# A 2-group construction from an extension of the 3-loop group \(\varOmega ^3G\)

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## Abstract

We define a 3-loop group \({\varOmega ^3G}\) as a subgroup of smooth maps from a 3-ball to a Lie group *G*, and then construct a 2-group based on an automorphic action on the Mickelsson–Faddeev extension of \({\varOmega ^3G}\). In this, we follow the strategy of Murray et al. (J Lie Theory 27(4):1151–1177, 2017), who earlier described a similar construction for one-dimensional loops. The three-dimensional situation presented here is further complicated by the fact that the 3-loop group extension is not central.

## Keywords

Loop group Lie 2-group Mickelsson–Faddeev extension## Mathematics Subject Classification

22E67 81R10 18D35## 1 Introduction

There is a growing interest in generalizing Lie groups and algebras to higher-dimensional objects in the sense of category theory. In particular, such smooth categorical groups would be valuable as new mathematical objects that would aid in (re)defining and building fundamental physics. One driving notion is that of the string group and its geometric realizations, which are closely tied to the concept of loop groups.

A loop group generalization in this vein has been built in [2], in which a specific relation between Lie 2-algebras and Lie 2-groups was constructed using groups of based paths. This is shown to lead to a geometric realization of the string group. More generally, there is a no-go theorem which states that the 2-group generalization coming from the direct Lie 2-algebra construction for a simple Lie group *G* allows for a smooth structure only in a very specific situation [1]. The way around this is to construct an infinite-dimensional Lie 2-group whose Lie 2-algebra is *equivalent* to the desired Lie 2-algebra.

This approach of using categorical equivalences instead of isomorphisms seems to be crucial in constructing such higher objects. With the string 2-group model in mind, the loop group setting can be extended to quasi-periodic paths [8]. Yet there is a topological obstruction for building a strict 2-group model on this group; the remedy is to flatten the paths around the base point in such a way that there is a categorical equivalence to the original 2-group. This gives a *coherent* 2-group model for the string group, which—in contrast to the model in [2]—also admits the action of the circle group \(S^1{\cong }SO(2)\).

Our aim in this paper is simple: following the strategy from [8], we construct a similar 2-group by considering the group of based maps \({\varOmega ^3G}\) from the 3-sphere to a Lie group *G*. To begin with, we study the Mickelsson–Faddeev cocycle and the related Lie group extension [6]. This cocycle is crucially dependent on defining Lie algebra valued 1-forms over the domain—in particular, the extension is not central.^{1}

On this 3-loop group extension, we impose the action of the group of smooth maps \({B^3_{\flat }G}\) from the 3-ball to the Lie group *G* with the additional condition that the maps are flattened on the boundary \(S^2\). We show that it is possible to build a crossed module starting with the groups \({B^3_{\flat }G}\) and \({\varOmega ^3G}\) in a similar fashion to the loop group described in [8]. The main difference is that the extension of the 3-loop group is not central. This complicates the lifting of the automorphic action \({B^3_{\flat }G}\rightarrow {\mathrm {aut}({\varOmega ^3G})}\) to the extension, and furthermore, we cannot define a crossed module directly from these groups. Hence, we need to extend the acting group \({B^3_{\flat }G}\) by the group \({{{\text {Map}}({B^3_{\flat }G},S^1)}}/S^1\). Then, from the automorphic action of \({\widehat{{B^3_{\flat }G}}}\) on the 3-loop group extension \({\widehat{{\varOmega ^3G}}}\) and the natural projection between these two groups we gain a crossed module and thus a strict 2-group.

One of the strengths of the original loop group construction is that it allows the action of the circle group. For 3-loops, one would like to have the corresponding action of *SO*(4), but there seems to be no straightforward way to incorporate this into the picture presented here. The extension of the 3-loop group necessitates a fixed point in \(B^3\)—namely, the contracted boundary \(S^2\)—and this choice cannot be equivariant under the symmetry action.

*Notation and conventions* Unless otherwise stated, *G* is a simply connected and finite-dimensional Lie group. The group identity element is denoted by *e* throughout, and we write \({{\mathcal {P}}(G)}\) for the group of based paths in *G*. The Lie algebra of *G* is denoted by \({\mathfrak {g}}\). We identify the 1-sphere \(S^1\) with the circle group. All the groups of functions taking values in a Lie group are considered as infinite-dimensional Fréchet Lie groups, see [9, Sec. II].

In places, there is an implicit assumption to consider the identity component of a given group in case the connectedness for the whole group is not available.

