The notion of Calabi-Yau category is a horizontal categorification of that of Frobenius algebra – a Frobenius algebroid . Their name derives from the fact that the definition of Calabi-Yau categories have been originally studied as an abstract version of the derived category of coherent sheaves on a Calabi-Yau manifold.
to the ground field, such that for all objects the induced pairing
is symmetric and non-degenerate.
this is symmetric in that
this is cyclically invariant in that for all elements is the respective hom-complexes we have
Then is a CY -category in a naive way:
the non-binary composition maps are all trivial;
the pairing is given by Serre duality (one needs also a choice of trivialization of the canonical bundle of )
The Fukaya category associated with a symplectic manifold . But see mathoverflow for more discussion: http://mathoverflow.net/questions/13114/are-fukaya-categories-calabi-yau-categories
string topology: for a compact simply connected oriented manifold, its cohomology is naturally a Calabi-Yau -category with a single object. The structure comes from the homological perturbation lemma. One could also use the dg algebra of cochains .
Calabi-Yau -categories classify TCFTs. This remarkable result is what actually one should expect. Indeed, TCFTs originally arose as an abstract version of the CFTs constructed from sigma-models whose targets are Calabi-Yau spaces.
Cho, Lee. Notes on Kontsevich-Soibelman’s theorem about cyclic A-infinity algebras http://arxiv.org/abs/1002.3653