalization of gauge theory in which the four dimensional spacetime is a stratified space $X$ immersed into a Calabi-Yau fourfold $Z$. The local model ${\bf M}_{k}({\vec n})$ of the corresponding instanton moduli space is the moduli space of charge $k$ (noncommutative) instantons on origami spacetimes. There, $X$ is modelled on a union of (up to six) coordinate complex planes ${\bf C}^{2}$ intersecting in $Z$ modelled on ${\bf C}^{4}$. The instantons are shared by the collection of four dimensional gauge theories sewn along two dimensional defect surfaces and defect points. We also define several quiver versions ${\bf M}_{\bf k}^{\gamma}({\vec{\bf n}})$ of ${\bf M}_{k}({\vec n})$, motivated by the considerations of sewn gauge theories on orbifolds ${\bf C}^{4}/{\Gamma}$. The geometry of the spaces ${\bf M}_{\bf k}^{\gamma}({\vec{\bf n}})$, more specifically the compactness of the set of torus-fixed points, for various tori, underlies the non-perturbative Dyson-Schwinger identities recently found to be satisfied by the correlation functions of $qq$-characters viewed as local gauge invariant operators in the ${\mathcal{N}}=2$ quiver gauge theories. The cohomological and K-theoretic operations defined using ${\bf M}_{k}({\vec n})$ and their quiver versions as correspondences provide the geometric counterpart of the $qq$-characters, line and surface defects.

Hilbert schemes of points of [formula omitted] as quiver varieties

In a previous paper, a realization of the moduli space of framed torsion-free sheaves on Hirzebruch surfaces in terms of monads was given. We build upon that result to construct ADHM data for the Hilbert scheme of points of the total space of the line bundles O(−n) on P1, for n≥1, i.e., the resolutions of the singularities of type 1n(1,1). Basically by implementing a version of the special McKay correspondence, this ADHM description is in turn used to realize these Hilbert schemes as irreducible connected components of quiver varieties. We obtain in this way new examples of quiver varieties which are not of the Nakajima type.

Stable reflexive sheaves and localization

We study moduli spaces N of rank 2 stable reflexive sheaves on P3. Fixing Chern classes c1, c2, and summing over c3, we consider the generating function Zrefl(q) of Euler characteristics of such moduli spaces. The action of the torus T on P3 lifts to N and we classify all sheaves in NT. This leads to an explicit expression for Zrefl(q). Since c3 is bounded below and above, Zrefl(q) is a polynomial. We find a simple formula for its leading term when c1=−1.Next, we study moduli spaces of rank 2 stable torsion free sheaves on P3 and consider the generating function of Euler characteristics of such moduli spaces. We give an expression for this generating function in terms of Zrefl(q) and Euler characteristics of Quot schemes of certain T-equivariant reflexive sheaves, which are studied elsewhere. Many techniques of this paper apply to any toric 3-fold. In general, Zrefl(q) depends on the choice of polarization which leads to wall-crossing phenomena. We briefly illustrate this in the case of P2×P1.

On the rationality of Nagaraj–Seshadri moduli space

We show that each of the irreducible components of moduli of rank 2 torsion-free sheaves with odd Euler characteristic over a reducible nodal curve is rational.

On a relative Mumford–Newstead theorem

In this paper, we prove a relative version of the classical Mumford–Newstead theorem for a family of smooth curves degenerating to a reducible curve with a simple node. We also prove a Torelli-type theorem by showing that certain moduli spaces of torsion free sheaves on a reducible curve allow us to recover the curve from the moduli space.

Some Examples of Tilt-Stable Objects on Threefolds

We investigate properties and describe examples of tilt-stable objects on a smooth complex projective threefold. We give a structure theorem on slope semistable sheaves of vanishing discriminant, and describe certain Chern classes for which every slope semistable sheaf yields a Bridgeland semistable object of maximal phase. Then, we study tilt stability as the polarization ω gets large, and give sufficient conditions for tilt-stability of sheaves of the following two forms: 1) twists of ideal sheaves or 2) torsion-free sheaves whose first Chern class is twice a minimum possible value.

Defects in cohomological gauge theory and Donaldson–Thomas invariants

Donaldson-Thomas theory on a Calabi-Yau can be described in terms of a certain sixdimensional cohomological gauge theory. We introduce a certain class of defects in this gauge theory which generalize surface defects in four dimensions. These defects are associated with divisors and are defined by prescribing certain boundary conditions for the gauge fields. We discuss generalized instanton moduli spaces when the theory is defined with a defect and propose a generalization of Donaldson-Thomas invariants. These invariants arise by studying torsion free coherent sheaves on Calabi-Yau varieties with a certain parabolic structure along a divisor, determined by the defect. We discuss the case of the affine space as a concrete example. In this case the moduli space of parabolic sheaves admits an alternative description in terms of the representation theory of a certain quiver. The latter can be used to compute the invariants explicitly via equivariant localization. We also briefly discuss extensions of our work to other higher dimensional field theories.

Framed sheaves on root stacks and supersymmetric gauge theories on ALE spaces

We develop a new approach to the study of supersymmetric gauge theories on ALE spaces using the theory of framed sheaves on root toric stacks, which illuminates relations with gauge theories on R4 and with two-dimensional conformal field theory. We construct a stacky compactification of the minimal resolution Xk of the Ak−1 toric singularity C2/Zk, which is a projective toric orbifold Xk such that Xk∖Xk is a Zk-gerbe. We construct moduli spaces of torsion free sheaves on Xk which are framed along the compactification gerbe. We prove that this moduli space is a smooth quasi-projective variety, compute its dimension, and classify its fixed points under the natural induced toric action. We use this construction to compute the partition functions and correlators of chiral BPS operators for N=2 quiver gauge theories on Xk with nontrivial holonomies at infinity. The partition functions are computed with and without couplings to bifundamental matter hypermultiplets and expressed in terms of toric blowup formulas, which relate them to the corresponding Nekrasov partition functions on the affine toric open subsets of Xk. We compare our new partition functions with previous computations, explore their connections to the representation theory of affine Lie algebras, and find new constraints on fractional instanton charges in the coupling to fundamental matter. We show that the partition functions in the low energy limit are characterized by the Seiberg–Witten curves, and in some cases also by suitable blowup equations involving Riemann theta-functions on the Seiberg–Witten curve with characteristics related to the nontrivial holonomies.

