∞-Lie theory

cohomology

spin geometry

string geometry

# Contents

## Idea

The string 2-group is a smooth 2-group-refinement of the topological group called the string group. It is the ∞-group extension induced by the smooth/stacky version of the first fractional Pontryagin class/second Chern class.

## Definition

A string 2-group extension $\mathrm{String}\left(G\right)$ is defined for every simple simply connected compact Lie group $G$, such as the spin group $G=\mathrm{Spin}\left(n\right)$ or the special unitary group $G=\mathrm{SU}\left(n\right)$ (for non-low $n$).

Since string structures arise predominantly as higher analogs of spin structures, the default choice is $G=\mathrm{Spin}$ and in that case one usually just writes $\mathrm{String}=\mathrm{String}\left(\mathrm{Spin}\right)$, for short.

Recall first that the string group in Top is one step in the Whitehead tower of the orthogonal group.

###### Definition

For $n\in ℕ$ let $\mathrm{Spin}\left(n\right)$ denote the spin group, regarded as a topological group. Write $B\mathrm{Spin}\left(n\right)\in$ Top for its classifying space and

$\frac{1}{2}{p}_{1}:B\mathrm{Spin}\left(n\right)\to {B}^{4}ℤ$\frac{1}{2}p_1 : B Spin(n) \to B^4 \mathbb{Z}

for a representative of the characteristic class called the first fractional Pontryagin class. Its homotopy fiber in the (∞,1)-topos Top $\simeq$ ∞Grpd is denoted $B\mathrm{String}\left(n\right):=BO\left(n\right)⟨7⟩$

$\begin{array}{ccc}B\mathrm{String}\left(n\right)& \to & *\\ ↓& & ↓\\ B\mathrm{Spin}\left(n\right)& \stackrel{\frac{1}{2}{p}_{1}}{\to }& {B}^{4}ℤ\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ B String(n) &\to& * \\ \downarrow && \downarrow \\ B Spin(n) &\stackrel{\frac{1}{2} p_1}{\to}& B^4 \mathbb{Z} } \,.

The loop space

$\mathrm{String}\left(n\right):=\Omega B\mathrm{String}\left(n\right)$String(n) := \Omega B String(n)

is the ∞-group-object in Top called the string group.

Write now

$\left(\Pi ⊣\mathrm{Disc}⊣\Gamma ⊣\mathrm{coDisc}\right):\mathrm{Smooth}\infty \mathrm{Grpd}\stackrel{\stackrel{\Pi }{\to }}{\stackrel{\stackrel{\mathrm{Disc}}{←}}{\stackrel{\stackrel{\Gamma }{\to }}{\underset{\mathrm{coDisc}}{←}}}}\infty \mathrm{Grpd}\simeq \mathrm{Top}$(\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : Smooth\infty Grpd \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{coDisc}{\leftarrow}}}} \infty Grpd \simeq Top

for the (∞,1)-topos Smooth∞Grpd of smooth ∞-groupoids, regarded as a cohesive (∞,1)-topos over ∞Grpd.

###### Proposition

There is a lift through $\Pi$ of $\frac{1}{2}{p}_{1}$ to the smooth first fractional Pontryagin class

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

in Smooth∞Grpd, where

This is shown in (FSS).

###### Definition

Write $B\mathrm{String}\left(n\right)$ for the homotopy fiber of the smooth first fractional Pontryagin class

$\begin{array}{ccc}B\mathrm{String}& \to & *\\ ↓& & ↓\\ B\mathrm{Spin}& \stackrel{\frac{1}{2}{p}_{1}}{\to }& {B}^{3}U\left(1\right)\end{array}$\array{ \mathbf{B}String &\to& * \\ \downarrow && \downarrow \\ \mathbf{B}Spin &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}& \mathbf{B}^3 U(1) }

in Smooth∞Grpd. Its loop space object

$\mathrm{String}\left(n\right):=\Omega B\mathrm{String}\left(n\right)$String(n) := \Omega \mathbf{B}String(n)

is the smooth ∞-group called the smooth string 2-group.

