symmetric monoidal (∞,1)-category of spectra
This is (Ginzburg, def. 3.2.3).
This appears as (Ginzburg, def. 7.1.1).
For smooth connected variety which is projective over an affine variety, let be a tilting generator, def. 3.
Then is a Calabi-Yau algebra of dimension precisely if is a Calabi-Yau manifold of dimension .
This appears as (Ginzburg, prop. 3.3.1).
Then a Calabi-Yau object in is an algebra object equipped with an -equivariant morphism
from the Hochschild homology , satisfying the condition that the composite morphism
exhibits as its own dual object .
Such an algebra object is called a Calabi-Yau algebra object.
This is (Lurie 09, example 4.2.8).
|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)|
Jacob Lurie, section 4.2 of On the Classification of Topological Field Theories (arXiv:0905.0465)