nLab
Calabi-Yau object

Definition

Definition

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

• a dualizable object;

• a morphism $\eta :\dim (X):{\mathrm{ev}}_X\circ {\mathrm{circoev}}_X\to {\mathrm{Id}}_x$ 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 ${\mathrm{ev}}_X$ and ${\mathrm{coev}}_X$.

This is (Lurie, def. 4.2.6).

Examples

Calabi-Yau algebras

Example

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

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

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

satisfying the condition that the composite morphism

$$A\otimes A\simeq {\int }_{S^0}A\to {\int }_{S^1}A\stackrel{\mathrm{tr}}{\to }1$$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 $(\mathrm{\infty },2)$-functors

$$Z:{\mathrm{Bord}}_2^{\mathrm{nc}}\to 𝒞$$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 $𝒞$.

This is Lurie, 4.2.11.

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

Related concepts

• Calabi-Yau category

References

Section 4.2 of

• Jacob Lurie, On the Classification of Topological Field Theories