nLab framed little 2-disk operad

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-dimensional 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

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

This is due to Severa 09, Giansiracusa-Salvatore 09). See also (Valette, slide 35).

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).

Related concepts

• framed little 2-disk operad

• framed little n-disk operad

• BD operad

algebraic deformation quantization

dimension	classical field theory	Lagrangian BV quantum field theory	factorization algebra of observables
general $n$	P-n algebra	BD-n algebra?	E-n algebra
$n = 0$	Poisson 0-algebra	BD-0 algebra? = BD algebra	E-0 algebra? = pointed space
$n = 1$	P-1 algebra = Poisson algebra	BD-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

• Jeffrey Giansiracusa, Paolo Salvatore, Formality of the framed little 2-discs operad and semidirect products , in: Homotopy theory of function spaces and related topics, Cont. Math. 519, AMS, pp. 115-121 (arxiv 0911.4428)

and

• Pavol Severa, Formality of the chain operad of framed little disks, (arxiv AT 0902.3576)

Discussion of homotopy BV-algebras is in

• Imma Galvez-Carrillo, Andy Tonks, Bruno Valette?, Homotopy Batalin-Vilkovisky algebras (arXiv:0907.2246)

see also

• Imma Galvez-Carrillo, Vassily Gorbounov, Andy Tonks, Homotopy Gerstenhaber structures and vertex algebras, math/0611231.QA

Slides of a talk summarizing this are at

• Bruno Valette?, Homotopy Batalin-Vilkovisky algebras (pdf)