nLab Calabi-Yau algebra

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. .

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).

$0$-Calabi-Yau algebras

A cochain dg-algebra over $k$ is $0$-Calabi-Yau iff it is Koszul and $Tor^0_A(k_A,{}_A k)$ is a symmetric coalgebra. Proven in

• J.-W. He, X.-F. Mao, Connected cochain DG algebras of Calabi-Yau dimension 0, Proc. Amer. Math. Soc. 145 (2017) 937–953 doi

It follows that a Koszul dg-algebra is $0$-Calabi-Yau iff its Ext-algebra is symmetric Frobenius.

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

Related concepts

• Calabi-Yau object

  • Calabi-Yau algebra, Calabi-Yau manifold

  • Calabi-Yau category

• pre-Calabi-Yau algebra

References

• Victor Ginzburg, Calabi-Yau algebras (arXiv:0612139)

• Jacob Lurie, section 4.2 of On the Classification of Topological Field Theories (arXiv:0905.0465)

An adaptation of this notion to the setting of spectra in stable homotopy theory is studied in

• Ralph L. Cohen, Inbar Klang, Twisted Calabi–Yau ring spectra, string topology, and gauge symmetry, Tunisian J. Math. 2:1 (2020) doi