## Properties

Write

$\mid -\mid :=\mid \Pi \left(-\right)\mid :\mathrm{Smooth}\infty \mathrm{Grpd}\stackrel{\Pi }{\to }\infty \mathrm{Grpd}\stackrel{\mid -\mid }{\to }\mathrm{Top}$\vert - \vert := \vert\Pi(-)\vert : Smooth \infty Grpd \stackrel{\Pi}{\to} \infty Grpd \stackrel{\vert - \vert}{\to} Top
###### Proposition

The smooth string 2-group, def. 2, indeed maps under $\mid -\mid$ to the topological string group:

$\mid B\mathrm{String}\left(n\right)\mid \simeq B\mathrm{String}\left(n\right)\phantom{\rule{thinmathspace}{0ex}}.$\vert \mathbf{B}String(n) \vert \simeq B String(n) \,.
###### Proof

Since ${B}^{3}U\left(1\right)$ is presented by a simplicial presheaf that is degreewise presented by a paracompact smooth manifold (a finite product of the circle group with itself), it follows from the general properties of $\Pi$ discussed at Smooth∞Grpd that $\Pi$ preserves the homotopy fiber of $\frac{1}{2}{p}_{1}$.

## Presentations

Several explicit presentations of the string Lie 2-group are known.

### By Lie integration of the string Lie 2-algebra

We discuss a presentation of the smooth string 2-group by Lie integration of the skeletal version of the string Lie 2-algebra.

Recall the identification of L-∞ algebras $𝔤$ with their dual Chevalley-Eilenberg algebras $\mathrm{CE}\left(𝔤\right)$.

###### Definition

Write

$\mu :=⟨-,\left[-,-\right]⟩:\mathrm{𝔰𝔬}\left(n\right)\to {b}^{2}ℝ$\mu := \langle - ,[-,-]\rangle : \mathfrak{so}(n) \to b^2 \mathbb{R}

for the canonical degree-3 cocycle in the Lie algebra cohomology of the special orthogonal group, normalized such that the 3-form

${\Omega }^{•}\left(\mathrm{Spin}\left(n\right)\right)↩\mathrm{CE}\left(\mathrm{𝔰𝔬}\left(n\right)\right)\stackrel{\mu }{←}\mathrm{CE}\left({b}^{2}ℝ\right)$\Omega^\bullet(Spin(n)) \hookleftarrow CE(\mathfrak{so}(n)) \stackrel{\mu}{\leftarrow} CE(b^2 \mathbb{R})

represents the image in de Rham cohomology of a generators of the integral cohomology group ${H}^{3}\left(G,ℤ\right)\simeq ℤ$.

Define the string Lie 2-algebra

$\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}\left(n\right):=\mathrm{𝔰𝔬}\left(n{\right)}_{\mu }$\mathfrak{string}(n) := \mathfrak{so}(n)_\mu

to be given by the Chevalley-Eilenberg algebra

$\mathrm{CE}\left(\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}\left(n\right)\right):={\wedge }^{•}\left(\mathrm{𝔰𝔬}\left(n{\right)}^{*}\oplus ⟨b⟩,{d}_{\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}}\right)$CE(\mathfrak{string}(n)) := \wedge^\bullet ( \mathfrak{so}(n)^* \oplus \langle b\rangle , d_{\mathfrak{string}})

which is that of $\mathrm{𝔰𝔬}\left(n\right)$ with a single generator $b$ in degree 3 adjoined and the differential given by

${d}_{\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}}{\mid }_{\mathrm{𝔰𝔬}\left(n{\right)}^{*}}={d}_{\mathrm{𝔰𝔬}\left(n\right)};$d_{\mathfrak{string}}|_{\mathfrak{so}(n)^*} = d_{\mathfrak{so}(n)};
${d}_{\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}}:b↦\mu \phantom{\rule{thinmathspace}{0ex}}.$d_{\mathfrak{string}} : b \mapsto \mu \,.
###### Proposition

We have a pullback square in ${L}_{\infty }\mathrm{Alg}$

$\begin{array}{ccc}\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}\left(n\right)& \to & ebℝ\\ ↓& & ↓\\ \mathrm{𝔰𝔬}\left(n\right)& \stackrel{\mu }{\to }& {b}^{2}ℝ\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ \mathfrak{string}(n) &\to& e b \mathbb{R} \\ \downarrow && \downarrow \\ \mathfrak{so}(n) &\stackrel{\mu}{\to}& b^2 \mathbb{R} } \,.

