nLab
Fivebrane group

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

\cdots \to Fivebrane group \to string group \to spin group \to special orthogonal group \to orthogonal group.


Contents

Definition

The Fivebrane group Fivebrane(n)Fivebrane(n) is defined to be, as a topological group, the 7-connected cover of the String group String(n)String(n), for any nn \in \mathbb{N}.

Notice that String(n)String(n) itself if the 3-connected cover of Spin(n)Spin(n), which is itself is the simply connected cover of the special orthogonal group SO(n)SO(n), which in turn is the connected component (of the identity) of the orthogonal group O(n)O(n). Hence Fivebrane(n)Fivebrane(n) is one element in the Whitehead tower of O(n)\mathrm{O}(n):

Fivebrane(n)String(n)Spin(n)SO(n)O(n). \cdots \to Fivebrane(n) \to String(n) \to Spin(n) \to SO(n) \to \mathrm{O}(n) \,.

The homotopy groups of O(n)O(n) are for kk \in \mathbb{N} and for sufficiently large nn

π 8k+0(O) = 2 π 8k+1(O) = 2 π 8k+2(O) =0 π 8k+3(O) = π 8k+4(O) =0 π 8k+5(O) =0 π 8k+6(O) =0 π 8k+7(O) =. \array{ \pi_{8k+0}(O) & = \mathbb{Z}_2 \\ \pi_{8k+1}(O) & = \mathbb{Z}_2 \\ \pi_{8k+2}(O) & = 0 \\ \pi_{8k+3}(O) & = \mathbb{Z} \\ \pi_{8k+4}(O) & = 0 \\ \pi_{8k+5}(O) & = 0 \\ \pi_{8k+6}(O) & = 0 \\ \pi_{8k+7}(O) & = \mathbb{Z} } \,.

By co-killing these groups step by step one gets

cokillthis toget π 0(O) = 2 SO π 1(O) = 2 Spin π 2(O) =0 π 3(O) = String π 4(O) =0 π 5(O) =0 π 6(O) =0 π 7(O) = Fivebrane. \array{ cokill this &&&& to get \\ \\ \pi_{0}(O) & = \mathbb{Z}_2 &&& SO \\ \pi_{1}(O) & = \mathbb{Z}_2 &&& Spin \\ \pi_{2}(O) & = 0 \\ \pi_{3}(O) & = \mathbb{Z} &&& String \\ \pi_{4}(O) & = 0 \\ \pi_{5}(O) & = 0 \\ \pi_{6}(O) & = 0 \\ \pi_{7}(O) & = \mathbb{Z} &&& Fivebrane } \,.

Further information

…should eventually go here. For the time being have a look at Fivebrane structure.

nn012345678910111213141516
Whitehead tower of orthogonal grouporientationspinstringfivebraneninebrane
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 term fivebrane group and the role of this topological group in quantum anomaly cancellaton conditions in dual heterotic string theory was found by Hisham Sati and appeared in

The term shortly after was picked up in

The refinement to fivebrane principal infinity-connections, hence differential fivebrane structures was then discussed in

Discussion in a comprehensive context is in section 5 of

Revised on November 13, 2013 00:47:05 by Urs Schreiber (82.169.114.243)