Results 1  10
of
86
Noncommutative DonaldsonThomas theory and the conifold
, 2008
"... Given a quiver algebra A with relations defined by a superpotential, this paper defines a set of invariants of A counting framed cyclic Amodules, analogous to rank1 Donaldson–Thomas invariants of Calabi–Yau threefolds. For the special case when A is the noncommutative crepant resolution of the th ..."
Abstract

Cited by 65 (0 self)
 Add to MetaCart
Given a quiver algebra A with relations defined by a superpotential, this paper defines a set of invariants of A counting framed cyclic Amodules, analogous to rank1 Donaldson–Thomas invariants of Calabi–Yau threefolds. For the special case when A is the noncommutative crepant resolution of the threefold ordinary double point, it is proved using torus localization that the invariants count certain pyramidshaped partitionlike configurations, or equivalently infinite dimer configurations in the square dimer model with a fixed boundary condition. The resulting partition function admits an infinite product expansion, which factorizes into the rank1 Donaldson–Thomas partition functions of the commutative crepant resolution of the singularity and its flop. The different partition functions are speculatively interpreted as counting stable objects in the derived category of Amodules under different stability conditions; their relationship should then be an instance of wall crossing in the space of stability conditions on this triangulated category.
Noncommutative Counterparts of the Springer Resolution
, 2006
"... Springer resolution of the set of nilpotent elements in a semisimple Lie algebra plays a central role in geometric representation theory. A new structure on this variety has arisen in several representation theoretic constructions, such as the (local) geometric Langlands duality and modular represe ..."
Abstract

Cited by 47 (3 self)
 Add to MetaCart
Springer resolution of the set of nilpotent elements in a semisimple Lie algebra plays a central role in geometric representation theory. A new structure on this variety has arisen in several representation theoretic constructions, such as the (local) geometric Langlands duality and modular representation theory. It is also related to some algebrogeometric problems, such as the derived equivalence conjecture and description of T. Bridgeland’s space of stability conditions. The structure can be described as a noncommutative counterpart of the resolution, or as a tstructure on the derived category of the resolution. The intriguing fact that the same tstructure appears in these seemingly disparate subjects has strong technical consequences for modular representation theory.
Cluster tilting for onedimensional hypersurface singularities
 Adv. Math
"... Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete d ..."
Abstract

Cited by 39 (15 self)
 Add to MetaCart
Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological method using higher almost split sequences and results from birational geometry. We obtain a large class of 2CY tilted algebras which are finite dimensional symmetric and satisfies τ 2 = id. In particular, we compute 2CY tilted algebras for simple/minimally elliptic curve singuralities.
Cluster tilting for higher Auslander algebras
"... Abstract. The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representationfinite algebras and Auslander algebras. The nAuslanderReiten translation functor τn plays an important role in the study of ncluster tilting subcategories. We study the ca ..."
Abstract

Cited by 32 (10 self)
 Add to MetaCart
Abstract. The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representationfinite algebras and Auslander algebras. The nAuslanderReiten translation functor τn plays an important role in the study of ncluster tilting subcategories. We study the category Mn of preinjectivelike modules obtained by applying τn to injective modules repeatedly. We call a finite dimensional algebra Λ ncomplete if Mn = add M for an ncluster tilting object M. Our main result asserts that the endomorphism algebra EndΛ(M) is (n + 1)complete. This gives an inductive construction of ncomplete algebras. For example, any representationfinite hereditary algebra Λ (1) is 1complete. Hence the Auslander algebra Λ (2) of Λ (1) is 2complete. Moreover, for any n ≥ 1, we have an ncomplete algebra Λ (n) which has an ncluster tilting object M (n) such that Λ (n+1) = End Λ (n)(M (n)). We give the presentation of Λ (n) by a quiver with relations. We apply our results to construct ncluster tilting subcategories of derived categories of ncomplete algebras. Contents 1. Our results 3 1.1. ncluster tilting in module categories 4
Crystal Melting and Toric CalabiYau Manifolds
"... We construct a statistical model of crystal melting to count BPS bound states of D0 and D2 branes on a single D6 brane wrapping an arbitrary toric CalabiYau threefold. The threedimensional crystalline structure is determined by the quiver diagram and the brane tiling which characterize the low ene ..."
Abstract

