Lie bialgebras and the cyclic homology of $A_{\infty }$ structures in topology

Abstract

$A_{\infty }$ categories are a mathematical structure that appears in topological field theory, string topology, and symplectic topology. This paper studies the cyclic homology of a Calabi-Yau $A_{\infty }$ category, and shows that it is naturally an equivariant topological conformal field theory, and in particular, contains an involutive Lie bialgebra. Applications of the theory to string topology and the Fukaya category are given; in particular, it is shown that there is a Lie bialgebra homomorphism from the cyclic cohomology of the Fukaya category of a symplectic manifold with contact type boundary to the linearized contact homology of the boundary.

arXiv comment24 pages. Added new references. Comments are welcome

References in corpus (10)

• Batalin-Vilkovisky algebras and two-dimensional topological field theories
• Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves
• Cyclic symmetry and adic convergence in LagrangianFloer theory
• Two-dimensional topological gravity and equivariant cohomology
• On the cyclic homology of ringed spaces and schemes
• Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I
• Modular operads and Batalin-Vilkovisky geometry
• Homotopy algebras and noncommutative geometry
• Noncommutative geometry and compactifications of the moduli space of curves
• Characteristic classes of $\ai$-algebras

Cited by in corpus (1)

• Cyclic Homology of Fukaya Categories and the Linearized Contact Homology