nLab
string 2-group

Idea

The string 2-group is a smooth 2-group incarnation of the topological group called the string group.

Motivation from quantum physics

Since, by definition, the topological string group has vanishing third homotopy group, it cannot be a (finite dimensional) Lie group (because these all have nontrivial ${\pi }_3$ as soon as they are nonabelian). However, for various applications it is necessary to have a smooth model of the string group. The string 2-group provides this.

This is best understood by comparison with the situation one step down in the Whitehead tower of $O(n)$: using the spin group instead of the string group. A priori the spin group is defined as the topological group that is the universal cover of the special orthogonal group. This alone is sufficient to talk about spin structure of an oriented $n$-dimensional manifold $X$: this is a lift of the classifying map $X\to ℬ\mathrm{SO}(n)$ – a morphism of topological spaces – of the tangent bundle $TX$ of $X$ through the projection $ℬ\mathrm{Spin}(n)\to ℬ\mathrm{SO}(n)$.

And this can be said entirely within Top. In quantum physics, the existence of such a lift is necessary in order for spinning? particles to have a consistent kinematics on $X$. However, the dynamics of such spinning particles is encoded by a richer structure:

the spin group naturally has the structure not just of a topological group but of a Lie group. This means that a spin structure on $X$ may be exhibited by a smooth $\mathrm{Spin}(n)$-principal bundle. Being a smooth bundle for a Lie group, there is a notion of connection on a bundle. The choice of such refines the nonabelian cohomology class given by the spin structure $X\to ℬ\mathrm{Spin}(n)$ to a class in differential nonabelian cohomology.

Such a differential refinement of topological nonabelian cohomology is what the string 2-group admits and which the plain string group in the form of a topological group does not admit:

The kinematics of the “spinning string” called the heterotic string? requires that the spin structure $X\to ℬ\mathrm{Spin}(n)$ lifts to a string structure $X\to ℬ\mathrm{String}(n)$, again so far just as a lift in Top.

Such a lift classifies a topological string group-principal bundle on $X$. But the dynamics of the string is determined by a differential refinement of this nonabelian cohomology class (aspects of this are described at twisted differential String- and Fivebrane structures): it is given by a smooth $\mathrm{String}(n)$-principal 2-bundle with connection.

In order to make sense of this one needs an incarnation and refinement of the topological string group inside an (infinity,1)-topos of Lie infinity-groupoids. This is what the string 2-group accomplishes.

As an integration of the String Lie 2-algebra

The string Lie 2-group is the result of applying Lie integration to the String Lie 2-algebra for the case that the Lie algebra 3-cocycle this is normalized so that its image as a left-invariant 3-form on the spin group is the image in deRham cohomology of the generator of the degree integral cohomology group $H^3(\mathrm{Spin}(n),\mathbb{Z})\simeq \mathbb{Z}$.

…

In terms of Whitehead towers in a smooth $(\mathrm{\infty }\mathrm{,1})$-topos

Urs Schreiber: the following should be the right general-nonsense point of view, but strictly speaking I haven’t fully provided complete proof of this perspective, just evidence. See the technical discussion further below.

Just as the topological string group is the element above the spin group in the Whitehead tower of $O(n)$ inside the (∞,1)-topos Top – in terms of deloopings

$$(\cdots \to ℬ\mathrm{String}(n)\to ℬ\mathrm{Spin}(n)\to ℬ\mathrm{SO}(n)\to ℬO(n))\in \mathrm{Top},$$( \cdots \to \mathcal{B} String(n) \to \mathcal{B} Spin(n) \to \mathcal{B}SO(n) \to \mathcal{B}O(n) ) \in Top \,,

so the string 2-group is the Whitehead tower element above $\mathrm{Spin}(n)$, but now regarded in an (∞,1)-topos $H$ of smooth ∞-groupoids. In terms of deloopings:

$$(\cdots \to B\mathrm{String}(n)\to B\mathrm{Spin}(n)\to B\mathrm{SO}(n)\to BO(n))\in H.$$( \cdots \to \mathbf{B} String(n) \to \mathbf{B} Spin(n) \to \mathbf{B}SO(n) \to \mathbf{B}O(n) ) \in \mathbf{H} \,.

It is in $H$ the homotopy fiber of the morphism

$$\frac{1}{2}{p}_1:B\mathrm{Spin}(n)\stackrel{}{\to }{B}^{3}U(1)$$\frac{1}{2}p_1 : \mathbf{B} Spin(n) \stackrel{}{\to} \mathbf{B}^3 U(1)

that is the smooth group cocycle that integrates the normalized canonical Lie algebra 3-cocycle ${\mu }_3\in \mathrm{CE}(\mathrm{𝔰𝔬}(n))$.

