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fivebrane 6-group

Context

Higher 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

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Higher spin geometry

String theory

Contents

Idea

The fivebrane 6-group Fivebrane(n)Fivebrane(n) is a smooth version of the topological space that appears in the second step of the Whitehead tower of the orthogonal group.

It is a lift of this through the geometric realization functor Π:\Pi : ∞LieGrpd \to ∞Grpd.

One step below the fivebrane 6-group in the Whitehead tower is the string Lie 2-group.

For the time being see the discussions at

smooth Whitehead tower

and the Motivation section at

infinity-Chern-Weil theory

for more background.

Definition

In the (∞,1)-topos H=\mathbf{H} = ∞LieGrpd we have a smooh refinement of the second fractional Pontryagin class

16p 2:BString(n)B 7/ \frac{1}{6} \mathbf{p}_2 : \mathbf{B} String(n) \to \mathbf{B}^7 \mathbb{R}/\mathbb{Z}

defined on the delooping of the string Lie 2-group.

The delooping BFivebrane(n)\mathbf{B}Fivebrane(n) of the fivebrane 6-group is the principal ∞-bundle classified by this in H\mathbf{H}, that is the homotopy fiber

BFivebrane(n) * BString(n) 16p 2 B 7/. \array{ \mathbf{B} Fivebrane(n) &\to& {*} \\ \downarrow && \downarrow \\ \mathbf{B}String(n) &\stackrel{\frac{1}{6}\mathbf{p}_2}{\to}& \mathbf{B}^7 \mathbb{R}/\mathbb{Z} } \,.

Construction

Along the lines of the description at Lie integration and string 2-group, in a canonical model for H\mathbf{H} the morphism 16p 2\frac{1}{6}\mathbf{p}_2 is given by a morphism out of a resolution BQ\mathbf{B}Q of BString(n)\mathbf{B}String(n) that is built in degree k7k \leq 7 from smooth kk-simplices in the Lie group Spin(n)Spin(n). This morphism assigns to a 7-simplex ϕ:Δ Diff 7Spin(n)\phi : \Delta^7_{Diff} \to Spin(n) the integral

Δ Diff 7ϕ *μ 7/ \int_{\Delta^7_{Diff}} \phi^* \mu_7 \;\;\in \mathbb{R}/\mathbb{Z}

of the degree 7 Lie algebra cocycle μ 7\mu_7 of the special orthogonal Lie algebra 𝔰𝔬(n)\mathfrak{so}(n) which is normalized such that its pullback to String(n)String(n) (..explain…) is the deRham image of the generator in integral cohomology there.

More in detail, a resolution of BString(n)\mathbf{B}String(n) is given by the coskeleton

cosk 7(Q 7hom(Δ Diff 7,G)×(U(1)) 87654 Q 4hom(Δ Diff 4,G)×(U(1)) 20 Q 3hom(Δ Diff 3,G)×(U(1)) 4 hom(Δ Diff 2,G)×U(1) hom(Δ Diff 1,G) *) \mathbf{cosk}_7 \left( \array{ Q_7 \subset hom(\Delta^7_{Diff}, G) \times (U(1))^{8 \cdot 7 \cdot 6 \cdot 5 \cdot 4} \\ \downarrow \downarrow \downarrow\downarrow \downarrow \downarrow \downarrow \downarrow \\ \vdots \\ \downarrow \downarrow \downarrow\downarrow \downarrow \downarrow \\ Q_4 \subset hom(\Delta^4_{Diff}, G) \times (U(1))^{20} \\ \downarrow \downarrow \downarrow\downarrow \downarrow \\ Q_3 \subset hom(\Delta^3_{Diff}, G) \times (U(1))^4 \\ \downarrow \downarrow \downarrow\downarrow \\ hom(\Delta^2_{Diff}, G) \times U(1) \\ \downarrow \downarrow \downarrow \\ hom(\Delta^1_{Diff}, G) \\ \downarrow \downarrow \\ * } \right)

where the subobjects are those consisting of 3-simplices in GG with 2-faces labeled in U(1)U(1) such that the integral of μ 3\mu_3 over the 3-simplex in /\mathbb{R}/\mathbb{Z} is the signed product of these labels.

(…)

fivebrane 6-group \to string 2-group \to spin group \to special orthogonal group \to orthogonal group

References

The topological fivebrane group with its interpretation in dual heterotic string theory was discussed in

and the smooth fivebrane 6-group was indicated. The latter is discussed in more detail in section 4.1 of

Revised on August 29, 2013 17:31:16 by Urs Schreiber (89.204.130.26)