symmetric monoidal (∞,1)-category of spectra
from -copies of the 2-ball to itself, which restrict on each component to a map that is a combination of
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.
This is due to (Getzler).
The framed little disk operad is formal.
This means that there is a zig-zag of quasi-isomorphisms
Accordingly one makes the following definition:
If is an -operad, then the sequence of quotient spaces forms a symmetric operad equivalent to the frame little disks operad.
This is (Wahl, lemma 1.5.17).
|dimension||classical field theory||Lagrangian BV quantum field theory||factorization algebra of observables|
|general||P-n algebra||BD-n algebra?||E-n algebra|
|Poisson 0-algebra||BD-0 algebra? = BD algebra||E-0 algebra? = pointed space|
|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 was shown in
Discussion of homotopy BV-algebras is in
Slides of a talk summarizing this are at