## 2 3-loop group and an automorphic action

*G*. Since every

*n*-sphere can be given as the quotient \(B^n/S^{n-1}\) where the boundary of the

*n*-ball is contracted to a point, we define by analogy the flattened group

*3-loop group*defined as

*SO*(4) acts on the sphere group \(S^3G\), it does not act on \({\varOmega ^3G}\) since the boundary of \(B^3\) is the contracted fixed point and the action cannot be extended to \(S^3\).

*LG*, there is the split exact sequence

*G*:

### 2.1 Abelian extension \({\widehat{{\varOmega ^3G}}}\) and lifting automorphisms of \({\varOmega ^3G}\) to its extension

*X*, there is an extension by the Abelian group of smooth maps \({{\text {Map}}({{\text {Map}}(X,G)},S^1)}\) [11, pp. 66–67]. In particular, we consider an Abelian extension [6]

^{2}

More generally, we have the following proposition.

### Proposition 1

*K*be a simply connected Lie group such that \(K = H/N\), where \(N \subset H\) is a normal subgroup of a simply connected Lie group

*H*for which the first three integral cohomology groups are trivial. Denote by \(\pi : H \rightarrow K\) the natural projection so that the fibration has

*N*as the canonical fibre. Then, given a cohomology class \([\omega ] \in {\mathrm {H}}^3(K,{\mathbb {Z}})\), there is a 2-form \(\theta \) on

*H*given by the pullback \(\pi ^*\omega = {\mathrm{d}}\theta \) such that the following holds:

- 1.
\(\theta \) is closed in vertical directions and defines a 2-cocycle on the Lie algebra \({\mathfrak {n}}\) of

*N*with values in \({{\text {Map}}(H,{\mathbb {R}})}\). - 2.
\([\theta ]\) in \({\mathrm {H}}^2({{\mathfrak {n}},{{\text {Map}}(H,{\mathbb {R}})}})\) is uniquely determined by \(\omega \).

- 3.
\([\theta ]\) in \({\mathrm {H}}^2({{\mathfrak {n}},{{\text {Map}}(H,{\mathbb {R}})}})\) is invariant with respect to right action \(r_f^*\), where \(f\in H\).

### Proof

The right action \(r^*_f\) is composed from the right action on the argument in \({{\text {Map}}(H, {\mathbb {R}})}\) and the (right) conjugation on \({\mathfrak {n}}.\) Recall that the Lie algebra of *N* generates vector fields on *H* through the right action of *N*. Evaluation of a 2-form \(\theta \) (which is closed in vertical directions) along these vector fields gives the 2-cocycle in the Lie algebra \({\mathfrak {n}}\) with values in smooth functions since the output of any smooth form evaluated along smooth vector fields is a smooth function.

Consider \(\theta ' = \theta + \psi \), where \(\psi \) is a closed 2-form. Since \({\mathrm {H}}^2(H,{\mathbb {Z}})\) is trivial, the 2-form \(\psi \) is also exact. Hence, the 2-cocycle \(\theta '-\theta \) in *N* is a coboundary, and \(\theta \) defines a 2-cocycle on the Lie algebra of *N* with values in \({{\text {Map}}(H,{\mathbb {R}})}\).

*H*, where \(t\in [0,1]\) with \(f(0) = e\) and \(f(1) = f\). Then, the cohomology class \([\omega ]\) is invariant by the right action in

*K*by \(\pi (f)\):

*H*is trivial, we can write for some 1-form \(\alpha \)

### Remark 1

The proposition can be applied to 3-loops with \(K=S^2G\), \(N=S_e^3G\) and \(H={{\mathcal {P}}(S^2G)}\). From this, we can squeeze out a general formula for computing the coboundary (see Appendix A).

### Remark 2

*K*. In the gauge theoretic framework, the group

*K*models the moduli space of based gauge transformations which parameterizes fermionic Fock spaces [12]. There is a natural principal bundle

*G*. The Dixmier–Douady class in \({\mathrm {H}}^3({\mathcal {A}}/{\mathcal {G}}_e)\) represents the obstruction to lifting the \({\mathcal {G}}_e\)-bundle to its extension by \({{\text {Map}}({\mathcal {A}},S^1)}\). This is also the topological origin of the Lie algebra cocycle (1) [4].

We now the lift the action to the Lie group level while retaining the cohomological invariance of the corresponding 2-cocycle. To this end, we need further results concerning cohomology; more details on the following can be found in [7, 10].

