The notion of Calabi-Yau category is a horizontal categorification of that of Frobenius algebra – a Frobenius algebroid. Their name derives from the fact that the definition of Calabi-Yau categories have been originally studied as an abstract version of the derived category of coherent sheaves on a Calabi-Yau manifold.
A Calabi-Yau category is a Vect-enriched category equipped for each object with a trace-like map
to the ground field, such that for all objects the induced pairing
given by
is symmetric and non-degenerate.
[Question: this 1-categorical definition seems to allow for different Frobenius structures on the endomorphism algebras of isomorphic objects. Would it be better to define it as a dinatural transformation from the Hom-functor to the constant functor with value the ground field ?]
A Calabi-Yau category with a single object is the same (or rather: is equivalently) the pointed monoid delooping of a Frobenius algebra.
A Calabi-Yau -category of dimension is an A-∞ category equipped with, for each pair of objects, a morphism of chain complexes
such that
this is non-degenerate and is symmetric in that
for the symmetry isomorphism of the symmetric monoidal category of chain complexes;
this is cyclically invariant in that for all elements is the respective hom-complexes we have
Let be a smooth projective Calabi-Yau variety of dimension . Write for the bounded derived category of that of coherent sheaves on .
Then is a CY -category in a naive way:
the non-binary composition maps are all trivial;
the pairing is given by Serre duality (one needs also a choice of trivialization of the canonical bundle of )
This is however not the morally correct CY -structure associated with a Calabi-Yau. A correct choice is, for example, the Dolbeault dg-enhancement of the derived category
(Costello 04, 7.2, Costello 05, 2.2)
The Fukaya category associated with a symplectic manifold . But see this MO discussion for more.
string topology: for a compact simply connected oriented manifold, its cohomology is naturally a Calabi-Yau -category with a single object. The structure comes from the homological perturbation lemma. One could also use the dg algebra of cochains .
Calabi-Yau -categories classify TCFTs. This remarkable result is what actually one should expect. Indeed, TCFTs originally arose as an abstract version of the CFTs constructed from sigma-models whose targets are Calabi-Yau spaces.
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) |
Kevin Costello, Topological conformal field theories and Calabi-Yau categories (arXiv:math/0412149)
Kevin Costello, The Gromov-Witten potential associated to a TCFT (arXiv:0509264)
Maxim Kontsevich, Yan Soibelman. Notes on A-infinity algebras, A-infinity categories and non-commutative geometry, arXiv:math/0606241
Lee Cho, Notes on Kontsevich-Soibelman’s theorem about cyclic A-infinity algebras (arXiv:1002.3653)
Jacob Lurie, section 4.2 of On the Classification of Topological Field Theories (arXiv:0905.0465)
A relative version is defined for functors instead of categories,
Last revised on August 9, 2024 at 17:13:35. See the history of this page for a list of all contributions to it.