Remark. Notice that in the existing literature the smooth group cocycles of a Lie group are defined to be simplicial morphisms out of

$$(\cdots G\times G\stackrel{\overrightarrow{\to }}{\to }G\overrightarrow{\to }*).$$(\cdots G\times G\stackrel{\stackrel{\to}{\to}}{\to}G\stackrel{\to}{\to}{*}) \,.

With that definition, $\frac{1}{2}{p}_1$ does not exist as a smooth cocycle. But, while that definition is copied verbatim from the correct definition in Top, it is not the right general definition for smooth group cohomology. Instead, the right definition is given by the general nonsense of cohomology in an (infinity,1)-topos $H$: a smooth group cocycle on the Lie group $G$ is a morphism out of its delooping but regarded in $H$. This $H$ in turn is modeled by simplicial presheaves which for the present purposes we may assume to be simplicial diffeological spaces. But in that context there are considerably “bigger” resolutions of $BG$ than the one above. (In fact the one above is just $BG$ in $H$ itself). The 3-cocycle $\frac{1}{2}{p}_1$ is a morphism out of one of these bigger resolutions. More on that below.

…

So the string 2-group is a smooth 2-group incarnation of…

Constructions

There are various equivalent constructions that should eventually be described here in detail. For the time being this here is very incomplete and – notably – biased. But it should improve eventally.

As a weak Lie 2-group

… Henriques…

As a strict Lie 2-group

A realization of the string 2-group as a strict 2-group internal to diffeological spaces was given in

• Baez, Crans, Schreiber, Stevenson, (arXiv)

This is one of three different (there should be more), weakly equivalent such strict 2-group internal to diffeological space models that are discussed in the (to date unpublished)

• nactwist, section 5.2.3

(This particular section, and its results, are joint work of Urs Schreiber and Danny Stevenson).

We have the following pattern of routes through Lie integration:

$$\begin{array}{ccccccc}\mathrm{StrLie}\omega \mathrm{Grpd}& & & & \mathrm{StrLie}\omega \mathrm{Grpd}& \stackrel{\simeq }{\leftarrow }& \mathrm{LieCrsdCmplx}\\ {\uparrow }^{{\Pi }_{n}S\mathrm{CE}}& & & & \uparrow & & {\uparrow }^{\exp (-)}\\ L_{\mathrm{\infty }}\mathrm{Algebras}& & \leftarrow & & \mathrm{Str}{L}_{\mathrm{\infty }}\mathrm{Algebras}& \to & \mathrm{DiffCrsdCmplx}\end{array}$$\array{ StrLie \omega Grpd &&&& StrLie \omega Grpd &\stackrel{\simeq}{\leftarrow}& LieCrsdCmplx \\ \uparrow^{\Pi_n S CE} &&&& \uparrow && \uparrow^{\exp(-)} \\ L_\infty Algebras && \leftarrow&& Str L_\infty Algebras &\to& DiffCrsdCmplx }

Here $\mathrm{StrLie}\omega \mathrm{Grpd}$ is strict omega-groupoids internal to diffeological spaces, $\mathrm{LieCrsCmplx}$ is accordingly smooth crossed complexes , $L_{\mathrm{\infty }}\mathrm{Algebra}$ is all L-infinity algebras and $\mathrm{Str}{L}_{\mathrm{\infty }}\mathrm{Algebra}$ is strict $L_{\mathrm{\infty }}$-algebras. The vertical morphism on the right is term-wise ordinary Lie integration. The other vertical morphisms take an L-infinity algebra, form the sheaf on Diff of flat ∞-Lie algebroid differential forms, and then take path n-groupoid ${\Pi }_n(-)$ of that.

For the String-case this yields

$$\begin{array}{ccccccc}{\Pi }_2({\Omega }_{\mathrm{fl}}^{•}(-,{\mathrm{𝔰𝔬}}_{{\mu }_3}))& \stackrel{\simeq }{\mapsto }& B{\mathrm{String}}_{\mathrm{Mick}}& \stackrel{\simeq }{\mapsto }& B{\mathrm{String}}_{\mathrm{BCSS}}& \leftarrow \mid & (\hat{\Omega }\mathrm{Spin}\to P\mathrm{Spin})\\ \uparrow & & & \nearrow & \uparrow & & \uparrow \\ {\mathrm{𝔰𝔬}}_{{\mu }_3}& & \stackrel{\simeq }{\mapsto }& & \mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}& \mapsto & (\hat{\Omega }\mathrm{𝔰𝔬}\to P\mathrm{𝔰𝔬})\end{array},$$\array{ \Pi_2(\Omega^\bullet_{fl}(-,\mathfrak{so}_{\mu_3})) &\stackrel{\simeq}{\mapsto}& \mathbf{B} String_{Mick} &\stackrel{\simeq}{\mapsto}& \mathbf{B} String_{BCSS} &\leftarrow|& (\hat \Omega Spin \to P Spin) \\ \uparrow &&&\nearrow& \uparrow && \uparrow \\ \mathfrak{so}_{\mu_3} &&\stackrel{\simeq}{\mapsto}&& \mathfrak{string} &\mapsto& (\hat \Omega \mathfrak{so} \to P \mathfrak{so}) } \,,

where

• ${\mathrm{𝔰𝔬}}_{{\mu }_3}$ denotes the weak, skeletal String Lie 2-algebra

• $\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}$ its equivalent strict version given by BCSS

• the diagonal morphism is the construction in BCSS.

