FQFT and cohomology
Types of quantum field thories
For a symmetric monoidal (infinity,2)-category, a Calabi-Yau object in is
This is (Lurie 09, def. 4.2.6).
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).
A version of the cobordism hypothesis says that symmetric monoidal -functors
out of a version of the (infinity,2)-category of cobordisms where all 2-cobordisms have at least one outgoing (ingoing) boundary component, are equivalently given by their value on the point, which is a Calabi-Yau object in .
This is (Lurie 09, theorem 4.2.11).
Here the trace condition translates to the cobordism which is the “disappearance of a circle”.
Its would-be adjoint, the “appearance of a circle” is not included in .
|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)|