Calabi-Yau category



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 CC equipped for each object cCc \in C with a trace-like map

Tr C:C(c,c)k Tr_C : C(c,c) \to k

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

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

given by

f,g=Tr(gf) \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.


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

, a,b:C(a,b)C(b,a)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,aσ a,b \langle - , - \rangle_{a,b} = \langle - , - \rangle_{b,a} \circ \sigma_{a,b}

    for σ a,b:C(a,b)C(b,a)C(b,a)C(a,b)\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 (α i)(\alpha_i) is the respective hom-complexes we have

    m n1(α 0α n2),α n1=(1) (n+1)+|α 0| i=1 n1|α i|m n1(α 1α n2),α 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 \,.


Of A A_\infty CY-categories

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

    Then D b(X)D^b(X) is a CY A A_\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 XX)

This is however not the morally correct CY A A_\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.


Calabi-Yau A A_\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.


Revised on July 21, 2011 22:22:41 by Urs Schreiber (