# Contents

## 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

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

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

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

given by

$\langle f,g \rangle = Tr(g \circ f)$

is symmetric and non-degenerate.

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

### $(\infty,1)$-categorical

A Calabi-Yau $A_\infty$-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

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

such that

1. this is symmetric in that

$\langle - , - \rangle_{a,b} = \langle - , - \rangle_{b,a} \circ \sigma_{a,b}$

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

$\langle m_{n-1}(\alpha_0 \otimes \cdots \otimes \alpha_{n-2}), \alpha_{n-1} \rangle = (-1)^{(n+1)+ |\alpha_0| \sum_{i = 1}^{n-1}|\alpha_i|} \langle m_{n-1}(\alpha_1 \otimes \cdots \otimes \alpha_{n-2}), \alpha_0 \rangle \,.$

## Examples

### Of $A_\infty$ 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 $A_\infty$-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 $A_\infty$-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.

## Properties

Calabi-Yau $A_\infty$-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.

## References

Revised on July 21, 2011 22:22:41 by Urs Schreiber (82.113.99.58)