*H*be a Lie group. Recall that an Abelian Lie group

*A*is a smooth

*H*-module, if it is a

*H*-module and the action map \(H\times A \rightarrow A\) is smooth. Assuming that

*N*is a normal subgroup of

*H*, the group of

*N*-invariant elements of

*A*is denoted by

*H*-submodule of

*A*.

### Definition 1

(Refined cohomology [10, Appendix D]) Let us denote *smooth group cohomology* by \({\mathrm {H}}^n_s(N,A)\) and *continuous Lie algebra cohomology* by \({\mathrm {H}}^n_c({\mathfrak {n}},{\mathfrak {a}})\). The gist is that the maps \(N^n \rightarrow A\) (resp. \({\mathfrak {n}}^n \rightarrow {\mathfrak {a}}\)) are smooth (resp. continuous). The smoothness and continuity are defined locally in a neighbourhood of the identity.

*smoothly cohomologically invariant with respect to H*if there is a map

*smoothly invariant cohomology classes of N with values in A*.

### Theorem 1

*N*be a connected Lie group and \(A {\cong } {\mathfrak {a}}/\varGamma _A\) a smooth

*N*-module, where \(\varGamma _A \subset {\mathfrak {a}}\) is a discrete subgroup of the sequentially complete locally convex space \({\mathfrak {a}}\). Then, there is an exact sequence

*identity component*of \({\varOmega ^3G}\) in case non-connectedness becomes an issue.

Assuming \(\pi _4(G)\) is trivial, \(H = {B^3_{\flat }G}\), \(N = {\varOmega ^3G}\) and \(A = {{{\text {Map}}({B^3_{\flat }G},S^1)}}\) as above, the sequence of Theorem 1 gives a monomorphism \({\mathrm {H}}^2_s(N,A) \rightarrow {\mathrm {H}}^2_c({\mathfrak {n}},{\mathfrak {a}})\); the Lie algebra \({\mathfrak {a}}\) of *A* is of the form \({{\text {Map}}({B^3_{\flat }G},i{\mathbb {R}})}\), with the choice of \(\varGamma _A = {\mathbb {Z}}\). In the light of Eq. 2, the Lie algebra cocycle is smoothly cohomologically invariant with respect to right adjoint action by \(f\in {B^3_{\flat }G}\), and by above monomorphism the same holds on the group level.

It is then natural to ask whether we can lift the action of \({B^3_{\flat }G}\) on \({\varOmega ^3G}\) to its extension \({\widehat{{\varOmega ^3G}}}\). Given any element \(f \in {B^3_{\flat }G}\), we know that there is a smooth action by conjugation both on \({\varOmega ^3G}\) and the fibre \({{{\text {Map}}({B^3_{\flat }G},S^1)}}\). However, a map from \({B^3_{\flat }G}\) to \({\mathrm {aut}({\widehat{{\varOmega ^3G}}})}\) obtained in this fashion is not a group homomorphism; this can be seen by a direct computation already on the Lie algebra level. Generally, we have the following proposition.

### Proposition 2

*H*be a Lie group,

*N*a connected normal Lie subgroup of

*H*, \(\theta \in {\mathrm {Z}}^2_s(N,A)\) a smooth 2-cocycle and \({\widehat{N}}\) the corresponding Lie group extension by an Abelian group

*A*. Then, the smooth group homomorphism \(\psi : H \rightarrow {\mathrm {aut}(A)} \times {\mathrm {aut}(N)}\) lifts to a smooth homomorphism \({\widehat{\psi }}:H\rightarrow {\mathrm {aut}({\widehat{N}},A)}\) if and only if

- 1.
\(\theta \) is smoothly cohomologically invariant with respect to

*H*, and - 2.
the corresponding cohomology class \([{\mathrm{d}}_{\psi }\phi ] \in {\mathrm {H}}^2_s(H,{\mathrm {Z}}^1_s(N,A))\) is trivial, where the 1-cocycle \(\phi \) is defined via \({\mathrm{d}}_N (\phi (h)) = h.\theta - \theta \) for any \(h \in H\).

The group \({\mathrm {aut}({\widehat{N}},A)}\) in Proposition 2 is formed from automorphisms of the extension \({\widehat{N}}\) that preserve the split Lie subgroup *A*. The map \({\mathrm{d}}_{\psi }\) is the coboundary operator in the cochain complex of maps \(f : H^p \rightarrow {\mathrm {C}}_s^1(N,A)\), defined in relation to the action of the group *H* on \({\mathrm {C}}_s^1(N,A)\) by \(h.f = \psi (h).f\) (in [7], the corresponding notation is \({\mathrm{d}}_{S_{\psi }}\), where \(S_{\psi }\) denotes the action).

