Results 11  20
of
140
Formality of canonical symplectic complexes and Frobenius manifolds
 Internat. Math. Res. Notices
, 1998
"... It is shown that the de Rham complex of a symplectic manifold M satisfying the hard Lefschetz condition is formal. Moreover, it is shown that the differential GerstenhaberBatalinVilkoviski algebra associated to such a symplectic structure gives rise, along the lines explained in the papers of Bara ..."
Abstract

Cited by 44 (3 self)
 Add to MetaCart
(Show Context)
It is shown that the de Rham complex of a symplectic manifold M satisfying the hard Lefschetz condition is formal. Moreover, it is shown that the differential GerstenhaberBatalinVilkoviski algebra associated to such a symplectic structure gives rise, along the lines explained in the papers of Barannikov and Kontsevich [alggeom/9710032] and Manin [math/9801006], to the structure of a Frobenius manifold on the de Rham cohomology of M. It was shown in [1] (see also [4] for detailed exposition and proofs) that the formal moduli space of solutions to the MaurerCartan equations modulo gauge equivalence associated to a very special class of differential GerstenhaberBatalinVilkoviski (dGBV) algebras, carries a natural structure of a Frobenius
Three constructions of Frobenius manifolds: a comparative study
"... The paper studies three classes of Frobenius manifolds: Quantum Cohomology (topological sigma–models), unfolding spaces of singularities (K. Saito’s theory, Landau–Ginzburg models), and the recent Barannikov–Kontsevich construction starting with the Dolbeault complex of a Calabi–Yau manifold and con ..."
Abstract

Cited by 43 (0 self)
 Add to MetaCart
(Show Context)
The paper studies three classes of Frobenius manifolds: Quantum Cohomology (topological sigma–models), unfolding spaces of singularities (K. Saito’s theory, Landau–Ginzburg models), and the recent Barannikov–Kontsevich construction starting with the Dolbeault complex of a Calabi–Yau manifold and conjecturally producing the B–side of the Mirror Conjecture in arbitrary dimension. Each known construction provides the relevant Frobenius manifold with an extra structure which can be thought of as a version of“non–linear cohomology”. The comparison of thesestructures sheds some light on the general Mirror Problem: establishing isomorphisms between Frobenius manifolds of different classes. Another theme is the study of tensor products of Frobenius manifolds, corresponding respectively to the Künneth formula in Quantum Cohomology, direct sum of singularities in Saito’s theory, and presumably, the tensor product of the differential Gerstenhaber–Batalin–Vilkovisky algebras. We extend the initial Gepner’s construction of mirrors to the context of Frobenius manifolds and formulate the
Semiinfinite Hodge structures and mirror symmetry for projective spaces, preprint
, 10
"... Abstract. We express total set of rational GromovWitten invariants of CP n via periods of variations of semiinfinite Hodge structure associated with their mirror partners. For this explicit example we give detailed description of general construction of solutions to WDVVequation from variations o ..."
Abstract

Cited by 40 (2 self)
 Add to MetaCart
Abstract. We express total set of rational GromovWitten invariants of CP n via periods of variations of semiinfinite Hodge structure associated with their mirror partners. For this explicit example we give detailed description of general construction of solutions to WDVVequation from variations of semiinfinite Hodge structures of CalabiYau type which was suggested in a proposition from our previous paper ([B2] proposition 6.5). Contents
On the relation between open and closed topological strings.” hepth/0405232
, 2004
"... Abstract. We discuss the relation between open and closed string correlators using topological string theories as a toy model. We propose that one can reconstruct closed string correlators from the ..."
Abstract

Cited by 40 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We discuss the relation between open and closed string correlators using topological string theories as a toy model. We propose that one can reconstruct closed string correlators from the
Topological strings in generalized complex space
, 2006
"... A twodimensional topological sigmamodel on a generalized CalabiYau target space X is defined. The model is constructed in BatalinVilkovisky formalism using only a generalized complex structure J and a pure spinor ρ on X. In the present construction the algebra of Qtransformations automatically ..."
Abstract

