nLab fivebrane Lie 6-algebra

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.

Related concepts

• special orthogonal Lie algebra

• string Lie 2-algebra

• fivebrane Lie 6-algebra

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

• SSSI (web)

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

• SSSII (web)