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See at cobordism hypothesis – For non-compact cobordisms.
For a symmetric monoidal (infinity,2)-category, a Calabi-Yau object in is
a morphism in which is equivariant with respect to the canonical ∞-action of the circle group on and which is the counit for an adjunction between the evaluation map and coevaluation map .
This is (Lurie 09, def. 4.2.6).
Let be a good symmetric monoidal (∞,1)-category. Write for the symmetric monoidal (∞,2)-category whose objects are algebra objects in and whose morphisms are bimodule objects.
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 .
This is closely related to the description of 2d TQFT as TCFTs (Lurie 09, theorem 4.2.13).
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) |
Last revised on December 19, 2023 at 18:56:30. See the history of this page for a list of all contributions to it.