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

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

2010-02-15

arxiv.org/abs/1002.2939v3

Mathematics

Algebraic Topology

24 pages. Added new references. Comments are welcome

Scientific paper

$A_\infty$ 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.

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