# 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

${\mathrm{Tr}}_{C}:C\left(c,c\right)\to k$Tr_C : C(c,c) \to k

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

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

given by

$⟨f,g⟩=\mathrm{Tr}\left(g\circ f\right)$\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.

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

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

$⟨-,-{⟩}_{a,b}:C\left(a,b\right)\otimes C\left(b,a\right)\to k\left[d\right]$\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 ${\sigma }_{a,b}:C\left(a,b\right)\otimes C\left(b,a\right)\to C\left(b,a\right)\otimes C\left(a,b\right)$ the symmetry isomorphism of the symmetric monoidal category of chain complexes;

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

$⟨{m}_{n-1}\left({\alpha }_{0}\otimes \cdots \otimes {\alpha }_{n-2}\right),{\alpha }_{n-1}⟩=\left(-1{\right)}^{\left(n+1\right)+\mid {\alpha }_{0}\mid \sum _{i=1}^{n-1}\mid {\alpha }_{i}\mid }⟨{m}_{n-1}\left({\alpha }_{1}\otimes \cdots \otimes {\alpha }_{n-2}\right),{\alpha }_{0}⟩\phantom{\rule{thinmathspace}{0ex}}.$\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 ${D}^{b}\left(X\right)$ for the bounded derived category of that of coherent sheaves on $X$.

Then ${D}^{b}\left(X\right)$ 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)