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Calabi-Yau object

Context

Functorial quantum field theory

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

See at cobordism hypothesis – For non-compact cobordisms.

Definition

Definition

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

This is (Lurie, def. 4.2.6).

Examples

Calabi-Yau algebras

Example

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

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

tr: S 1A1 tr : \int_{S^1} A \to 1

satisfying the condition that the composite morphism

AA S 0A S 1Atr1 A \otimes A \simeq \int_{S^0} A \to \int_{S^1} A \stackrel{tr}{\to} 1

exhibits AA as its own dual A 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 (,2)(\infty,2)-functors

Z:Bord 2 nc𝒞 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\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 February 16, 2014 13:52:26 by Urs Schreiber (89.204.139.247)