Some New Components of the Moduli Scheme MP3(2; -1; 2; 0) of Stable Coherent Torsion Free Sheaves of Rank 2 on P3

In this paper we consider Giseker-Maruyama moduli scheme M := M P 3 (2; ¡ 1 ; 2 ; 0) of stable coherent torsion free sheaves of rank 2 with Chern classes c 1 = - 1, c 2 = 2, c 3 = 0 on 3-dimensional projective space P 3 . We will de¯ne two sets of sheaves M 1 and M 2 in M and we will prove that closures of M 1 and M 2 in M are irreducible components of dimensions 15 and 19, accordingly.

Stable Sheave Moduli of Rank 2 with Chern Classes c 1 = -1; c2 = 2; c3 = 0 on Q3

Рассматривается схема MQ( 2;-1; 2; 0 ) модулей стабильных пучков ранга 2 без кручения с классами Черна c1 = -1; c2 = 2; c3 = 0 на гладкой трехмерной проективной квадрике Q. МногообразиеMQ(-1; 2) модулей расслоений ранга 2 с классами Черна c1 = -1, c2 = 2 на Q было изучено Оттавиани и Шуреком в 1994 г. В 2007 г. автором было получено описание замыкания многообразия MQ (-1; 2) в схеме MQ(2;-1; 2; 0). В настоящей статье доказывается, что в MQ(2;-1; 2; 0) существует единственная неприводимая компонента, отличная от MQ(-1; 2), являющаяся рациональным многообразием размерности 10.

On an Isomorphism of Compactifications of Moduli Scheme of Vector Bundles

A morphism of the reduced Gieseker - Maruyama moduli functor (of semistable coherent torsion-free sheaves) on the surface to the reduced moduli functor of admissible semistable pairs with the same Hilbert polynomial, is constructed. It is shown that main components of reduced moduli scheme for semistable admissible pairs (( e S; e L ) ; e E ) are isomorphic to main components of the reduced Gieseker - Maruyama moduli scheme.

Monads for framed sheaves on Hirzebruch surfaces

Abstract We define monads for framed torsion-free sheaves on Hirzebruch surfaces and use them to construct moduli spaces for these objects. These moduli spaces are smooth algebraic varieties, and we show that they are fine by constructing a universal monad.

$${\mathcal{N} = 2}$$ N = 2 Quiver Gauge Theories on A-type ALE Spaces

We survey and compare recent approaches to the computation of the partition functions and correlators of chiral BPS observables in $\mathcal{N}=2$ gauge theories on ALE spaces based on quiver varieties and the minimal resolution $X_k$ of the $A_{k-1}$ toric singularity $\mathbb{C}^2/\mathbb{Z}_k$, in light of their recently conjectured duality with two-dimensional coset conformal field theories. We review and elucidate the rigorous constructions of gauge theories for a particular family of ALE spaces, using their relation to the cohomology of moduli spaces of framed torsion free sheaves on a suitable orbifold compactification of $X_k$. We extend these computations to generic $\mathcal{N}=2$ superconformal quiver gauge theories, obtaining in these instances new constraints on fractional instanton charges, a rigorous proof of the Nekrasov master formula, and new quantizations of Hitchin systems based on the underlying Seiberg-Witten geometry.

Euler characteristics of moduli spaces of torsion free sheaves on toric surfaces

Given a smooth toric variety \(X\), the action of the torus \(T\) lifts to the moduli space \({\mathcal {M}}\) of stable sheaves on \(X\). Using the pioneering work of Klyacho, a fairly explicit combinatorial description of the fixed point locus \({\mathcal {M}}^T\) can be given (as shown by earlier work of the author). In this paper, we apply this description to the case of torsion free sheaves on a smooth toric surface \(S\). A general expression for the generating function of the Euler characteristics of such moduli spaces is obtained. The generating function is expressed in terms of Euler characteristics of certain moduli spaces of stable configurations of linear subspaces appearing in classical GIT. The expression holds for any choice of \(S\), polarization, rank, and first Chern class. Specializing to various examples allows us to compute some new as well as known generating functions.

Stability and Fourier–Mukai transforms on elliptic fibrations

We systematically develop Bridgeland's [7] and Bridgeland–Maciocia's [10] techniques for studying elliptic fibrations, and identify criteria that ensure 2-term complexes are mapped to torsion-free sheaves under a Fourier–Mukai transform. As an application, we construct an open immersion from a moduli of stable complexes to a moduli of Gieseker stable sheaves on elliptic threefolds. As another application, we give various 1–1 correspondences between fibrewise semistable torsion-free sheaves and codimension-1 sheaves on Weierstrass surfaces.

A note on singular moduli spaces of sheaves on K3 surfaces

This paper studies deformations and birational maps between singular moduli spaces of torsion free semistable sheaves with 2-divisible Mukai vectors on K3 surfaces. It is showed that when the greatest common divisor of the rank and the first Chern class is 2, two such moduli spaces of the same dimension can be connected by deformations and birational maps.

Flips of moduli of stable torsion free sheaves with $c_1=1$ on $P^2$

Flips of moduli of stable torsion free sheaves with $c_1=1$ on $P^2$