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278
Quantization of Integrable Systems and Four Dimensional Gauge Theories
, 2009
"... We study four dimensional N = 2 supersymmetric gauge theory in the Ωbackground with the two dimensional N = 2 superPoincare invariance. We explain how this gauge theory provides the quantization of the classical integrable system underlying the moduli space of vacua of the ordinary four dimension ..."
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Cited by 115 (3 self)
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We study four dimensional N = 2 supersymmetric gauge theory in the Ωbackground with the two dimensional N = 2 superPoincare invariance. We explain how this gauge theory provides the quantization of the classical integrable system underlying the moduli space of vacua of the ordinary four dimensional N = 2 theory. The εparameter of the Ωbackground is identified with the Planck constant, the twisted chiral ring maps to quantum Hamiltonians, the supersymmetric vacua are identified with Bethe states of quantum integrable systems. This four dimensional gauge theory in its low energy description has two dimensional twisted superpotential which becomes the YangYang function of the integrable system. We present the thermodynamicBetheansatz like formulae for these functions and for the spectra of commuting Hamiltonians following the direct computation in gauge theory. The general construction is illustrated at the examples of the manybody systems, such as the periodic Toda chain, the elliptic CalogeroMoser system, and their relativistic versions, for which we present a complete characterization of the L²spectrum. We very briefly discuss the quantization of Hitchin system.
CLUSTER ALGEBRAS VIA CLUSTER CATEGORIES WITH INFINITEDIMENSIONAL MORPHISM SPACES
"... Abstract. We apply our previous work on cluster characters for Hominfinite cluster categories to the theory of cluster algebras. We give a new proof of Conjectures 5.4, 6.13, 7.2, 7.10 and 7.12 of Fomin and Zelevinsky’s Cluster algebras IV for skewsymmetric cluster algebras. We also construct an e ..."
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Cited by 55 (3 self)
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Abstract. We apply our previous work on cluster characters for Hominfinite cluster categories to the theory of cluster algebras. We give a new proof of Conjectures 5.4, 6.13, 7.2, 7.10 and 7.12 of Fomin and Zelevinsky’s Cluster algebras IV for skewsymmetric cluster algebras. We also construct an explicit bijection sending certain objects of the cluster category to the decorated representations of Derksen, Weyman and Zelevinsky, and show that it is compatible with mutations in both settings. Using this map, we give a categorical interpretation of the Einvariant and show that an arbitrary decorated representation with vanishing Einvariant is characterized by its gvector. Finally, we obtain a substitution formula for cluster characters of not necessarily rigid
Wall crossing in local Calabi Yau manifolds
, 2008
"... We study the BPS states of a D6brane wrapping the conifold and bound to collections of D2 and D0 branes. We find that in addition to the complexified Kähler parameter of the rigid P 1 it is necessary to introduce an extra real parameter to describe BPS partition functions and marginal stability wa ..."
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Cited by 47 (3 self)
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We study the BPS states of a D6brane wrapping the conifold and bound to collections of D2 and D0 branes. We find that in addition to the complexified Kähler parameter of the rigid P 1 it is necessary to introduce an extra real parameter to describe BPS partition functions and marginal stability walls. The supergravity approach to BPS statecounting gives a simple derivation of results of Szendrői concerning DonaldsonThomas theory on the noncommutative conifold. This example also illustrates some interesting limitations on the supergravity approach to BPS statecounting and wallcrossing.
Motivic degree zero Donaldson–Thomas invariants
, 2009
"... We refine the degree zero Donaldson–Thomas invariants of affine threespace C³ using the virtual motive on Hilb n (C³), the Hilbert scheme of n points on C³, defined via its description as a global degeneracy locus. Assuming a conjecture about vanishing cycles, we compute the motivic generating fu ..."
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Cited by 42 (2 self)
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We refine the degree zero Donaldson–Thomas invariants of affine threespace C³ using the virtual motive on Hilb n (C³), the Hilbert scheme of n points on C³, defined via its description as a global degeneracy locus. Assuming a conjecture about vanishing cycles, we compute the motivic generating function of C³ using a calculation involving the motive of the space of pairs of commuting matrices. We express the generating function of virtual motives of Hilbert schemes of points of an arbitrary smooth and quasiprojective threefold as a motivic exponential of the generating series of motives of projective spaces, obtaining an expression resembling Göttsche’s formula for an algebraic surface. As a consequence, we derive a formula for the generating series of the corresponding weight polynomials in terms of deformed MacMahon functions.
Derevied categories of small toric CalabiYau 3folds and counting invariants
"... We first construct a derived equivalence between a small crepant resolution of an affine toric CalabiYau 3fold and a certain quiver with a superpotential. Under this derived equivalence we establish a wallcrossing formula for the generating function of the counting invariants of perverse coherent ..."
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Cited by 38 (0 self)
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We first construct a derived equivalence between a small crepant resolution of an affine toric CalabiYau 3fold and a certain quiver with a superpotential. Under this derived equivalence we establish a wallcrossing formula for the generating function of the counting invariants of perverse coherent systems. As an application we provide certain equations on DonaldsonThomas, PandeharipandeThomas and Szendroi’s invariants.
Projectivity and birational geometry of Bridgeland moduli spaces
, 2012
"... ABSTRACT. We construct a family of nef divisor classes on every moduli space of stable complexes in the sense of Bridgeland. This divisor class varies naturally with the Bridgeland stability condition. For a generic stability condition on a K3 surface, we prove that this class is ample, thereby gene ..."
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Cited by 35 (2 self)
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ABSTRACT. We construct a family of nef divisor classes on every moduli space of stable complexes in the sense of Bridgeland. This divisor class varies naturally with the Bridgeland stability condition. For a generic stability condition on a K3 surface, we prove that this class is ample, thereby generalizing a result of Minamide, Yanagida, and Yoshioka. Our result also gives a systematic explanation of the relation between wallcrossing for Bridgelandstability and the minimal model program for the moduli space. We give three applications of our method for classical moduli spaces of sheaves on a K3 surface: 1. We obtain a region in the ample cone in the moduli space of Giesekerstable sheaves only depending on the lattice of the K3. 2. We determine the nef cone of the Hilbert scheme of n points on a K3 surface of Picard rank one when n is large compared to the genus. 3. We verify the “HassettTschinkel/Huybrechts/Sawon ” conjecture on the existence of a birational Lagrangian fibration for the Hilbert scheme in a new family of cases.
MMP FOR MODULI OF SHEAVES ON K3S VIA WALLCROSSING: NEF AND MOVABLE CONES, LAGRANGIAN FIBRATIONS
"... ABSTRACT. We use wallcrossing with respect to Bridgeland stability conditions to systematically study the birational geometry of a moduli space M of stable sheaves on a K3 surface X: (a) We describe the nef cone, the movable cone, and the effective cone of M in terms of the Mukai lattice of X. (b) ..."
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Cited by 35 (2 self)
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ABSTRACT. We use wallcrossing with respect to Bridgeland stability conditions to systematically study the birational geometry of a moduli space M of stable sheaves on a K3 surface X: (a) We describe the nef cone, the movable cone, and the effective cone of M in terms of the Mukai lattice of X. (b) We establish a longstanding conjecture that predicts the existence of a birational Lagrangian fibration on M whenever M admits an integral divisor class D of square zero (with respect to the BeauvilleBogomolov form). These results are proved using a natural map from the space of Bridgeland stability conditions Stab(X) to the cone Mov(X) of movable divisors on M; this map relates wallcrossing in Stab(X) to birational transformations of M. In particular, every minimal model of M appears as a moduli space of Bridgelandstable objects on X. CONTENTS