nLab Calabi-Yau algebra




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 AA a dg-algebra and NN a dg-bimodule over AA, write

N !:=RHom ABimod(N,AA) N^! := RHom_{A Bimod}(N, A \otimes A)

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


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

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

such that

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

This is (Ginzburg, def. 3.2.3).



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


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

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

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

  3. the endomorphism algebra End()=Hom(,)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

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

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


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

Then EndEnd \mathcal{E} is a Calabi-Yau algebra of dimension dd precisely if XX is a Calabi-Yau manifold of dimension dd.

This appears as (Ginzburg, prop. 3.3.1).

00-Calabi-Yau algebras

A cochain dg-algebra over kk is 00-Calabi-Yau iff it is Koszul and Tor A 0(k A, Ak)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 00-Calabi-Yau iff its Ext-algebra is symmetric Frobenius.

Relation to 2d extended TQFT and the Cobordism hypothesis


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 \colon \int_{S^1} A \to 1

from the Hochschild homology S 1AA AAA\int_{S^1} A \simeq A \otimes_{A \otimes A} A, 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 object A 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{}_kFrobenius algebra AAfolklore+(Abrams 96)
open topological string with closed string bulk theoryVect k{}_kFrobenius algebra AA with trace map BZ(A)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{}_k) over field kkseparable 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(S)(\mathbf{S}) for a symmetric monoidal (∞,1)-category S\mathbf{S}Calabi-Yau object in S\mathbf{S}(Lurie 09, section 4.2)


Last revised on January 27, 2023 at 09:02:20. See the history of this page for a list of all contributions to it.