See string Lie 2-algebra for more discussion.

###### Proposition

The Lie integration of $\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}\left(n\right)$ yields a presentation of the smooth String 2-group, def. 2

${\mathrm{cosk}}_{3}\mathrm{exp}\left(\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}\left(n\right)\right)\simeq B\mathrm{String}\left(n\right)\phantom{\rule{thinmathspace}{0ex}}.$\mathbf{cosk}_3 \exp(\mathfrak{string}(n)) \simeq \mathbf{B} String(n) \,.

This is essentially the model considered in (Henriques), discussed here in the context of Smooth∞Grpd as described in (FSS).

###### Proof

We observe the image under Lie integration of the ${L}_{\infty }$-algebra pullback diagram from prop. 3 is a pullback diagram in $\left[{\mathrm{CartSp}}_{\mathrm{smooth}}^{\mathrm{op}},\mathrm{sSet}{\right]}_{\mathrm{proj}}$ that presents the defining homotopy fiber. Before applying the coskeleton operation we have immediately

$\mathrm{exp}\left(-\right)\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}\left(\begin{array}{ccc}\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}\left(n\right)& \to & ebℝ\\ ↓& & ↓\\ \mathrm{𝔰𝔬}\left(n\right)& \stackrel{\mu }{\to }& {b}^{2}ℝ\end{array}\right)\phantom{\rule{thickmathspace}{0ex}}↦\phantom{\rule{thickmathspace}{0ex}}\left(\begin{array}{ccc}\mathrm{exp}\left(\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}\left(n\right)\right)& \to & \mathrm{exp}\left(ebℝ\right)\\ ↓& & ↓\\ \mathrm{exp}\left(\mathrm{𝔰𝔬}\left(n\right)\right)& \stackrel{\mu }{\to }& \mathrm{exp}\left({b}^{2}ℝ\right)\end{array}\right)$\exp(-) \; :\; \left( \array{ \mathfrak{string}(n) &\to& e b \mathbb{R} \\ \downarrow && \downarrow \\ \mathfrak{so}(n) &\stackrel{\mu}{\to}& b^2 \mathbb{R} } \right) \;\mapsto \; \left( \array{ \exp(\mathfrak{string}(n)) &\to& \exp(e b \mathbb{R}) \\ \downarrow && \downarrow \\ \exp(\mathfrak{so}(n)) &\stackrel{\mu}{\to}& \exp(b^2 \mathbb{R}) } \right)

such that on the right we still have a pullback diagram.

We discuss the descent o this pullback diagram along the projection $\mathrm{exp}\left(\mathrm{𝔰𝔬}\left(n\right)\right)\to {\mathrm{cosk}}_{3}\mathrm{exp}\left(\mathrm{𝔰𝔬}\left(n\right)\right)$.

Notice from Lie integration the weak equivalence

${\int }_{{\Delta }^{•}}:\mathrm{exp}\left({b}^{n}ℝ\right)\simeq {B}^{n+1}{ℝ}_{c}\phantom{\rule{thinmathspace}{0ex}}.$\int_{\Delta^\bullet} : \exp(b^n \mathbb{R}) \simeq \mathbf{B}^{n+1}\mathbb{R}_c \,.

Let $I$ be the set of maps $\partial \Delta \left[4\right]\to \mathrm{exp}\left({b}^{2}ℝ\right)$ that fit into a diagram

$\begin{array}{ccc}\partial \Delta \left[4\right]& \to & \mathrm{exp}\left({b}^{2}ℝ\right)\\ ↓& & {↓}^{{\int }_{{\Delta }^{•}}}\\ & & {B}^{3}{ℝ}_{c}\\ ↓& & ↓\\ \Delta \left[4\right]& \to & {B}^{3}\left(ℤ\to ℝ{\right)}_{c}\end{array}$\array{ \partial \Delta[4] &\to& \exp(b^2 \mathbb{R}) \\ \downarrow && \downarrow^{\mathrlap{\int_{\Delta^\bullet}}} \\ && \mathbf{B}^3 \mathbb{R}_c \\ \downarrow && \downarrow \\ \Delta[4] &\to& \mathbf{B}^3 (\mathbb{Z} \to \mathbb{R})_c }

(closed 3-forms on 3-balls whose integral is an integer).

