symmetric monoidal (∞,1)-category of spectra
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
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.
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.
The homology of the framed little 2-disk operad in chain complexes is the BV-operad $BV$ the operad for BV-algebras:
This is due to (Getzler).
The framed little disk operad is formal in characteristic zero.
This means that there is a zig-zag of quasi-isomorphisms
This is due to (Giansiracusa-Salvatore-Severa 09)). See also (Valette, slide 35).
Accordingly one makes the following definition:
The operad for homotopy BV-algebras is any cofibrant resolution of $BV \simeq H_\bullet(fD_2)$, or equivalently of $C_\bullet(fD_2)$.
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.
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).
framed little 2-disk 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 |
The framed little 2-disk operad was introduced in
For the relation to ribbons see
The formality of $fD_2$ was shown in
and
Discussion of homotopy BV-algebras is in
see also
Slides of a talk summarizing this are at