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Definition

The string group String(n)String(n) is defined to be, as a topological group, the 3-connected cover of the Spin group Spin(n)Spin(n), for any nn \in \mathbb{N}.

Notice that Spin(n)Spin(n) 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 String(n)String(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 next higher connected group is called the Fivebrane group.

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 } \,.

Definition by co-killing of π 3\pi_3

More in detail this means the following.

First notice that since by construction π i(Spin(n))=0\pi_i(\mathcal{B}Spin(n)) = 0 for 0i30 \leq i \leq 3, by the Hurewicz theorem we have for the degree 4 integral cohomology group of the classifying space BSpin(n)B Spin(n) that

H 4(BSpin(n))π 4(BSpin(n)). H^4(B Spin(n)) \simeq \pi_4(B Spin(n)) \simeq \mathbb{Z} \,.

The generator of this group is called the fractional first Pontryagin class and denoted

12p 1:BSpin(n)B 4K(,4) \frac{1}{2}p_1 : B Spin(n) \to B^4 \mathbb{Z} \simeq K(\mathbb{Z},4)

because the ordinary first Pontryagin class p 1:SO(n)K(,4)p_1 : \mathcal{B} SO(n) \to K(\mathbb{Z},4) fits into a diagram

BSpin(n) 12p 1 B 4 2 BSO(n) p 1 B 4, \array{ B Spin(n) &\stackrel{\frac{1}{2} p_1}{\to}& B^4 \mathbb{Z} \\ \downarrow && \downarrow^{\cdot 2} \\ B SO(n) &\stackrel{p_1}{\to}& B^4 \mathbb{Z} } \,,

where the right vertical morphism comes from multiplication by 2 in \mathbb{Z}.

This says that after being pulled back to Spin(n)\mathcal{B} Spin(n) the first Pontryagin class is 2 times the generator of the degree 4 integral cohomology group of Spin(n)\mathcal{B}Spin(n) and hence that generator is called one half of p 1p_1, denoted 12p 1\frac{1}{2}p_1 (by slight abuse of notation).

The delooping of the String-group as a topological group is the homotopy fiber of this fractional Pontyagin class, i.e. the homotopy pullback

BString(n) * BSpin(n) 4 \array{ B String(n) &\to& {*} \\ \downarrow && \downarrow \\ B Spin(n) &\to& \mathcal{B}^4 \mathbb{Z} }

in Top.

In other words: BString(n)B String(n) is the U(1)U(1)- 2-gerbe or B 2U(1)B^2 U(1) principal ∞-bundle on BSpin(n)B Spin(n) whose class is 12p 1H 4(BSpin(n),4)\frac{1}{2}p_1 \in H^4(B Spin(n), 4).

Models

As a topological group

There is a model due to Stolz and Teichner in ‘What is an elliptic object?’…

As a smooth 2-group

While Spin(n)Spin(n) is not just a topological group but a (finite dimensional) Lie group, String(n)String(n) cannot have the structure of a finite dimensional Lie group, due to the fact that the third homotopy group is nontrivial for every (finite dimensional) Lie group, while for π 3(String(n))=0\pi_3(String(n)) = 0 by the very definition of String(n)String(n).

David Roberts: This raises an interesting question: is there an infinite dimensional Lie group which is a 3-connected cover? This seems very hard to do…. Thomas Nikolaus: Yes, there is an infinite dimensional group, as shown in our article listed in the references below.

But there are smooth models of String(n)String(n) in the form of 2-groups. See string 2-group.

Role in string theory

The reason for the name is that in string theory, for (blah) to be well-defined, it is necessary for the structure group of (blah) to lift to (blah).

See String structure.

If one considers passing to the (free) loop space of spacetime and then doing quantum mechanics, the requirement of the previous paragraph is that the structure group lifts to … (cite Killingback, Mickelsson, Schreiber, Witten,…)

Generalization to other groups

One may consider the universal 3-connected cover of any general compact, simple and simply connected Lie group GG, in complete analogy to the case G=Spin(n)G = Spin(n). Accordingly one speaks of string-groups String GString_G.

Of these the case G=G = E8 is the other one relevant in string theory: see Green-Schwarz mechanism.

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

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
groupsymboluniversal coversymbolhigher coversymbol
orthogonal groupO(n)\mathrm{O}(n)Pin groupPin(n)Pin(n)Tring groupTring(n)Tring(n)
special orthogonal groupSO(n)SO(n)Spin groupSpin(n)Spin(n)String groupString(n)String(n)
Lorentz groupO(n,1)\mathrm{O}(n,1)\,Spin(n,1)Spin(n,1)\,\,
anti de Sitter groupO(n,2)\mathrm{O}(n,2)\,Spin(n,2)Spin(n,2)\,\,
Narain groupO(n,n)O(n,n)
Poincaré groupISO(n,1)ISO(n,1)\,\,\,\,
super Poincaré groupsISO(n,1)sISO(n,1)\,\,\,\,

References

Originally the String-group was just known by its generic name: with O8\mathcal{B} O \langle 8 \rangle being the topologist’s notation for the 7-connected cover of the delooping/classifying space O\mathcal{B}O of the group OO.

When it was realized that lifts of the structure maps XOX \to \mathcal{B}O of the tangent bundle of a manifold XX through the projection O8O\mathcal{B}O\langle 8 \rangle \to \mathcal{B}O – now called a String structure – play the same role in string theory as a Spin structure does in ordinary quantum mechanics, the term String group for Ω(O8)\Omega (\mathcal{B}O\langle 8 \rangle) was suggested.

Following some inquiries by Jim Stasheff and confirmed in private email by Haynes Miller it seems that the first one to propose the term StringString group for the group known to topologists as Ω(O8)\Omega (\mathcal{B}O\langle 8\rangle) was Haynes Miller.

A model of the string group by local nets of fermions is discussed in

Many more models exist by now in terms of geometric realization of a model for the string 2-group. See there for more references.

A good review is in the introduction of

In

is is shown that the topological string group does admit a Frechet manifold Lie group structure.

Revised on November 13, 2013 01:02:22 by Urs Schreiber (82.169.114.243)