Write

$\mathrm{exp}\left({b}^{2}ℝ/ℤ\right):={\mathrm{cosk}}_{3}\left(\left(I×\Delta \left[4\right]\right)\coprod _{I×\partial \Delta \left[4\right]}{\mathrm{cosk}}_{3}\mathrm{exp}\left({b}^{2}ℝ\right)\right)$\exp(b^2 \mathbb{R}/\mathbb{Z}) := \mathbf{cosk}_3 \left( (I \times \Delta[4])\coprod_{I \times \partial \Delta[4]} \mathbf{cosk_3} \exp(b^2 \mathbb{R}) \right)

for the result of filling all these by 4-cells. Similarly define $\mathrm{exp}\left(ebℝ/ℤ\right)$.

Then applying the coskeleton functor to the above pullback diagram and using the projection (FSS)

$\begin{array}{ccc}\mathrm{exp}\left(\mathrm{𝔰𝔬}\left(n\right)\right)& \stackrel{\mathrm{exp}\left(\mu \right)}{\to }& \mathrm{exp}\left({b}^{2}ℝ\right)\\ ↓& & ↓\\ {\mathrm{cosk}}_{3}\mathrm{exp}\left(\mathrm{𝔰𝔬}\left(n\right)\right)& \stackrel{\frac{1}{2}{p}_{1}}{\to }& \mathrm{exp}\left({b}^{2}ℝ/ℤ\right)\end{array}$\array{ \exp(\mathfrak{so}(n)) &\stackrel{\exp(\mu)}{\to}& \exp(b^2 \mathbb{R}) \\ \downarrow && \downarrow \\ \mathbf{cosk}_3\exp(\mathfrak{so}(n)) &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}& \exp(b^2 \mathbb{R}/\mathbb{Z}) }

we get the diagram

$\begin{array}{ccc}{\mathrm{cosk}}_{3}\left(\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}\left(n\right)\right)& \to & \mathrm{exp}\left(ebℝ/ℤ\right)\\ ↓& & ↓\\ {\mathrm{cosk}}_{3}\mathrm{exp}\left(\mathrm{𝔰𝔬}\right)& \stackrel{\frac{1}{2}{p}_{1}}{\to }& \mathrm{exp}\left({b}^{2}ℝ/ℤ\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ \mathbf{cosk}_3 (\mathfrak{string}(n)) &\to& \exp(e b \mathbb{R}/\mathbb{Z}) \\ \downarrow && \downarrow \\ \mathbf{cosk}_3 \exp(\mathfrak{so}) &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}& \exp(b^2 \mathbb{R}/\mathbb{Z}) } \,.

This is again a pullback diagram of a fibration resolution of the point inclusion, hence presents the homotopy fiber in question.

### By strict Lie $2$-group

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

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)

(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:

$\begin{array}{ccccccc}\mathrm{StrLie}\omega \mathrm{Grpd}& & & & \mathrm{StrLie}\omega \mathrm{Grpd}& \stackrel{\simeq }{←}& \mathrm{LieCrsdCmplx}\\ {↑}^{{\Pi }_{n}S\mathrm{CE}}& & & & ↑& & {↑}^{\mathrm{exp}\left(-\right)}\\ {L}_{\infty }\mathrm{Algebras}& & ←& & \mathrm{Str}{L}_{\infty }\mathrm{Algebras}& \to & \mathrm{DiffCrsdCmplx}\end{array}$\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 , ${L}_{\infty }\mathrm{Algebra}$ is all L-infinity algebras and $\mathrm{Str}{L}_{\infty }\mathrm{Algebra}$ is strict ${L}_{\infty }$-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 ${\Pi }_{n}\left(-\right)$ of that.

