nLab string group

under construction

Definition

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

Notice that for $n\ge3$, $Spin(n)$ itself is the simply connected cover of the special orthogonal group $SO(n)$, which in turn is the connected component (of the identity) of the orthogonal group $O(n)$. Hence $String(n)$ is one element in the Whitehead tower of $\mathrm{O}(n)$:

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

The next higher connected group is called the Fivebrane group.

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

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

By co-killing these groups step by step one gets

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

Definition by co-killing of $\pi_3$

More in detail this means the following.

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

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

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

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

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

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

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

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

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

in Top.

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

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 $Spin(n)$ is not just a topological group but a (finite dimensional) Lie group, $String(n)$ 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 $\pi_3(String(n)) = 0$ by the very definition of $String(n)$.

However, one can define an infinite-dimensional Lie group with the correct properties to be a model of $String(n)$ (Nikolaus-Sachse-Wockel 2013).

There are also smooth models of $String(n)$ 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 $G$, in complete analogy to the case $G = Spin(n)$. Accordingly one speaks of string-groups $String_G$.

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

Related concepts

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

	$n$	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Whitehead tower of orthogonal group		orientation	spin group		string group				fivebrane group				ninebrane group
higher versions		special orthogonal group	spin group		string 2-group				fivebrane 6-group				ninebrane 10-group
homotopy groups of stable orthogonal group	$\pi_n(O)$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	0	$\mathbb{Z}$	0	0	0	$\mathbb{Z}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	0	$\mathbb{Z}$	0	0	0	$\mathbb{Z}$	$\mathbb{Z}_2$
stable homotopy groups of spheres	$\pi_n(\mathbb{S})$	$\mathbb{Z}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	$\mathbb{Z}_{24}$	0	0	$\mathbb{Z}_2$	$\mathbb{Z}_{240}$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$	$\mathbb{Z}_6$	$\mathbb{Z}_{504}$	0	$\mathbb{Z}_3$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2$	$\mathbb{Z}_{480} \oplus \mathbb{Z}_2$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2$
image of J-homomorphism	$im(\pi_n(J))$	0	$\mathbb{Z}_2$	0	$\mathbb{Z}_{24}$	0	0	0	$\mathbb{Z}_{240}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	0	$\mathbb{Z}_{504}$	0	0	0	$\mathbb{Z}_{480}$	$\mathbb{Z}_2$

• string orientation of tmf

group	symbol	universal cover	symbol	higher cover	symbol
orthogonal group	$\mathrm{O}(n)$	Pin group	$Pin(n)$	Tring group	$Tring(n)$
special orthogonal group	$SO(n)$	Spin group	$Spin(n)$	String group	$String(n)$
Lorentz group	$\mathrm{O}(n,1)$	$\,$	$Spin(n,1)$	$\,$	$\,$
anti de Sitter group	$\mathrm{O}(n,2)$	$\,$	$Spin(n,2)$	$\,$	$\,$
conformal group	$\mathrm{O}(n+1,t+1)$	$\,$
Narain group	$O(n,n)$
Poincaré group	$ISO(n,1)$	Poincaré spin group	$\widehat {ISO}(n,1)$	$\,$	$\,$
super Poincaré group	$sISO(n,1)$	$\,$	$\,$	$\,$	$\,$
superconformal group

References

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

When it was realized that lifts of the structure maps $X \to \mathcal{B}O$ of the tangent bundle of a manifold $X$ through the projection $\mathcal{B}O\langle 8 \rangle \to \mathcal{B}O$ – 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 $\Omega (\mathcal{B}O\langle 8 \rangle)$ 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 $String$ group for the group known to topologists as $\Omega (\mathcal{B}O\langle 8\rangle)$ was Haynes Miller.

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

• Stefan Stolz, Peter Teichner, The spinor bundle on loop space (2005) (pdf)

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

• Chris Schommer-Pries, Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-Group (arXiv:0911.2483)

In

• Thomas Nikolaus, Christoph Sachse, Christoph Wockel, A Smooth Model for the String Group, Int. Math. Res. Not. IMRN 16 (2013) 3678-3721, doi:10.1093/imrn/rns154, (arXiv:1104.4288)

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

For relation to conformal nets see

• Bas Janssens, Notes on Defects & String Group (pdf)