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 dg-algebra and a dg-bimodule over , write
for the dual -bimodule, where denotes the right derived hom-functor with respect to the model structure on dg-modules.
A homologically smooth dg-algebra is a Calabi-Yau algebra of dimension if there is a quasi-isomorphism of -bimodules
such that
This is (Ginzburg, def. 3.2.3).
Let be a smooth quasi-projective variety. Write for the derived category of bounded chain complexes of coherent sheaves over .
An object is called a tilting generator if the Ext-functor satisfies
for all ;
implies ;
the endomorphism algebra has finite Hochschild dimension.
This appears as (Ginzburg, def. 7.1.1).
For a tilting generator there is an equivalence of triangulated categories
to the derived category of modules over .
For smooth connected variety which is projective over an affine variety, let be a tilting generator, def. .
Then is a Calabi-Yau algebra of dimension precisely if is a Calabi-Yau manifold of dimension .
This appears as (Ginzburg, prop. 3.3.1).
A cochain dg-algebra over is -Calabi-Yau iff it is Koszul and is a symmetric coalgebra. Proven in
It follows that a Koszul dg-algebra is -Calabi-Yau iff its Ext-algebra is symmetric Frobenius.
Let be a good? symmetric monoidal (∞,1)-category. Write for the symmetric monoidal (∞,2)-category whose objects are algebra objects in and whose morphisms are bimodule objects.
Then a Calabi-Yau object in is an algebra object equipped with an -equivariant morphism
from the Hochschild homology , satisfying the condition that the composite morphism
exhibits as its own dual object .
Such an algebra object is called a Calabi-Yau algebra object.
This is (Lurie 09, example 4.2.8).
2d TQFT (“TCFT”) | coefficients | algebra structure on space of quantum states | |
---|---|---|---|
open topological string | Vect | Frobenius algebra | folklore+(Abrams 96) |
open topological string with closed string bulk theory | Vect | Frobenius algebra with trace map 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) over field | 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 for a symmetric monoidal (∞,1)-category | Calabi-Yau object in | (Lurie 09, section 4.2) |
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)
An adaptation of this notion to the setting of spectra in stable homotopy theory is studied in
Last revised on April 25, 2024 at 21:25:05. See the history of this page for a list of all contributions to it.