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CLUSTER ALGEBRAS, QUIVER REPRESENTATIONS AND TRIANGULATED CATEGORIES
"... Abstract. This is an introduction to some aspects of FominZelevinsky’s cluster algebras and their links with the representation theory of quivers and with CalabiYau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). I ..."
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Abstract. This is an introduction to some aspects of FominZelevinsky’s cluster algebras and their links with the representation theory of quivers and with CalabiYau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and recent results on the interpretation of mutations as derived equivalences. Contents
Infinity structure of Poincaré duality spaces
, 2006
"... We show that the complex C•X of rational simplicial chains on a compact and triangulated Poincaré duality space X of dimension d is an A ∞ coalgebra with ∞ duality. This is the structure required for an A ∞ version of the cyclic Deligne conjecture. One corollary is that the shifted Hochschild cohom ..."
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Cited by 9 (3 self)
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We show that the complex C•X of rational simplicial chains on a compact and triangulated Poincaré duality space X of dimension d is an A ∞ coalgebra with ∞ duality. This is the structure required for an A ∞ version of the cyclic Deligne conjecture. One corollary is that the shifted Hochschild cohomology HH •+d (C • X, C•X) of the cochain algebra C • X with values in C•X has a BV structure. This implies, if X is moreover simply connected, that the shifted homology H•+dLX of the free loop space admits a BV structure. An appendix by Dennis Sullivan gives a general local construction of ∞ structures.
Openclosed homotopy algebra in mathematical physics
 J. Math. Phys
"... In this paper we discuss various aspects of openclosed homotopy algebras (OCHAs) presented in our previous paper, inspired by Zwiebach’s openclosed string field theory, but that first paper concentrated on the mathematical aspects. Here we show how an OCHA is obtained by extracting the tree part o ..."
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Cited by 9 (2 self)
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In this paper we discuss various aspects of openclosed homotopy algebras (OCHAs) presented in our previous paper, inspired by Zwiebach’s openclosed string field theory, but that first paper concentrated on the mathematical aspects. Here we show how an OCHA is obtained by extracting the tree part of Zwiebach’s quantum openclosed string field theory. We clarify the explicit relation of an OCHA with Kontsevich’s deformation quantization and with the Bmodels of homological mirror symmetry. An explicit form of the minimal model for an OCHA is given as well as its relation to the perturbative expansion of openclosed string field theory. We show that our openclosed homotopy algebra gives us a general scheme for deformation of open string structures (A∞algebras) by closed strings (L∞algebras).
Topological conformal field theories and gauge theories
, 2006
"... This paper gives a construction, using heat kernels, of differential forms on the moduli space of metrised ribbon graphs, or equivalently on the moduli space of Riemann surfaces with boundary. The construction depends on a manifold with a bundle of Frobenius algebras, satisfying various conditions. ..."
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Cited by 7 (2 self)
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This paper gives a construction, using heat kernels, of differential forms on the moduli space of metrised ribbon graphs, or equivalently on the moduli space of Riemann surfaces with boundary. The construction depends on a manifold with a bundle of Frobenius algebras, satisfying various conditions. These forms satisfy gluing conditions which mean they form an open topological conformal field theory, i.e. a kind of open string theory. If the integral of these forms converged, it would yield the purely quantum part of the partition function of a ChernSimons type gauge theory. YangMills theory on
Homological mirror symmetry on noncommutative twotori
, 2004
"... Homological mirror symmetry is a conjecture that a category constructed in the Amodel and a category constructed in the Bmodel are equivalent in some sense. We construct a cyclic differential graded (DG) category of holomorphic vector bundles on noncommutative twotori ..."
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Cited by 4 (3 self)
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Homological mirror symmetry is a conjecture that a category constructed in the Amodel and a category constructed in the Bmodel are equivalent in some sense. We construct a cyclic differential graded (DG) category of holomorphic vector bundles on noncommutative twotori
Cyclic symmetry and adic convergence in Lagrangian Floer theory
, 2009
"... Abstract. In this paper we use continuous family of multisections of the moduli space of pseudo holomorphic discs to partially improve the construction of Lagrangian Floer cohomology of [13] in the case of R coefficient. Namely we associate cyclically symmetric filtered A ∞ algebra to every relative ..."
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Abstract. In this paper we use continuous family of multisections of the moduli space of pseudo holomorphic discs to partially improve the construction of Lagrangian Floer cohomology of [13] in the case of R coefficient. Namely we associate cyclically symmetric filtered A ∞ algebra to every relatively spin Lagrangian submanifold. We use the same trick to construct a local rigid analytic family of filtered A ∞ structure associated to a (family of) Lagrangian submanifolds. We include the study of homological algebra of pseudoisotopy of cyclic (filtered) A ∞ algebra. Contents
Superpotentials, A∞ Relations and WDVV Equations for Open Topological Strings
, 2004
"... We give a systematic derivation of the consistency conditions which constrain openclosed disk amplitudes of topological strings. They include the A ∞ relations (which generalize associativity of the boundary product of topological field theory), as well as certain homotopy versions of bulkboundary ..."
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We give a systematic derivation of the consistency conditions which constrain openclosed disk amplitudes of topological strings. They include the A ∞ relations (which generalize associativity of the boundary product of topological field theory), as well as certain homotopy versions of bulkboundary crossing symmetry and Cardy constraint. We discuss integrability of amplitudes with respect to bulk and boundary deformations, and write down the analogs of WDVV equations for the spacetime superpotential. We also study the structure of these equations from a string field theory point of view. As an application, we determine the effective superpotential for certain families of Dbranes in Btwisted topological minimal models, as a function of both closed and open string moduli. This provides an exact description of tachyon condensation in such models, which allows one to determine the truncation of the open string spectrum in a simple manner.
Counting pseudoholomorphic discs in CalabiYau 3 fold, submitted
"... Abstract. In this paper we define an invariant of a pair of 6 dimensional symplectic manifold with vanishing 1st Chern class and its Lagrangian submanifold with vanishing Maslov index. This invariant is a function on the set of the path connected components of the bounding cochains (solution of A in ..."
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Abstract. In this paper we define an invariant of a pair of 6 dimensional symplectic manifold with vanishing 1st Chern class and its Lagrangian submanifold with vanishing Maslov index. This invariant is a function on the set of the path connected components of the bounding cochains (solution of A infinity version of MaurerCartan equation of the filtered A infinity algebra associated to the Lagrangian submanifold). In the case when the Lagrangian submanifold is a rational homology sphere, it becomes a numerical invariant. This invariant depends on the choice of almost complex structure. The way how it depends on the almost complex structure is described by a wall crossing formula which involves moduli space of pseudoholomorphic spheres. Contents