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238
Homological Algebra of Mirror Symmetry
 in Proceedings of the International Congress of Mathematicians
, 1994
"... Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3dimensional CalabiYau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual Ca ..."
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Cited by 529 (3 self)
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Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3dimensional CalabiYau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual CalabiYau manifolds V, W of dimension n (not necessarily equal to 3) one has dim H p (V, Ω q) = dim H n−p (W, Ω q). Physicists conjectured that conformal field theories associated with mirror varieties are equivalent. Mathematically, MS is considered now as a relation between numbers of rational curves on such a manifold and Taylor coefficients of periods of Hodge structures considered as functions on the moduli space of complex structures on a mirror manifold. Recently it has been realized that one can make predictions for numbers of curves of positive genera and also on CalabiYau manifolds of arbitrary dimensions. We will not describe here the complicated history of the subject and will not mention many beautiful contsructions, examples and conjectures motivated
Feynman diagrams and lowdimensional topology
, 2006
"... We shall describe a program here relating Feynman diagrams, topology of manifolds, homotopical algebra, noncommutative geometry and several kinds of “topological physics”. The text below consists of 3 parts. The first two parts (topological sigma model and ChernSimons theory) are formally independ ..."
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Cited by 237 (3 self)
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We shall describe a program here relating Feynman diagrams, topology of manifolds, homotopical algebra, noncommutative geometry and several kinds of “topological physics”. The text below consists of 3 parts. The first two parts (topological sigma model and ChernSimons theory) are formally independent and could be read separately. The third part describes the common algebraic background of both theories.
CalabiYau algebras
"... Abstract. We introduce some new algebraic structures arising naturally in the geometry of CY manifolds and mirror symmetry. We give a universal construction of CY algebras in terms of a noncommutative symplectic DG algebra resolution. In dimension 3, the resolution is determined by a noncommutative ..."
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Cited by 156 (1 self)
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Abstract. We introduce some new algebraic structures arising naturally in the geometry of CY manifolds and mirror symmetry. We give a universal construction of CY algebras in terms of a noncommutative symplectic DG algebra resolution. In dimension 3, the resolution is determined by a noncommutative potential. Representation varieties of the CY algebra are intimately related to the set of critical points, and to the sheaf of vanishing cycles of the potential. Numerical invariants, like ranks of cyclic homology groups, are expected to be given by ‘matrix integrals ’ over representation varieties. We discuss examples of CY algebras involving quivers, 3dimensional McKay correspondence, crepant resolutions, Sklyanin algebras, hyperbolic 3manifolds and ChernSimons. Examples related to quantum Del Pezzo surfaces are discussed in [EtGi].
Modular Operads
 COMPOSITIO MATH
, 1994
"... We develop a "higher genus" analogue of operads, which we call modular operads, in which graphs replace trees in the definition. We study a functor F on the category of modular operads, the Feynman transform, which generalizes Kontsevich's graph complexes and also the bar constructi ..."
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Cited by 103 (5 self)
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We develop a "higher genus" analogue of operads, which we call modular operads, in which graphs replace trees in the definition. We study a functor F on the category of modular operads, the Feynman transform, which generalizes Kontsevich's graph complexes and also the bar construction for operads. We calculate the Euler characteristic of the Feynman transform, using the theory of symmetric functions: our formula is modelled on Wick's theorem. We give applications to the theory of moduli spaces of pointed algebraic curves.
RozanskyWitten invariants via Atiyah classes
 Compositio Math
, 1999
"... Recently, L.Rozansky and E.Witten [RW] associated to any hyperKähler manifold X an invariant of topological 3manifolds. In fact, their construction gives a system of weights cΓ(X) associated to 3valent graphs Γ and the corresponding invariant of a 3manifold Y is obtained as the sum ∑ cΓ(X)IΓ(Y) ..."