For the String-case this yields

$\begin{array}{ccccccc}{\Pi }_{2}\left({\Omega }_{\mathrm{fl}}^{•}\left(-,{\mathrm{𝔰𝔬}}_{{\mu }_{3}}\right)\right)& \stackrel{\simeq }{↦}& B{\mathrm{String}}_{\mathrm{Mick}}& \stackrel{\simeq }{↦}& B{\mathrm{String}}_{\mathrm{BCSS}}& ←\mid & \left(\stackrel{^}{\Omega }\mathrm{Spin}\to P\mathrm{Spin}\right)\\ ↑& & & ↗& ↑& & ↑\\ {\mathrm{𝔰𝔬}}_{{\mu }_{3}}& & \stackrel{\simeq }{↦}& & \mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}& ↦& \left(\stackrel{^}{\Omega }\mathrm{𝔰𝔬}\to P\mathrm{𝔰𝔬}\right)\end{array}\phantom{\rule{thinmathspace}{0ex}},$\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 ${\Pi }_{2}\left({\Omega }_{\mathrm{fl}}^{•}\left(-,{𝔤}_{{\mu }_{3}}\right)\right)$ has, notice, as morphism smooth paths in $\mathrm{Spin}\left(n\right)$ 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 $\mathrm{Spin}\left(n\right)$ 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 $\mathrm{Spin}\left(n\right)$ that are composed using the group product).

### As an automorphism 2-group of fermionic CFT

The string 2-group also appears as a certain automorphism 2-group inside the 3-category of fermionic conformal nets (Douglas-Henriques)

### As the automorphisms of the Wess-Zumino-Witten gerbe 2-connection

For $G$ a compact simply connected simple Lie group, there is the “WZW gerbe”, hence the circle 2-bundle with connection on $G$ whose curvature 3-form is the left invariant extension $⟨\theta \wedge \left[\theta \wedge \theta \right]⟩$ of the canonical Lie algebra 3-cocycle to the group

${ℒ}_{\mathrm{WZW}}\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}G⟶{B}^{2}\phantom{\rule{thinmathspace}{0ex}}.$\mathcal{L}_{WZW} \;\colon\; G \longrightarrow \mathbf{B}^2 \,.
###### Proposition

The string 2-group is the smooth 2-group of automorphism of ${ℒ}_{\mathrm{WZW}}$ which cover the left action of $G$ on itself (hence the “Heisenberg 2-group” of ${ℒ}_{\mathrm{WZW}}$ regarded as a prequantum 2-bundle)

$\mathrm{Aut}\left({ℒ}_{\mathrm{WZW}}\right)\simeq \mathrm{String}\left(G\right)\phantom{\rule{thinmathspace}{0ex}},$\mathbf{Aut}(\mathcal{L}_{WZW}) \simeq String(G) \,,

This is due to (Fiorenza-Rogers-Schreiber 13, section 6.2.1).

fivebrane 6-group $\to$ string 2-group $\to$ spin group $\to$ special orthogonal group $\to$ orthogonal group $↪$ general linear group

## References

A crossed module presentation of a topological realization of the string 2-group is implicit in

A realization of the string 2-group in ∞-groupoids internal to Banach spaces by Lie integration of the skeletal version of the string Lie 2-algebra is in

A realization of the string 2-group in strict 2-groups internal to Frechet manifolds by Lie integration of a strict Lie 2-algebra incarnation of the string Lie 2-algebra in in

• BCSS, From loop groups to 2-groups (web)

A realization of the string 2-group as a 2-group in finite-dimensional smooth manifolds in in

A discussion as an ∞-group object in Smooth∞Grpd and the realization of the smooth first fractional Pontryagin class is in

and in section 4.1 of

A 2-group model which has a smoothening of the topological string group in lowest degree has been given in

A construction explicitly in terms of the “basic” bundle gerbe on $G$ is discussed in

Via fermionic nets/2-Clifford algebra:

The realization of the string 2-group as the Heisenberg 2-group of the WZW gerbe is due to

Revised on November 6, 2013 08:54:48 by Urs Schreiber (145.116.129.122)