Contents

Definition

Definition

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

• a morphism $\eta : dim(X) : ev_X \circ coev_X \to Id_x$ in $\Omega_x C$ which is equivariant with respect to the canonical action of the circle group 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, 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

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

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 $A^\vee$.

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

This is (Lurie, example 4.2.8).

Properties

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, 4.2.11.

This is closely related to the description of TCFTs (Lurie, theorem 4.2.13).

References

Section 4.2 of

Revised on July 6, 2014 06:39:45 by Urs Schreiber (192.76.8.26)