symmetric monoidal (∞,1)-category of spectra
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.
For $A$ a dg-algebra and $N$ a dg-bimodule over $A$, write
for the dual $A$-bimodule, where $RHom$ denotes the right derived hom-functor with respect to the model structure on dg-modules.
A homologically smooth dg-algebra $A$ is a Calabi-Yau algebra of dimension $d$ if there is a quasi-isomorphism of $A$-bimodules
such that
This is (Ginzburg, def. 3.2.3).
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 $X$.
An object $\mathcal{E} \in D^b(Coh X)$ is called a tilting generator if the Ext-functor satisfies
$Ext^i(\mathcal{E}, \mathcal{E}) = 0$ for all $i \gt 0$;
$Ext^\bullet(\mathcal{E},\mathcal{F}) = 0$ implies $\mathcal{F} = 0$;
the endomorphism algebra $End(\mathcal{E}) = Hom(\mathcal{E},\mathcal{E})$ has finite Hochschild dimension.
This appears as (Ginzburg, def. 7.1.1).
For $\mathcal{E}$ a tilting generator there is an equivalence of triangulated categories
to the derived category of modules over $End(\mathcal{E})$.
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).
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
It follows that a Koszul dg-algebra is $0$-Calabi-Yau iff its Ext-algebra is symmetric Frobenius.
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
from the Hochschild homology $\int_{S^1} A \simeq A \otimes_{A \otimes A} A$, satisfying the condition that the composite morphism
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).
Calabi-Yau algebra, Calabi-Yau manifold
Victor Ginzburg, Calabi-Yau algebras (arXiv:0612139)
Jacob Lurie, section 4.2 of On the Classification of Topological Field Theories (arXiv:0905.0465)
Last revised on January 27, 2023 at 09:02:20. See the history of this page for a list of all contributions to it.