nLab string 2-group

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 $String(G)$ is defined for every simple simply connected compact Lie group $G$, such as the spin group $G = Spin(n)$ or the special unitary group $G = SU(n)$ (for non-low $n$).

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

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

Definition

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

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

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 $B String(n) := B O(n)\langle 7 \rangle$

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

The loop space

$$String(n) := \Omega B String(n)$$

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

Write now

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

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

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

in Smooth∞Grpd, where

• $\mathbf{B}Spin$ is the delooping Lie groupoid of $Spin(n)$ regarded as a Lie group (see Smooth ∞-groupoids – Cohesive ∞-groups – Lie groups);

• $\mathbf{B}^3 U(1)$ is the three-fold delooping of the circle group, regarded as a Lie group (see Smooth ∞-groupoids – Cohesive ∞-groups – Circle Lie n-group);

• $\frac{1}{2}\mathbf{p}_1$ is the image under Lie integration of the canonical cocycle

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

  on the orthogonal Lie algebra.

This is shown in (FSS).

Definition

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

$$\array{ \mathbf{B}String &\to& * \\ \downarrow && \downarrow \\ \mathbf{B}Spin &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}& \mathbf{B}^3 U(1) }$$

in Smooth∞Grpd. Its loop space object

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

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

Properties

Write

$$\vert - \vert := \vert\Pi(-)\vert : Smooth \infty Grpd \stackrel{\Pi}{\to} \infty Grpd \stackrel{\vert - \vert}{\to} Top$$

for the intrinsic geometric realization in Smooth∞Grpd.

Proposition

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

$$\vert \mathbf{B}String(n) \vert \simeq B String(n) \,.$$

Proof

Since $\mathbf{B}^3 U(1)$ 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 $\frac{1}{2}\mathbf{p}_1$.

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 $\mathfrak{g}$ with their dual Chevalley-Eilenberg algebras $CE(\mathfrak{g})$.

Definition

Write

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

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

$$\Omega^\bullet(Spin(n)) \hookleftarrow CE(\mathfrak{so}(n)) \stackrel{\mu}{\leftarrow} CE(b^2 \mathbb{R})$$

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

Define the string Lie 2-algebra

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

to be given by the Chevalley-Eilenberg algebra

$$CE(\mathfrak{string}(n)) := \wedge^\bullet ( \mathfrak{so}(n)^* \oplus \langle b\rangle , d_{\mathfrak{string}})$$

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

$$d_{\mathfrak{string}}|_{\mathfrak{so}(n)^*} = d_{\mathfrak{so}(n)};$$

$$d_{\mathfrak{string}} : b \mapsto \mu \,.$$

Proposition

We have a pullback square in $L_\infty Alg$

$$\array{ \mathfrak{string}(n) &\to& e b \mathbb{R} \\ \downarrow && \downarrow \\ \mathfrak{so}(n) &\stackrel{\mu}{\to}& b^2 \mathbb{R} } \,.$$

See string Lie 2-algebra for more discussion.

Proposition

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

$$\mathbf{cosk}_3 \exp(\mathfrak{string}(n)) \simeq \mathbf{B} String(n) \,.$$

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. is a pullback diagram in $[CartSp_{smooth}^{op}, sSet]_{proj}$ that presents the defining homotopy fiber. Before applying the coskeleton operation we have immediately

$$\exp(-) \; :\; \left( \array{ \mathfrak{string}(n) &\to& e b \mathbb{R} \\ \downarrow && \downarrow \\ \mathfrak{so}(n) &\stackrel{\mu}{\to}& b^2 \mathbb{R} } \right) \;\mapsto \; \left( \array{ \exp(\mathfrak{string}(n)) &\to& \exp(e b \mathbb{R}) \\ \downarrow && \downarrow \\ \exp(\mathfrak{so}(n)) &\stackrel{\mu}{\to}& \exp(b^2 \mathbb{R}) } \right)$$

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

We discuss the descent o this pullback diagram along the projection $\exp(\mathfrak{so}(n)) \to \mathbf{cosk}_3 \exp(\mathfrak{so}(n))$.

Notice from Lie integration the weak equivalence

$$\int_{\Delta^\bullet} : \exp(b^n \mathbb{R}) \simeq \mathbf{B}^{n+1}\mathbb{R}_c \,.$$

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

$$\array{ \partial \Delta[4] &\to& \exp(b^2 \mathbb{R}) \\ \downarrow && \downarrow^{\mathrlap{\int_{\Delta^\bullet}}} \\ && \mathbf{B}^3 \mathbb{R}_c \\ \downarrow && \downarrow \\ \Delta[4] &\to& \mathbf{B}^3 (\mathbb{Z} \to \mathbb{R})_c }$$

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

Write

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

for the result of filling all these by 4-cells. Similarly define $\exp(e b \mathbb{R}/\mathbb{Z})$.

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

$$\array{ \exp(\mathfrak{so}(n)) &\stackrel{\exp(\mu)}{\to}& \exp(b^2 \mathbb{R}) \\ \downarrow && \downarrow \\ \mathbf{cosk}_3\exp(\mathfrak{so}(n)) &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}& \exp(b^2 \mathbb{R}/\mathbb{Z}) }$$

we get the diagram

$$\array{ \mathbf{cosk}_3 (\mathfrak{string}(n)) &\to& \exp(e b \mathbb{R}/\mathbb{Z}) \\ \downarrow && \downarrow \\ \mathbf{cosk}_3 \exp(\mathfrak{so}) &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}& \exp(b^2 \mathbb{R}/\mathbb{Z}) } \,.$$

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)

• 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:

$$\array{ StrLie \omega Grpd &&&& StrLie \omega Grpd &\stackrel{\simeq}{\leftarrow}& LieCrsdCmplx \\ \uparrow^{\Pi_n S CE} &&&& \uparrow && \uparrow^{\exp(-)} \\ L_\infty Algebras && \leftarrow&& Str L_\infty Algebras &\to& DiffCrsdCmplx }$$

Here $StrLie \omega Grpd$ is strict omega-groupoids internal to diffeological spaces, $LieCrsCmplx$ is accordingly smooth crossed complexes , $L_\infty Algebra$ is all L-infinity algebras and $Str L_\infty Algebra$ 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 $\Pi_n(-)$ of that.

