group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
spin geometry, string geometry, fivebrane geometry …
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.
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.
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
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 $B String(n) := B O(n)\langle 7 \rangle$
The loop space
is the ∞-group-object in Top called the string group.
Write now
for the (∞,1)-topos Smooth∞Grpd of smooth ∞-groupoids, regarded as a cohesive (∞,1)-topos over ∞Grpd.
There is a lift through $\Pi$ of $\frac{1}{2} p_1$ to the smooth first fractional Pontryagin class
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
on the orthogonal Lie algebra.
This is shown in (FSS).
Write $\mathbf{B}String(n)$ for the homotopy fiber of the smooth first fractional Pontryagin class
in Smooth∞Grpd. Its loop space object
is the smooth ∞-group called the smooth string 2-group.
Write
for the intrinsic geometric realization in Smooth∞Grpd.
The smooth string 2-group, def. 2, indeed maps under $\vert-\vert$ to the topological string group:
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 $\frac{1}{2}\mathbf{p}_1$.
Several explicit presentations of the string Lie 2-group are known.
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 $CE(\mathfrak{g})$.
Write
for the canonical degree-3 cocycle in the Lie algebra cohomology of the special orthogonal group, normalized such that the 3-form
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
to be given by the Chevalley-Eilenberg algebra
which is that of $\mathfrak{so}(n)$ with a single generator $b$ in degree 3 adjoined and the differential given by
We have a pullback square in $L_\infty Alg$
See string Lie 2-algebra for more discussion.
The Lie integration of $\mathfrak{string}(n)$ yields a presentation of the smooth String 2-group, def. 2
This is essentially the model considered in (Henriques), discussed here in the context of Smooth∞Grpd as described in (FSS).
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
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
Let $I$ be the set of maps $\partial \Delta[4] \to \exp(b^2 \mathbb{R})$ that fit into a diagram
(closed 3-forms on 3-balls whose integral is an integer).
Write
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)
we get the diagram
This is again a pullback diagram of a fibration resolution of the point inclusion, hence presents the homotopy fiber in question.
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:
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
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).
The string 2-group also appears as a certain automorphism 2-group inside the 3-category of fermionic conformal nets (Douglas-Henriques)
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
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)
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 $\hookrightarrow$ general linear group
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
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