nLab Calabi-Yau object

Idea

See at cobordism hypothesis – For non-compact cobordisms.

Definition

Definition

For $C$ a symmetric monoidal (infinity,2)-category, a Calabi-Yau object in $C$ is

• a dualizable object $X$;

• a morphism $\eta \,\colon\, dim(X) \equiv ev_X \circ coev_X \longrightarrpw Id_x$ in $\Omega_x C$ which is equivariant with respect to the canonical ∞-action of the circle group $SO(2)$ on $dim(X)$ and which is the counit for an adjunction between the evaluation map $ev_X$ and coevaluation map $coev_X$.

This is (Lurie 09, def. 4.2.6).

Examples

Calabi-Yau algebras

Example

Let $\mathbf{S}$ be a good symmetric monoidal (∞,1)-category. Write $Alg(\mathbf{S})$ for the symmetric monoidal (∞,2)-category whose objects are algebra objects in $\mathbf{S}$ and whose morphisms are bimodule objects.

Then a Calabi-Yau object in $Alg(\mathbf{S})$ is an algebra object $A$ equipped with an $SO(2)$-equivariant morphism

$$tr \colon \int_{S^1} A \to 1$$

from the Hochschild homology $\int_{S^1} A \simeq A \otimes_{A \otimes A} A$, satisfying the condition that the composite morphism

$$A \otimes A \simeq \int_{S^0} A \to \int_{S^1} A \stackrel{tr}{\to} 1$$

exhibits $A$ as its own dual object $A^\vee$.

Such an algebra object is called a Calabi-Yau algebra object.

This is (Lurie 09, example 4.2.8).

Properties

Relation to extended 2d TQFT (TCFT) and the Cobordism hypothesis

A version of the cobordism hypothesis says that symmetric monoidal $(\infty,2)$-functors

$$Z : Bord_2^{nc} \to \mathcal{C}$$

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 $\mathcal{C}$.

This is (Lurie 09, theorem 4.2.11).

Here the trace condition translates to the cobordism which is the “disappearance of a circle”.

$$\array{ && \longleftarrow \\ & \swarrow && \nwarrow \\ & \searrow && \nearrow \\ && \longrightarrow } \;\;\;\;\;\; \Rightarrow \;\;\;\;\;\; \ast$$

Its would-be adjoint, the “appearance of a circle” is not included in $Bord_2^{nc}$.

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${}_k$	Frobenius algebra $A$	folklore+(Abrams 96)
open topological string with closed string bulk theory	Vect${}_k$	Frobenius algebra $A$ with trace map $B \to Z(A)$ 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${}_k$) over field $k$	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$(\mathbf{S})$ for a symmetric monoidal (∞,1)-category $\mathbf{S}$	Calabi-Yau object in $\mathbf{S}$	(Lurie 09, section 4.2)

Related concepts

• Calabi-Yau category

• 3d Calabi-Yau object

References

• Jacob Lurie, section 4.2 of On the Classification of Topological Field Theories (arXiv:0905.0465)