M Theory Lesson 93

Back in Lesson 21 we looked at how holography suggests turning pants diagrams into little disc diagrams. The reverse process would be an operad inclusion map taking little discs to tube-like trees. This is in fact what was considered by Getzler in the paper recently discussed at The Cafe. In this paper, Getzler shows that BV algebras come from a little disc operad which can be included in an operad of moduli of genus zero surfaces (the pants pictures).

But it is not the ordinary little discs operad, where discs may be mapped into a larger disc allowing for translations and a dilation of each little disc. The BV disc operad also allows for a rotation of each little disc. That is, we allow a $\mathbb{C}^{\times}$ action on the disc. This extension of the usual little discs operad may be viewed in terms of braid groups. In fact, the little disc and framed little disc operads are just homologies:

$LD(k) = H_{\ast} (\mathbb{P}_{k})$, $FLD(k) = H_{\ast} (\mathbb{Z}^{k} \times \mathbb{P}_{k})$

where $\mathbb{P}_{k}$ is the pure braid group on $k$ strands. There is a fibration of the framed operad over the usual disc operad with a torus fibre $T^{k}$ which represents all the phases of the framed little discs. The factor of $\mathbb{Z}^{k}$ actually comes from a ribbon braid group, with generators for twisting the ribbons. This associates the rotation of a little disc with a twisting of ribbons, as previously discussed. Recall also that the connection to the topological moduli operad imbues the index $k$ (number of strands) with a dimensional meaning, since the dimension of the moduli spaces increases with the number of punctures $k$.