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fivebrane Lie 6-algebra

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

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

μ 3:𝔤b 2 \mu_3 : \mathfrak{g} \to b^2 \mathbb{R}
μ 7:𝔤b 6. \mu_7 : \mathfrak{g} \to b^6 \mathbb{R} \,.

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

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

But μ 7\mu_7 is still also a ∞-Lie algebra cocycle on 𝔰𝔱𝔯𝔦𝔫𝔤\mathfrak{string}:

μ 7:𝔰𝔱𝔯𝔦𝔫𝔤b 6. \mu_7 : \mathfrak{string} \to b^6 \mathbb{R} \,.

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

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

Properties

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

Under Lie integration the Lie 6-algebra 𝔣𝔦𝔳𝔢𝔟𝔯𝔞𝔫𝔢\mathfrak{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)