For the String-case this yields

$$\array{ \Pi_2(\Omega^\bullet_{fl}(-,\mathfrak{so}_{\mu_3})) &\stackrel{\simeq}{\mapsto}& \mathbf{B} String_{Mick} &\stackrel{\simeq}{\mapsto}& \mathbf{B} String_{BCSS} &\leftarrow|& (\hat \Omega Spin \to P Spin) \\ \uparrow &&&\nearrow& \uparrow && \uparrow \\ \mathfrak{so}_{\mu_3} &&\stackrel{\simeq}{\mapsto}&& \mathfrak{string} &\mapsto& (\hat \Omega \mathfrak{so} \to P \mathfrak{so}) } \,,$$

where

• $\mathfrak{so}_{\mu_3}$ denotes the weak, skeletal String Lie 2-algebra

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

• the diagonal morphism is the construction in BCSS.

• the strict 2-groupoid $\Pi_2(\Omega^\bullet_{fl}(-,\mathfrak{g}_{\mu_3}))$ has, notice, as morphism smooth paths in $Spin(n)$ that are composed by concatenation

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

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

As a finite-dimensional weak Lie 2-group

(Schommer-Pries)

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 $\langle \theta \wedge [\theta \wedge \theta]\rangle$ of the canonical Lie algebra 3-cocycle to the group

$$\mathcal{L}_{WZW} \;\colon\; G \longrightarrow \mathbf{B}^2U(1) \,.$$

Proposition

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

$$\mathbf{Aut}(\mathcal{L}_{WZW}) \simeq String(G) \,,$$

This is due to (Fiorenza-Rogers-Schreiber 13, section 2.6.1).

Related concepts

• string Lie 2-algebra

• string Lie algebroid

• Platonic 2-group

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

• spin group, spin structure, twisted spin structure

• spin^c, spin^c structure, twisted spin^c structure

• string group, string 2-group, string structure, twisted string structure

  string cobordism

  stringor bundle

• T-duality 2-group

• string^c 2-group

• fivebrane group, fivebrane 6-group

References

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

• Stephan Stolz, Peter Teichner, What is an elliptic object? (pdf)

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

• Andre Henriques, Integrating $L_\infty$-algebras, Compositio Mathematica, Volume 144, Issue 4 July 2008 , pp. 1017-1045 (arXiv:math/0603563, doi:10.1112/S0010437X07003405)

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

• John Baez, Alissa Crans, Urs Schreiber, Danny Stevenson, From loop groups to 2-groups, Homology Homotopy Appl. Volume 9, Number 2 (2007), 101-135. (arXiv:math/0504123, euclid:hha/1201127333)

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

• Chris Schommer-Pries, Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-Group Geometry & Topology 15 (2011) 609-676 [arXiv:0911.2483, doi:10.2140/gt.2011.15.609]

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

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Cech cocycles for differential characteristic classes (web)

and in section 4.1 of

• Urs Schreiber, differential cohomology in a cohesive topos

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

• Thomas Nikolaus, Christoph Sachse, Christoph Wockel: A Smooth Model for the String Group, International Mathematics Research Notices, 2013 16 (2013) 3678-3721 [arXiv:1104.4288, doi:10.1093/imrn/rns154]

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

• Konrad Waldorf, A Construction of String 2-Group Models using a Transgression-Regression Technique, Contemp. Math. 584 (2012) 99-115 [arXiv:1201.5052, doi:10.1090/conm/584/11588]

Via fermionic nets/2-Clifford algebra:

• Chris Douglas, André Henriques, Geometric string structures (TringWP)

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

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, Higher geometric prequantum theory, 2013 (arXiv:1304.0236)

A model of the string 2-group using the smooth free loop space (instead of the based loop space) is dicussed in

• Michael Murray, David Roberts, Christoph Wockel, Quasi-periodic paths and a string 2-group model from the free loop group (arXiv:1702.01514)

Discussion in the context of matrix factorizations and equivariant K-theory:

• Daniel S. Freed, Constantin Teleman, Dirac families for loop groups as matrix factorizations, arxiv/1409.6051

Further on 2-group-extensions by the circle 2-group:

of tori:

• Nora Ganter, Categorical Tori, SIGMA 14 (2018), 014, 18 (arXiv:1406.7046)

of finite subgroups of SU(2) (to Platonic 2-groups):

• Narthana Epa, Nora Ganter, Platonic and alternating 2-groups, Higher Structures 1(1):122-146, 2017 (arXiv:1605.09192, hs:30)

Discussion of a general definition of smooth string group extensions $A\rightarrow \mathrm{String}(H)\rightarrow H$ of a compact simply connected Lie group $H$, with $A$ not necessarily chosen to be $\mathbf{B}U(1)$ but only of the same homotopy type, in

• Severin Bunk, Principal $\infty$-Bundles and Smooth String Group Models, arXiv:2008.12263