# nLab framed little 2-disk operad

Contents

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## Definition

###### Definition

The framed little 2-disk operad is the operad $fD_2$ in Top whose topological space $fD_2(n)$ of $n$-ary operations is the space of maps

$\coprod_n D \to D$

from $n$-copies of the 2-ball to itself, which restrict on each component to a map that is a combination of

• a translation

• a dilatation

• a rotation

of the disk (regarded via its standard embedding $D \hookrightarrow \mathbb{R}^2$ into the 2-simensional Cartesian space) such that the images of all disks are disjoint.

###### Remark

This differs from the little 2-disk operad by the fact that rotations of the disks are admitted. Under passing to chains and then to homology, this operation gives rise to the BV-operator in a BV-algebra. See Properties below.

## Properties

###### Theorem

The homology of the framed little 2-disk operad in chain complexes is the BV-operad $BV$ the operad for BV-algebras:

$BV \simeq H_\bullet(fD_2) \,.$

This is due to (Getzler).

###### Theorem

The framed little disk operad is formal in characteristic zero.

This means that there is a zig-zag of quasi-isomorphisms

$C_\bullet(fD_2) \stackrel{\simeq}{\leftarrow} \stackrel{\simeq}{\to} \cdots \stackrel{\simeq}{\leftarrow} \stackrel{\simeq}{\to} H_\bullet(fD_2) \,.$

Accordingly one makes the following definition:

###### Definition

The operad for homotopy BV-algebras is any cofibrant resolution of $BV \simeq H_\bullet(fD_2)$, or equivalently of $C_\bullet(fD_2)$.

###### Definition

Write $R \beta_j$ for the ribbon braid group? on $j$ elements and $P R \beta_j$ for the kernel of the surjection $R \beta_j \to \Sigma_j$ onto the symmetric group.

Say that a ribbon operad? $P$ is an $R_\infty$-operad if the ribbon braid group?s act freely and properly on $P$ and if each topological space $P(k)$ is contractible.

###### Theorem

If $P$ is an $R_\infty$-operad, then the sequence of quotient spaces $\{P(n)/P R \beta_n\}$ forms a symmetric operad equivalent to the frame little disks operad.

This is (Wahl, lemma 1.5.17).

algebraic deformation quantization

dimensionclassical field theoryLagrangian BV quantum field theoryfactorization algebra of observables
general $n$P-n algebraBD-n algebra?E-n algebra
$n = 0$Poisson 0-algebraBD-0 algebra? = BD algebraE-0 algebra? = pointed space
$n = 1$P-1 algebra = Poisson algebraBD-1 algebra?E-1 algebra? = A-∞ algebra

## References

The framed little 2-disk operad was introduced in

• Ezra Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories , Comm. Math. Phys. 159 (1994), no. 2, 265–285. (arXiv)

For the relation to ribbons see

• Nathalie Wahl?, Ribbon braids and related operads PhD thesis, Oxford (2001) (pdf).

The formality of $fD_2$ was shown in

and

Discussion of homotopy BV-algebras is in