nLab string 2-group (Rev #50, changes)

Showing changes from revision #49 to #50: Added | ~~Removed~~ | ~~Chan~~ged

Context

Higher Lie theory

Cohomology

Higher spin geometry

spin geometry, string geometry, fivebrane geometry …

Ingredients

Spin geometry

spin geometry

rotation groups in low dimensions:

Dynkin label	sp. orth. group	spin group	pin group	semi-spin group
	SO(2)	Spin(2)	Pin(2)
B1	SO(3)	Spin(3)	Pin(3)
D2	SO(4)	Spin(4)	Pin(4)
B2	SO(5)	Spin(5)	Pin(5)
D3	SO(6)	Spin(6)
B3	SO(7)	Spin(7)
D4	SO(8)	Spin(8)		SO(8)
B4	SO(9)	Spin(9)
D5	SO(10)	Spin(10)
B5	SO(11)	Spin(11)
D6	SO(12)	Spin(12)
	$\vdots$	$\vdots$
D8	SO(16)	Spin(16)		SemiSpin(16)
	$\vdots$	$\vdots$
D16	SO(32)	Spin(32)		SemiSpin(32)

String geometry

string geometry

Fivebrane geometry

Ninebrane geometry

ninebrane 10-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. 2, 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. 2

\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 $[CartSp_{smooth}^{op}, sSet]_{proj}$ 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

fivebrane 6-group $\to$ string 2-group $\to$ spin group $\to$ special orthogonal group $\to$ orthogonal group $\hookrightarrow$ general linear 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](http://arxiv.org/abs/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, Volume: 2013 , Issue: 16 , 2013 (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](http://arxiv.org/abs/1201.5052), doi:10.1090/conm/584/11588]

Via fermionic nets/2-Clifford algebra:

Chris Douglas, André Henriques, Geometric string structures ( ~~pdf~~ 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

Revision on August 26, 2024 at 09:17:57 by perezl.alonso See the history of this page for a list of all contributions to it.

nLab string 2-group (Rev #50, changes)

Context

Higher Lie theory

Cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Higher spin geometry

Ingredients

Spin geometry

String geometry

Fivebrane geometry

Ninebrane geometry

String theory

Ingredients

Critical string models

Extended objects

Topological strings

Backgrounds

Phenomenology

Contents

Idea

Definition

Definition

Proposition

Definition

Properties

Proposition

Proof

Presentations

By Lie integration of the string Lie 2-algebra

Definition

Proposition

Proposition

Proof

By strict Lie 22-group

As a finite-dimensional weak Lie 2-group

As an automorphism 2-group of fermionic CFT

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

Proposition

Related concepts

References

By strict Lie $2$ -group