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

$${\mu }_3:𝔤\to b^2ℝ$$\mu_3 : \mathfrak{g} \to b^2 \mathbb{R}

$${\mu }_7:𝔤\to b^6ℝ.$$\mu_7 : \mathfrak{g} \to b^6 \mathbb{R} \,.

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

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

But ${\mu }_7$ is still also a ∞-Lie algebra cocycle on $\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}$:

$${\mu }_7:\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}\to b^6ℝ.$$\mu_7 : \mathfrak{string} \to b^6 \mathbb{R} \,.

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

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

Properties

The Chevalley-Eilenberg algebra $\mathrm{CE}(\mathrm{𝔣𝔦𝔳𝔢𝔟𝔯𝔞𝔫𝔢})$ 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.

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 $\mathrm{\infty }$-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)