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 non-degenerate and 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 )
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.
|2d TQFT (“TCFT”)||coefficients||algebra structure on space of quantum states|
|open topological string||Vect||Frobenius algebra||folklore+(Abrams 96)|
|open topological string with closed string bulk theory||Vect||Frobenius algebra with trace map and Cardy condition||(Lazaroiu 00, Moore-Segal 02)|
|non-compact open topological string||Ch(Vect)||Calabi-Yau A-∞ algebra||(Kontsevich 95, Costello 04)|
|non-compact open topological string with various D-branes||Ch(Vect)||Calabi-Yau A-∞ category||“|
|non-compact open topological string with various D-branes and with closed string bulk sector||Ch(Vect)||Calabi-Yau A-∞ category with Hochschild cohomology||“|
|local closed topological string||2Mod(Vect) over field||separable symmetric Frobenius algebras||(SchommerPries 11)|
|non-compact local closed topological string||2Mod(Ch(Vect))||Calabi-Yau A-∞ algebra||(Lurie 09, section 4.2)|
|non-compact local closed topological string||2Mod for a symmetric monoidal (∞,1)-category||Calabi-Yau object in||(Lurie 09, section 4.2)|
Lee Cho, Notes on Kontsevich-Soibelman’s theorem about cyclic A-infinity algebras (arXiv:1002.3653)
Jacob Lurie, section 4.2 of On the Classification of Topological Field Theories (arXiv:0905.0465)