Contents

# 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 $SO(2)$ 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”)coefficientsalgebra structure on space of quantum states
open topological stringVect${}_k$Frobenius algebra $A$folklore+(Abrams 96)
open topological string with closed string bulk theoryVect${}_k$Frobenius algebra $A$ with trace map $B \to Z(A)$ and Cardy condition(Lazaroiu 00, Moore-Segal 02)
non-compact open topological stringCh(Vect)Calabi-Yau A-∞ algebra(Kontsevich 95, Costello 04)
non-compact open topological string with various D-branesCh(Vect)Calabi-Yau A-∞ category
non-compact open topological string with various D-branes and with closed string bulk sectorCh(Vect)Calabi-Yau A-∞ category with Hochschild cohomology
local closed topological string2Mod(Vect${}_k$) over field $k$separable symmetric Frobenius algebras(SchommerPries 11)
non-compact local closed topological string2Mod(Ch(Vect))Calabi-Yau A-∞ algebra(Lurie 09, section 4.2)
non-compact local closed topological string2Mod$(\mathbf{S})$ for a symmetric monoidal (∞,1)-category $\mathbf{S}$Calabi-Yau object in $\mathbf{S}$(Lurie 09, section 4.2)