nLab fivebrane 6-group

Contents

Context

Higher Lie theory

∞-Lie theory (higher geometry)

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 smooth 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. Strictly speaking, we need n>6n\gt 6, since for low nn, String(n)String(n) is not 6-connected. See orthogonal group for a table of the relevant homotopy groups.

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.

(…)

nn012345678910111213141516
Whitehead tower of orthogonal grouporientationspin groupstring groupfivebrane groupninebrane group
higher versionsspecial orthogonal groupspin groupstring 2-groupfivebrane 6-groupninebrane 10-group
homotopy groups of stable orthogonal groupπ n(O)\pi_n(O) 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z} 2\mathbb{Z}_2
stable homotopy groups of spheresπ n(𝕊)\pi_n(\mathbb{S})\mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_2 24\mathbb{Z}_{24}00 2\mathbb{Z}_2 240\mathbb{Z}_{240} 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 2 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2 6\mathbb{Z}_6 504\mathbb{Z}_{504}0 3\mathbb{Z}_3 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 480 2\mathbb{Z}_{480} \oplus \mathbb{Z}_2 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2
image of J-homomorphismim(π n(J))im(\pi_n(J))0 2\mathbb{Z}_20 24\mathbb{Z}_{24}000 240\mathbb{Z}_{240} 2\mathbb{Z}_2 2\mathbb{Z}_20 504\mathbb{Z}_{504}000 480\mathbb{Z}_{480} 2\mathbb{Z}_2

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

Last revised on May 26, 2022 at 22:58:10. See the history of this page for a list of all contributions to it.