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55
Derived Categories of Quadric Fibrations and Intersections of Quadrics
, 2005
"... We construct a semiorthogonal decomposition of the derived category of coherent sheaves on a quadric fibration consisting of several copies of the derived category of the base of the fibration and the derived category of coherent sheaves of modules over the sheaf of even parts of the Clifford algeb ..."
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Cited by 51 (12 self)
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We construct a semiorthogonal decomposition of the derived category of coherent sheaves on a quadric fibration consisting of several copies of the derived category of the base of the fibration and the derived category of coherent sheaves of modules over the sheaf of even parts of the Clifford algebras on the base corresponding to this quadric fibration generalizing the Kapranov’s description of the derived category of a single quadric. As an application we verify that the noncommutative algebraic variety (P(S 2 W ∗), B0), where B0 is the universal sheaf of even parts of Clifford algebras, is Homologically Projectively Dual to the projective space P(W) in the double Veronese embedding P(W) → P(S 2 W). Using the properties of the Homological Projective Duality we obtain a description of the derived category of coherent sheaves on a complete intersection of any number of quadrics.
Ideal classes of the Weyl algebra and noncommutative projective geometry
, 2001
"... Let R be the set of isomorphism classes of ideals in the Weyl algebra A = A1(C), and let C be the set of isomorphism classes of triples (V, X, Y), where V is a finitedimensional (complex) vector space, and X, Y are endomorphisms of V such that [X, Y]+I has rank 1. Following a suggestion of L. Le B ..."
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Cited by 26 (3 self)
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Let R be the set of isomorphism classes of ideals in the Weyl algebra A = A1(C), and let C be the set of isomorphism classes of triples (V, X, Y), where V is a finitedimensional (complex) vector space, and X, Y are endomorphisms of V such that [X, Y]+I has rank 1. Following a suggestion of L. Le Bruyn, we define a map θ: R → C by appropriately extending an ideal of A to a sheaf over a quantum projective plane, and then using standard methods of homological algebra. We prove that θ is inverse to a bijection ω: C → R constructed in [BW] by a completely different method. The main step in the proof is to show that θ is equivariant with respect to natural actions of the group G = Aut(A) on R and C: for that we have to study also the extensions of an ideal to certain weighted quantum projective planes. Along the way, we find an elementary description of θ.
Instanton expansion of noncommutative gauge theory in two dimensions
 Commun. Math. Phys
"... We show that noncommutative gauge theory in two dimensions is an exactly solvable model. A cohomological formulation of gauge theory defined on the noncommutative torus is used to show that its quantum partition function can be written as a sum over contributions from classical solutions. We derive ..."
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Cited by 24 (6 self)
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We show that noncommutative gauge theory in two dimensions is an exactly solvable model. A cohomological formulation of gauge theory defined on the noncommutative torus is used to show that its quantum partition function can be written as a sum over contributions from classical solutions. We derive an explicit formula for the partition function of YangMills theory defined on a projective module for arbitrary noncommutativity parameter θ which is manifestly invariant under gauge Morita equivalence. The energy observables are shown to be smooth functions of θ. The construction of noncommutative instanton contributions to the path integral is described in some detail. In general, there are infinitely many gauge inequivalent contributions of fixed topological charge, along with a finite number of quantum fluctuations about each instanton. The associated moduli spaces are combinations of symmetric products of an ordinary twotorus whose orbifold singularities are not resolved by noncommutativity. In particular, the weak coupling limit of the gauge theory is independent of θ and computes the symplectic volume of the moduli space
Quantum tori, mirror symmetry and deformation
"... Mathematical models of dualities in quantum physics is a very interesting and intriguing area of research. It became clear after the work of Kontsevich (see [Ko1][Ko3]) that the “right ” framework for general duality theorems ..."
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Cited by 23 (3 self)
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Mathematical models of dualities in quantum physics is a very interesting and intriguing area of research. It became clear after the work of Kontsevich (see [Ko1][Ko3]) that the “right ” framework for general duality theorems
Antiselfdual YangMills equations on noncommutative spacetime
 J. Geom. Phys
"... By replacing the ordinary product with the so called ⋆product, one can construct an analogue of the antiselfdual YangMills (ASDYM) equations on the noncommutative R4. Many properties of the ordinary ASDYM equations turn out to be inherited by the ⋆product ASDYM equation. In particular, the twis ..."
