Contents

# 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 $\mathfrak{g}$ be the special orthogonal Lie algebra. The first two ∞-Lie algebra cocycles on it are in degree 3 and 7.

$\mu_3 : \mathfrak{g} \to b^2 \mathbb{R}$
$\mu_7 : \mathfrak{g} \to b^6 \mathbb{R} \,.$

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

$b \mathbb{R} \to \mathfrak{string} \to \mathfrak{so} \,.$

But $\mu_7$ is still also a ∞-Lie algebra cocycle on $\mathfrak{string}$:

$\mu_7 : \mathfrak{string} \to b^6 \mathbb{R} \,.$

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

$b^5 \mathbb{R} \to \mathfrak{fivebrane} \to \mathfrak{string} \,.$

## Properties

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

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

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