Lie bialgebras and the cyclic homology of A_∞ structures in topology

2010/02/15 by Xiaojun Chen, Chen, Xiaojun
Mathematics · Physics and Astronomy · #Algebraic Topology (math.AT) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Symplectic Geometry (math.SG) #math-ph #math.AT #math.MP #math.SG

paper · pdf · doi:10.48550/arxiv.1002.2939

24 pages. Added new references. Comments are welcome

arxiv created 2010/04/22 · arxiv updated 2010/04/23

Abstract

A_∞ 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_∞ 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.

Related

• Cyclic Homology of Fukaya Categories and the Linearized Contact Homology
• Topological conformal field theories and Calabi-Yau categories
• Algebraic string topology from the neighborhood of infinity
• On the structure of open equivariant topological conformal field theories
• Symplectic A_∞-algebras and string topology operations