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Cited by 81 (1 self)
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Recently, L.Rozansky and E.Witten [RW] associated to any hyperKähler manifold X an invariant of topological 3manifolds. In fact, their construction gives a system of weights cΓ(X) associated to 3valent graphs Γ and the corresponding invariant of a 3manifold Y is obtained as the sum ∑ cΓ(X)IΓ(Y) where IΓ(Y) is the standard integral of the product of linking forms. So the new ingredient is the system of invariants cΓ(X) of hyperKähler manifolds X, one for each trivalent graph Γ. They are obtained from the Riemannian curvature of the hyperKähler metric. In this paper we give a reformulation of the cΓ(X) in simple cohomological terms which involve only the underlying holomorphic symplectic manifold. The idea is that we can replace the curvature by the Atiyah class [At] which is the cohomological obstruction to the existence of a global holomorphic connection. The role of what in [RW] is called “Bianchi identities in hyperKähler geometry ” is played here by an identity for the square of the Atyiah class expressing the existence of the fiber bundle of second order jets. The analogy between the curvature and the structure constants of a Lie algebra observed in [RW] in fact holds even without any symplectic structure, and we study the
Noncommutative symplectic geometry, quiver varieties, and operads
"... to Liza Quiver varieties have recently appeared in various different areas of Mathematics such as representation theory of KacMoody algebras and quantum groups, instantons on 4manifolds, and resolutions Kleinian singularities. In this paper, we show that many important affine quiver varieties, e.g ..."
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Cited by 61 (9 self)
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to Liza Quiver varieties have recently appeared in various different areas of Mathematics such as representation theory of KacMoody algebras and quantum groups, instantons on 4manifolds, and resolutions Kleinian singularities. In this paper, we show that many important affine quiver varieties, e.g., the CalogeroMoser space, can be imbedded as coadjoint orbits in the dual of an appropriate infinite dimensional Lie algebra. In particular, there is an infinitesimally transitive action of the Lie algebra in question on the quiver variety. Our construction is based on an extension of Kontsevich’s formalism of ‘noncommutative Symplectic geometry’. We show that this formalism acquires its most adequate and natural formulation in the much more general framework of Pgeometry, a ‘noncommutative geometry ’ for an algebra over an arbitrary cyclic Koszul operad.
Noncommutative Geometry and Quiver algebras
"... We develop a new framework for noncommutative differential geometry based on double derivations. This leads to the notion of moment map and of Hamiltonian reduction in noncommutative symplectic geometry. For any smooth associative algebra B, we define its noncommutative cotangent bundle T ∗ B, which ..."
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Cited by 48 (13 self)
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We develop a new framework for noncommutative differential geometry based on double derivations. This leads to the notion of moment map and of Hamiltonian reduction in noncommutative symplectic geometry. For any smooth associative algebra B, we define its noncommutative cotangent bundle T ∗ B, which is a basic example of noncommutative symplectic manifold. Applying Hamiltonian reduction to noncommutative cotangent bundles gives an interesting class of associative algebras, Π = Π(B), that includes preprojective algebras associated with quivers. Our formalism of noncommutative Hamiltonian reduction provides the space Π/[Π,Π] with a Lie algebra structure, analogous to the Poisson bracket on the zero fiber of the moment map. In the special case where Π is the preprojective algebra associated with a quiver of nonDynkin type, we give a complete description of the Gerstenhaber algebra structure on the Hochschild cohomology of Π in terms of the Lie algebra Π/[Π,Π].
Cyclic operads and cyclic homology
 in &quot;Geometry, Topology and Physics,&quot;International
, 1995
"... The cyclic homology of associative algebras was introduced by Connes [4] and Tsygan [22] in order to extend the classical theory of the Chern character to the noncommutative setting. Recently, there has been increased interest in more general algebraic structures than associative algebras, characte ..."
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Cited by 45 (3 self)
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The cyclic homology of associative algebras was introduced by Connes [4] and Tsygan [22] in order to extend the classical theory of the Chern character to the noncommutative setting. Recently, there has been increased interest in more general algebraic structures than associative algebras, characterized by the presence of several algebraic operations. Such structures appear, for example, in homotopy theory [18], [3] and topological field theory [9]. In this paper, we extend the formalism of cyclic homology to this more general framework. This extension is only possible under certain conditions which are best explained using the concept of an operad. In this approach to universal algebra, an algebraic structure is described by giving, for each n ≥ 0, the space P(n) of all nary expressions which can be formed from the operations in the given algebraic structure, modulo the universally valid identities. Permuting the arguments of the expressions gives an action of the symmetric group Sn on P(n). The sequence P = {P(n)} of these Snmodules, together with the natural composition structure on them, is the operad describing our class of algebras. In order to define cyclic homology for algebras over an operad P, it is necessary that P is what we call a cyclic operad: this means that the action of Sn on P(n) extends to an action of Sn+1 in a way compatible with compositions (see Section 2). Cyclic