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Cited by 22 (0 self)
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By replacing the ordinary product with the so called ⋆product, one can construct an analogue of the antiselfdual YangMills (ASDYM) equations on the noncommutative R4. Many properties of the ordinary ASDYM equations turn out to be inherited by the ⋆product ASDYM equation. In particular, the twistorial interpretation of the ordinary ASDYM equations can be extended to the noncommutative R4, from which one can also derive the fundamental strutures for integrability such as a zerocurvature representation, an associated linear system, the RiemannHilbert problem, etc. These properties are further preserved under dimensional reduction to the principal chiral field model and Hitchin’s Higgs pair equations. However, some structures relying on finite dimensional linear algebra break down in the ⋆product analogues.
Dualizing complexes and perverse sheaves on noncommutative ringed schemes
, 2002
"... Let (X, A) be a separated differential quasicoherent ringed scheme of finite type over a field k. We prove that there exists a rigid dualizing complex over A. The proof consists of two main parts. In the algebraic part we study differential filtrations on rings, and use the results obtained to sh ..."
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Cited by 22 (9 self)
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Let (X, A) be a separated differential quasicoherent ringed scheme of finite type over a field k. We prove that there exists a rigid dualizing complex over A. The proof consists of two main parts. In the algebraic part we study differential filtrations on rings, and use the results obtained to show that a rigid dualizing complex exists on every affine open set in X. In the geometric part of the proof we construct a perverse tstructure on the derived category of bimodules, and this allows us to glue the affine rigid dualizing complexes to
Sklyanin algebras and Hilbert schemes of points
"... We construct projective moduli spaces for torsionfree sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual (commutative) projective ..."
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Cited by 20 (2 self)
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We construct projective moduli spaces for torsionfree sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual (commutative) projective plane P 2. The generic noncommutative plane corresponds to the Sklyanin algebra S = Skl(E, σ) constructed from an automorphism σ of infinite order on an elliptic curve E ⊂ P 2. In this case, the fine moduli space of line bundles over S with first Chern class zero and Euler characteristic 1 − n provides a symplectic variety that is a deformation of the Hilbert scheme of n points on P² \ E.
Wilson’s Grassmannian and a noncommutative quadric
"... Let the group µ m of mth roots of unity act on the complex line by multiplication, inducing an action on Diff, the algebra of polynomial differential operators on the line. Following CrawleyBoevey and Holland, we introduce a multiparameter deformation, Dτ, of the smashproduct Diff#µ m. Our main re ..."
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Cited by 18 (2 self)
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Let the group µ m of mth roots of unity act on the complex line by multiplication, inducing an action on Diff, the algebra of polynomial differential operators on the line. Following CrawleyBoevey and Holland, we introduce a multiparameter deformation, Dτ, of the smashproduct Diff#µ m. Our main result provides natural bijections between (roughly speaking) the following spaces: (1) µ mequivariant version of Wilson’s adelic Grassmannian of rank r; (2) Rank r projective Dτmodules; (3) Rank r torsionfree sheaves on a ‘noncommutative quadric ’ P 1 ×τ P 1; (4) Disjoint union of Nakajima quiver varieties for the cyclic quiver with m vertices. The bijection between (1) and (2) is provided by a version of RiemannHilbert correspondence between Dmodules and sheaves. The bijections between (2), (3) and (4) were motivated by our previous work [BGK]. The resulting bijection between (1) and (4) reduces, in the very special case: r = 1 and µ m = {1} to the partition of (rank 1) adelic Grassmannian into a union of CalogeroMoser spaces, discovered by Wilson. This gives, in particular, a natural
Geometric construction of representations of affine algebras
 Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), 423–438, Higher Ed
, 2002
"... Let Γ be a finite subgroup of SL2(C). We consider Γfixed point sets in Hilbert schemes of points on the affine plane C 2. The direct sum of homology groups of components has a structure of a representation of the affine Lie algebra ̂g corresponding to Γ. If we replace homology groups by equivariant ..."
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Cited by 15 (1 self)
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Let Γ be a finite subgroup of SL2(C). We consider Γfixed point sets in Hilbert schemes of points on the affine plane C 2. The direct sum of homology groups of components has a structure of a representation of the affine Lie algebra ̂g corresponding to Γ. If we replace homology groups by equivariant Khomology groups, we get a representation of the quantum toroidal algebra Uq(L̂g). We also discuss a higher rank generalization and character formulas in terms of intersection homology groups.