Higher Lie theory
Formal Lie groupoids
Special and general types
Higher spin geometry
Critical string models
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 is defined for every simple simply connected compact Lie group , such as the spin group or the special unitary group (for non-low ).
Since string structures arise predominantly as higher analogs of spin structures, the default choice is and in that case one usually just writes , for short.
Recall first that the string group in Top is one step in the Whitehead tower of the orthogonal group.
For let denote the spin group, regarded as a topological group. Write 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 ∞Grpd is denoted
The loop space
is the ∞-group-object in Top called the string group.
for the (∞,1)-topos Smooth∞Grpd of smooth ∞-groupoids, regarded as a cohesive (∞,1)-topos over ∞Grpd.
This is shown in (FSS).
Write 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.
for the intrinsic geometric realization in Smooth∞Grpd.
The smooth string 2-group, def. 2, indeed maps under to the topological string group:
Since 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 discussed at Smooth∞Grpd that preserves the homotopy fiber of .
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 with their dual Chevalley-Eilenberg algebras .
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 .
Define the string Lie 2-algebra
to be given by the Chevalley-Eilenberg algebra
which is that of with a single generator in degree 3 adjoined and the differential given by
We have a pullback square in
See string Lie 2-algebra for more discussion.
The Lie integration of 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 -algebra pullback diagram from prop. 3 is a pullback diagram in 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 .
Notice from Lie integration the weak equivalence
Let be the set of maps that fit into a diagram
(closed 3-forms on 3-balls whose integral is an integer).
for the result of filling all these by 4-cells. Similarly define .
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.
By strict Lie -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:
Here is strict omega-groupoids internal to diffeological spaces, is accordingly smooth crossed complexes , is all L-infinity algebras and is strict -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 of that.
For the String-case this yields
denotes the weak, skeletal String Lie 2-algebra
its equivalent strict version given by BCSS
the diagonal morphism is the construction in BCSS.
the strict 2-groupoid has, notice, as morphism smooth paths in that are composed by concatenation
the 2-groupoid is a version of the String Lie 2-group that manifestly uses the Mickelsson cocycle? (morphism are paths in that are composed using the group product)
the 2-groupoid is the version given in BCSS (morhisms again are paths in that are composed using the group product).
As a finite-dimensional weak Lie 2-group
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 a compact simply connected simple Lie group, there is the “WZW gerbe”, hence the circle 2-bundle with connection on whose curvature 3-form is the left invariant extension of the canonical Lie algebra 3-cocycle to the group
This is due to (Fiorenza-Rogers-Schreiber 13, section 6.2.1).
fivebrane 6-group string 2-group spin group special orthogonal group orthogonal group 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
- 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 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