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framed little 2-disk operad

Contents

Definition

Definition

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

nDD \coprod_n D \to D

from nn-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 2D \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 BVBV the operad for BV-algebras:

BVH (fD 2). BV \simeq H_\bullet(fD_2) \,.

This is due to (Getzler).

Theorem

The framed little disk operad is formal.

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

C (fD 2)H (fD 2). 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 (Giansiracusa-Salvatore-Severa 09)). See also (Valette, slide 35).

Accordingly one makes the following definition:

Definition

The operad for homotopy BV-algebras is any cofibrant resolution of BVH (fD 2)BV \simeq H_\bullet(fD_2), or equivalently of C (fD 2)C_\bullet(fD_2).

Definition

Write Rβ jR \beta_j for the ribbon braid group? on jj elements and PRβ jP R \beta_j for the kernel of the surjection Rβ jΣ jR \beta_j \to \Sigma_j onto the symmetric group.

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

Theorem

If PP is an R R_\infty-operad, then the sequence of quotient spaces {P(n)/PRβ n}\{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 nnP-n algebraBD-n algebra?E-n algebra
n=0n = 0Poisson 0-algebraBD-0 algebra? = BD algebraE-0 algebra? = pointed space
n=1n = 1P-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 2fD_2 was shown in

and

Discussion of homotopy BV-algebras is in

see also

Slides of a talk summarizing this are at

Revised on January 25, 2013 16:23:47 by Urs Schreiber (89.204.153.176)