∞-Lie theory

# Contents

## Idea

The fivebrane Lie 6-algebra is the second step in the ∞-Lie algebra-Whitehead tower (read as the Whitehead tower in an (∞,1)-topos in ∞LieGrpd) of the special orthogonal group.

## Definition

Let $𝔤$ be the special orthogonal Lie algebra. The first two ∞-Lie algebra cocycles on it are in degree 3 and 7.

${\mu }_{3}:𝔤\to {b}^{2}ℝ$\mu_3 : \mathfrak{g} \to b^2 \mathbb{R}
${\mu }_{7}:𝔤\to {b}^{6}ℝ\phantom{\rule{thinmathspace}{0ex}}.$\mu_7 : \mathfrak{g} \to b^6 \mathbb{R} \,.

The extension classified by the first is the string Lie 2-algebra

$bℝ\to \mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}\to \mathrm{𝔰𝔬}\phantom{\rule{thinmathspace}{0ex}}.$b \mathbb{R} \to \mathfrak{string} \to \mathfrak{so} \,.

But ${\mu }_{7}$ is still also a ∞-Lie algebra cocycle on $\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}$:

${\mu }_{7}:\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}\to {b}^{6}ℝ\phantom{\rule{thinmathspace}{0ex}}.$\mu_7 : \mathfrak{string} \to b^6 \mathbb{R} \,.

The extension classified by this is the fivebrane Lie 6-algebra

${b}^{5}ℝ\to \mathrm{𝔣𝔦𝔳𝔢𝔟𝔯𝔞𝔫𝔢}\to \mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}\phantom{\rule{thinmathspace}{0ex}}.$b^5 \mathbb{R} \to \mathfrak{fivebrane} \to \mathfrak{string} \,.

## Properties

The Chevalley-Eilenberg algebra $\mathrm{CE}\left(\mathrm{𝔣𝔦𝔳𝔢𝔟𝔯𝔞𝔫𝔢}\right)$ is the relative Sullivan algebra obtained by gluing the two cocoycles.

Under Lie integration the Lie 6-algebra $\mathrm{𝔣𝔦𝔳𝔢𝔟𝔯𝔞𝔫𝔢}$ yields the fivebrane 6-group.

## References

As with many of these ∞-Lie algebra-constructions, the existence of the object itself, regarded dually as a dg-algebra is a triviality in rational homotopy theory, but the interpretation in $\infty$-Lie theory adds a new perspective to it. In this context the fivebrane Lie 6-algebra was introduced in

and its relation to fivebrane structures and quantum anomaly-cancellation in dual heterotic string theory was discussed in

Revised on October 25, 2010 14:53:55 by Urs Schreiber (131.211.232.186)