  $A_\infty$ 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.