• the strict 2-groupoid ${\Pi }_2({\Omega }_{\mathrm{fl}}^{•}(-,{𝔤}_{{\mu }_3}))$ has, notice, as morphism smooth paths in $\mathrm{Spin}(n)$ that are composed by concatenation

• the 2-groupoid $B{\mathrm{String}}_{\mathrm{Mick}}$ is a version of the String Lie 2-group that manifestly uses the Mickelsson cocycle? (morphism are paths in $\mathrm{Spin}(n)$ that are composed using the group product)

• the 2-groupoid $B{\mathrm{String}}_{\mathrm{BCSS}}$ is the version given in BCSS (morhisms again are paths in $\mathrm{Spin}(n)$ that are composed using the group product).

As a finite-dimensional weak Lie 2-group

…Schommer-Pries …

By co-killing of homotopy groups in the smooth $(\mathrm{\infty }\mathrm{,1})$-topos

…The smooth version of $\frac{1}{2}{p}_1:B\mathrm{Spin}(n)\to B^{3}R//Z$ is constructed as follows.

Use the model given by the model structure on simplicial presheaves on Diff or similar. All simplicial presheaves appearing are actually simplicial concrete sheaves hence simplicial diffeological spaces.

The cocycle is a span in this context out of an acyclic fibration over $B\mathrm{Spin}(n)$ (an anafunctor)

$$\begin{array}{ccc}BQ& \stackrel{f}{\to }& B^{3}R/Z\\ {\downarrow }^{\simeq }\\ B\mathrm{Spin}(n)\end{array}.$$\array{ \mathbf{B}Q &\stackrel{f}{\to}& \mathbf{B}^3 R/Z \\ \downarrow^{\simeq} \\ \mathbf{B}Spin(n) } \,.

The resolution $BQ$ may be chosen as follows: 1-cells are smooth paths in $\mathrm{Spin}(n)$ starting at the neutral element, 2-cells smooth 2-simplexes in $\mathrm{Spin}(n)$, 3-cells homotopy classes of smooth 3-simplexes. All other cells are degenerate. The projection to $B\mathrm{Spin}(n)$ takes paths to their endpoint. Because ${\pi }_2(\mathrm{Spin}(n))=0$ this is indeed a weak equivalence.

Then let ${\mu }_3\in {\Omega }^3(\mathrm{Spin}(n))$ be the above-mentioned de-Rham representative of the generator of $H^3(\mathrm{Spin}(n),\mathbb{Z})\simeq \mathbb{Z}$. The morphism $f$ in the above reads in 3-simplices $[\varphi :{\Delta }_{\mathrm{Diff}}^3\to \mathrm{Spin}(n)]$ and sends them to

$${\int }_{{\Delta }_{\mathrm{Diff}}^3}{\varphi }^{*}{\mu }_3\in R/Z.$$\int_{\Delta^3_{Diff}} \phi^* \mu_3 \;\; \in R/Z \,.

This is well defined because ${\mu }_3$ has integral periods.

That defines the smooth version of the cocycle. Notice how $Q$ is a considerably “bigger” resolution of $\mathrm{Spin}(n)$ than the familiar bar resolution, which doesn’t work here in the smooth case, as mentioned above.

Then using this the smooth string 2-group is defined abstractly as the homotopy fiber

$$\begin{array}{ccc}B\mathrm{String}& \to & *\\ \downarrow & & \downarrow \\ BQ& \stackrel{f}{\to }& B^{3}R/Z\\ {\downarrow }^{\simeq }\\ B\mathrm{Spin}(n)\end{array}.$$\array{ \mathbf{B} String &\to& {*} \\ \downarrow && \downarrow \\ \mathbf{B}Q &\stackrel{f}{\to}& \mathbf{B}^3 R/Z \\ \downarrow^{\simeq} \\ \mathbf{B}Spin(n) } \,.

Of course actually constructing this homotopy pullback (which is defined only up to weak equivalence of course) amounts to constructing one of the above models, or similar.

References

• Henriques

• BCSS

• Schommer-Pries

• etc …