nLab
Calabi-Yau category

Idea

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.

Definition

1-categorical

A Calabi-Yau category is a Vect-enriched category $C$ equipped for each object $c\in C$ with a trace-like map

$${\mathrm{Tr}}_C:C(c,c)\to k$$Tr_C : C(c,c) \to k

to the ground field, such that for all objects $d\in C$ the induced pairing

$$⟨-,-{⟩}_{c,d}:C(c,d)\otimes C(d,c)\to k$$\langle -,-\rangle_{c,d} : C(c,d) \otimes C(d,c) \to k

given by

$$⟨f,g⟩=\mathrm{Tr}(g\circ f)$$\langle f,g \rangle = Tr(g \circ f)

is symmetric and non-degenerate.

A Calabi-Yau category with a single object is the same (or rather the delooping) of a Frobenius algebra.

$(\mathrm{\infty },1)$-categorical

A Calabi-Yau $A_{\mathrm{\infty }}$-category of dimension $d\in \mathbb{N}$ is an A-∞ category $C$ equipped for each pair $a,b$ of objects a morphism of chain complexes

$$⟨-,-{⟩}_{a,b}:C(a,b)\otimes C(b,a)\to k[d]$$\langle -,-\rangle_{a,b} : C(a,b) \otimes C(b,a) \to k[d]

such that

1. this is symmetric in that

   $$⟨-,-{⟩}_{a,b}=⟨-,-{⟩}_{b,a}\circ {\sigma }_{a,b}$$\langle - , - \rangle_{a,b} = \langle - , - \rangle_{b,a} \circ \sigma_{a,b}

   for ${\sigma }_{a,b}:C(a,b)\otimes C(b,a)\to C(b,a)\otimes C(a,b)$ the symmetry isomorphism of the symmetric monoidal category of chain complexes;

2. this is cyclically invariant in that for all elements $({\alpha }_i)$ is the respective hom-complexes we have

   $$⟨m_{n-1}({\alpha }_0\otimes \cdots \otimes {\alpha }_{n-2}),{\alpha }_{n-1}⟩=(-1{)}^{(n+1)+\mid {\alpha }_0\mid \sum _{i=1}^{n-1}\mid {\alpha }_i\mid }⟨m_{n-1}({\alpha }_1\otimes \cdots \otimes {\alpha }_{n-2}),{\alpha }_0⟩.$$\langle m_{n-1}(\alpha_0 \otimes \cdots \otimes \alpha_{n-2}), \alpha_{n-1} \rangle = (-1)^{(n+1)+ |\alpha_0| \sum_{i = 1}^{n-1}|\alpha_i|} \langle m_{n-1}(\alpha_1 \otimes \cdots \otimes \alpha_{n-2}), \alpha_0 \rangle \,.

Examples

Of $A_{\mathrm{\infty }}$ CY-categories

• Let $X$ be a smooth projective Calabi-Yau variety of dimension $d$. Write $D^b(X)$ for the bounded derived category of that of coherent sheaves on $X$.

  Then $D^b(X)$ is a CY $A_{\mathrm{\infty }}$-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 $X$)

This is however not the morally correct CY $A_{\mathrm{\infty }}$-structure associated with a Calabi-Yau. A correct choice is, for example, the Dolbeault dg enhancement of the derived category; see section 2.2 of Cos05.

• The Fukaya category associated with a symplectic manifold $X$. But see mathoverflow for more discussion: http://mathoverflow.net/questions/13114/are-fukaya-categories-calabi-yau-categories

• string topology: for $X$ a compact simply connected oriented manifold, its cohomology $H^{•}(X)$ is naturally a Calabi-Yau $A_{\mathrm{\infty }}$-category with a single object. The $A_{\mathrm{\infty }}$ structure comes from the homological perturbation lemma. One could also use the dg algebra of cochains $C^{•}(X)$.

Properties

Calabi-Yau $A_{\mathrm{\infty }}$-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.

Related concepts

• Calabi-Yau object

References

• Kevin Costello, Topological conformal field theories and Calabi-Yau categories (arXiv:math/0509264)

• Kontsevich, Soibelman?. Notes on A-infinity algebras, A-infinity categories and non-commutative geometry http://arxiv.org/abs/math/0606241

• Cho, Lee. Notes on Kontsevich-Soibelman’s theorem about cyclic A-infinity algebras http://arxiv.org/abs/1002.3653