Cited by 37 (1 self)
 Add to MetaCart
A twodimensional topological sigmamodel on a generalized CalabiYau target space X is defined. The model is constructed in BatalinVilkovisky formalism using only a generalized complex structure J and a pure spinor ρ on X. In the present construction the algebra of Qtransformations automatically closes offshell, the model transparently depends only on J, the algebra of observables and correlation functions for topologically trivial maps in genus zero are easily defined. The extended moduli space appears naturally. The familiar action of the twisted N = 2 CFT can be recovered after a gauge fixing. In the open case, we consider an example of generalized deformation of complex structure by a holomorphic Poisson bivector β and recover holomorphic noncommutative Kontsevich ∗product.
Extended deformation functors
 Int. Math. Res. Not
"... We introduce a precise notion, in terms of some Schlessinger’s type conditions, of extended deformation functors which is compatible with most of recent ideas in the Derived Deformation Theory (DDT) program and with geometric examples. With this notion we develop the (extended) analogue of Schlessin ..."
Abstract

Cited by 29 (11 self)
 Add to MetaCart
(Show Context)
We introduce a precise notion, in terms of some Schlessinger’s type conditions, of extended deformation functors which is compatible with most of recent ideas in the Derived Deformation Theory (DDT) program and with geometric examples. With this notion we develop the (extended) analogue of Schlessinger and obstruction theories. The inverse mapping theorem holds for natural transformations of extended deformation functors and all such functors with finite dimensional tangent space are prorepresentable in the homotopy category. Finally we prove that the primary obstruction map induces a structure of graded Lie algebra on the tangent space. Mathematics Subject Classification (1991): 13D10, 14B10, 14D15.
Floer Homology and Mirror Symmetry I
, 1999
"... Abstract. In this survey article, we explain how the Floer homology of Lagrangian submanifold [Fl1],[Oh1] is related to (homological) mirror symmetry [Ko1],[Ko2]. Our discussion is based mainly on [FKO3]. 0. Introduction. This is the first of the two articles, describing a project in progress to stu ..."
Abstract

Cited by 28 (3 self)
 Add to MetaCart
(Show Context)
Abstract. In this survey article, we explain how the Floer homology of Lagrangian submanifold [Fl1],[Oh1] is related to (homological) mirror symmetry [Ko1],[Ko2]. Our discussion is based mainly on [FKO3]. 0. Introduction. This is the first of the two articles, describing a project in progress to study mirror symmetry and Dbrane using Floer homology of Lagrangian submanifold. The tentative goal, which we are far away to achiev, is to prove homological mirror symmetry conjecture by M. Kontsevich (see §3.) The final goal, which is yet very
Noncommutative homotopy algebras associated with open strings
 REV. MATH. PHYS
, 2003
"... We discuss general properties of A∞algebras and their applications to the theory of open strings. The properties of cyclicity for A∞algebras are examined in detail. We prove the decomposition theorem, which is a stronger version of the minimal model theorem, for A∞algebras and cyclic A∞algebras a ..."
Abstract

Cited by 28 (5 self)
 Add to MetaCart
(Show Context)
We discuss general properties of A∞algebras and their applications to the theory of open strings. The properties of cyclicity for A∞algebras are examined in detail. We prove the decomposition theorem, which is a stronger version of the minimal model theorem, for A∞algebras and cyclic A∞algebras and discuss various consequences of it. In particular it is applied to classical open string field theories and it is shown that all classical open string field theories on a fixed conformal background are cyclic A∞isomorphic to each other. The same results hold for classical closed string field theories, whose algebraic structure is governed by cyclic L∞algebras.
Spaces of stability conditions
"... Abstract. Stability conditions are a mathematical way to understand Πstability for Dbranes in string theory. Spaces of stability conditions seem to be related to moduli spaces of conformal field theories. This is a survey article describing what is currently known about spaces of stability conditi ..."
Abstract

Cited by 27 (3 self)
 Add to MetaCart
(Show Context)
Abstract. Stability conditions are a mathematical way to understand Πstability for Dbranes in string theory. Spaces of stability conditions seem to be related to moduli spaces of conformal field theories. This is a survey article describing what is currently known about spaces of stability conditions, and giving some pointers for future research. 1.