# nLab Calabi-Yau algebra

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

The notion of Calabi-Yau algebra is an algebraic incarnation of the notion of Calabi-Yau manifold and higher algebra-version of the notion of Frobenius algebra.

## Definition

###### Definition

For $A$ a dg-algebra and $N$ a dg-bimodule over $A$, write

$N^! := RHom_{A Bimod}(N, A \otimes A)$

for the dual $A$-bimodule, where $RHom$ denotes the right derived hom-functor with respect to the model structure on dg-modules.

###### Definition

A homologically smooth dg-algebra $A$ is a Calabi-Yau algebra of dimension $d$ if there is a quasi-isomorphism of $A$-bimodules

$f : A \stackrel{\simeq}{\to} A^![d]$

such that

$f \simeq f^![d] \,.$

This is (Ginzburg, def. 3.2.3).

## Properties

### General

Let $X$ be a smooth quasi-projective variety. Write $D^b(Coh X)$ for the derived category of bounded chain complexes of coherent sheaves over $X$.

###### Definition

An object $\mathcal{E} \in D^b(Coh X)$ is called a tilting generator if the Ext-functor satisfies

1. $Ext^i(\mathcal{E}, \mathcal{E}) = 0$ for all $i \gt 0$;

2. $Ext^\bullet(\mathcal{E},\mathcal{F}) = 0$ implies $\mathcal{F} = 0$;

3. the endomorphism algebra $End(\mathcal{E}) = Hom(\mathcal{E},\mathcal{E})$ has finite Hochschild dimension.

This appears as (Ginzburg, def. 7.1.1).

###### Remark

For $\mathcal{E}$ a tilting generator there is an equivalence of triangulated categories

$D^b(Coh X) \stackrel{\simeq}{\to} D^b(End(\mathcal{E})Mod)$

to the derived category of modules over $End(\mathcal{E})$.

###### Proposition

For $X$ smooth connected variety which is projective over an affine variety, let $\mathcal{E} in D^b(Coh X)$ be a tilting generator, def. 3.

Then $End \mathcal{E}$ is a Calabi-Yau algebra of dimension $d$ precisely if $X$ is a Calabi-Yau manifold of dimension $d$.

This appears as (Ginzburg, prop. 3.3.1).

### Relation to 2d extended TQFT and the Cobordism hypothesis

###### 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).

### Classification of 2d TQFT

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)

## References

Revised on November 18, 2014 08:48:57 by Urs Schreiber (217.155.201.6)