Cited by 31 (6 self)
 Add to MetaCart
(Show Context)
We construct a statistical model of crystal melting to count BPS bound states of D0 and D2 branes on a single D6 brane wrapping an arbitrary toric CalabiYau threefold. The threedimensional crystalline structure is determined by the quiver diagram and the brane tiling which characterize the low energy effective theory of D branes. The crystal is composed of atoms of different colors, each of which corresponds to a node of the quiver diagram, and the chemical bond is dictated by the arrows of the quiver diagram. BPS states are constructed by removing atoms from the crystal. This generalizes the earlier results on the BPS state counting to an arbitrary noncompact toric CalabiYau manifold. We point out that a proper understanding of the relation between the topological In type IIA superstring theory, supersymmetric bound states of D branes wrapping holomorphic cycles on a CalabiYau manifold give rise to BPS particles in four dimensions. In the past few years, remarkable connections have been found between the counting of such bound states and the topological string theory:
LEFSCHETZ DECOMPOSITIONS AND CATEGORICAL RESOLUTIONS OF SINGULARITIES
, 2006
"... Abstract. Let Y be a singular algebraic variety and let ˜ Y be a resolution of singularities of Y. Assume that the exceptional locus of ˜Y over Y is an irreducible divisor ˜Z in ˜Y. For every Lefschetz decomposition of ˜ Z we construct a triangulated subcategory ˜ D ⊂ D b ( ˜ Y) which gives a desin ..."
Abstract

Cited by 26 (9 self)
 Add to MetaCart
(Show Context)
Abstract. Let Y be a singular algebraic variety and let ˜ Y be a resolution of singularities of Y. Assume that the exceptional locus of ˜Y over Y is an irreducible divisor ˜Z in ˜Y. For every Lefschetz decomposition of ˜ Z we construct a triangulated subcategory ˜ D ⊂ D b ( ˜ Y) which gives a desingularization of D b (Y). If the Lefschetz decomposition is generated by a vector bundle tilting over Y then ˜ D is a noncommutative resolution, and if the Lefschetz decomposition is rectangular, then ˜ D is a crepant resolution. 1.
Categorical resolution of singularities
"... Abstract. Building on the concept of a smooth DG algebra we define the notion of a smooth derived category. We then propose the definition of a categorical resolution of singularities. Our main examples are concerned with a categorical resolution of the derived category of quasicoherent sheaves on ..."
Abstract

Cited by 23 (6 self)
 Add to MetaCart
Abstract. Building on the concept of a smooth DG algebra we define the notion of a smooth derived category. We then propose the definition of a categorical resolution of singularities. Our main examples are concerned with a categorical resolution of the derived category of quasicoherent sheaves on a scheme. We propose two kinds of such resolutions. Contents
EXCEPTIONAL SEQUENCES OF INVERTIBLE SHEAVES ON RATIONAL SURFACES
, 2008
"... In this article we consider exceptional sequences of invertible sheaves on smooth complete rational surfaces. We show that to every such sequence one can associate a smooth complete toric surface in a canonical way. We use this structural result to prove various theorems on exceptional and strongl ..."
Abstract

Cited by 22 (3 self)
 Add to MetaCart
(Show Context)
In this article we consider exceptional sequences of invertible sheaves on smooth complete rational surfaces. We show that to every such sequence one can associate a smooth complete toric surface in a canonical way. We use this structural result to prove various theorems on exceptional and strongly exceptional sequences of invertible sheaves on rational surfaces. We construct full strongly exceptional sequences for a large class of rational surfaces. For the case of toric surfaces we give a complete classification of full strongly exceptional sequences of invertible sheaves.
Reconstruction algebras of type A
, 2007
"... Abstract. This is the second in a series of papers which give an explicit description of the reconstruction algebra as a quiver with relations; these algebras arise naturally as geometric generalizations of preprojective algebras of extended Dynkin diagrams. This paper deals with dihedral groups G = ..."
Abstract

Cited by 19 (7 self)
 Add to MetaCart
(Show Context)
Abstract. This is the second in a series of papers which give an explicit description of the reconstruction algebra as a quiver with relations; these algebras arise naturally as geometric generalizations of preprojective algebras of extended Dynkin diagrams. This paper deals with dihedral groups G =�n,q for which all special CM modules have rank one, and we show that all but four of the relations on such a reconstruction algebra are given simply as the relations arising from a reconstruction algebra of type A. As a corollary, the reconstruction algebra reduces the problem of explicitly understanding the minimal resolution (=GHilb) to the same level of difficulty as the toric case. Contents