The first condition is fulfilled in the case of the 3-loop group, but the cohomology class given by \({\mathrm{d}}_\psi \phi \) is not trivial, and thus, the action of \({B^3_{\flat }G}\) does not lift to the extension. However, keeping in mind that our goal is a crossed module, we can reroute our approach through central extensions as will be explained in the next section. Let us conclude by gathering the results of this section in the following proposition:

### Proposition 3

### 2.2 The case of *SU*(2)

*f*in \({\varOmega ^3G}\) such that \(f(0) = e\) and \(f(1) = g\), when we consider only the identity component. This is a covering group for the component of the identity in \({\varOmega ^3G}\), and there is a natural projection to path points.

The right conjugation by \(f \in {B^3_{\flat }G}\) acting on any given path point does not change the homotopy. Thus, the right adjoint action of the group \({B^3_{\flat }G}\) on \({\varOmega ^3G}\) lifts to an automorphic action on the extension \({\mathcal {G}}\).

## 3 2-group from a crossed module

The aim of this paper to is construct an action groupoid that would fulfil the axioms of a crossed module, and thus define a strict 2-group [3]. Let us revisit the definitions.

### Definition 2

*Crossed module*) Let

*G*and

*H*be groups, and consider morphisms

*G*on

*H*, the diagrams correspond to the equations

### Definition 3

(*Smooth crossed module*) If the groups *G* and *H* in a crossed module \([\delta :H\rightarrow G]\) are Lie groups and the action defined by the morphism \(\alpha \) is smooth, the crossed module is called a Lie crossed module, or a smooth crossed module.

### Proposition 4

### Proof

### Remark 3

For the canonical central extension of the 1-loop group \(\varOmega G\), it holds that \(\pi _1({\widehat{\varOmega G}}) = 0\). A similar argument based on the long homotopy exact sequence can be used in the case of \({\varOmega ^3G}\).

*H*,

*N*and

*A*be as in Proposition 2 with the cocycle \(\theta \in {\mathrm {Z}}^2_s(N,A)\) corresponding to the Abelian extension \({\widehat{N}}\). Assuming that

*N*is split normal subgroup of

*H*and that \(A^N\) is a split Lie subgroup of

*A*, there is a commuting diagram [7, Lemma 4.3 and the preceding definitions] where \(\varGamma \) is the extension of

*H*by \({\mathrm {Z}}_s^1(N,A)\). Furthermore, there is a smooth automorphic action \(\varGamma \rightarrow {\mathrm {aut}({\widehat{N}})}\); the pair \(({\widehat{N}}, \varGamma )\) in fact has the structure of a smooth crossed module provided that the cohomology group \({\mathrm {H}}^1_s(N,A)\) is trivial [7, Prop. 4.5 and Thm. 4.6]. In particular, this condition ensures that \(\varGamma \) is a Lie group extension and that \({\mathrm {Z}}_s^1(N,A) {\cong }A/A^N\).

### Lemma 1

Let \(G=SU(p)\) with \(p\ge 3\). Then, the first cohomology group \({\mathrm {H}}^1_s({\varOmega ^3G},{{{\text {Map}}({B^3_{\flat }G},S^1)}})\) is trivial.

### Proof

Denote \(N = {\varOmega ^3G}\), \(A={{{\text {Map}}({B^3_{\flat }G},S^1)}}\) and \(H={{\text {Map}}(S^2,G)}\); note that we can identify *H* with \({B^3_{\flat }G}/N\). By the Bott periodicity, the cohomology of *H* in low degrees is generated by elements \(\alpha _{2k+1}\) in odd degrees \((k=1,2, \ldots )\). In particular, \({\mathrm {H}}^2_s(H, {\mathbb {Z}})\) vanishes for \(p \ge 3\) and there are no non-trivial circle bundles over *H*.

On the other hand, an element \(c_1\in {\mathrm {H}}^1_s(N, A)\) describes a circle bundle *Q* over \(H = {B^3_{\flat }G}/N\): elements in *Q* are equivalence classes of pairs \((g, \lambda ) \in {B^3_{\flat }G}\times S^1\) with the equivalence relation \((g, \lambda ) \sim (gu, c_1(g;u)\lambda )\). The bundle can be trivialized if and only if \(c_1(g; u)= f(gu)f(g)^{-1}\) for some \(f: {B^3_{\flat }G}\rightarrow S^1\). Thus, the group \({\mathrm {H}}_s^1(N,A)\) is trivial.\(\square \)

*G*by using the action of \({B^3_{\flat }G}\) on \({\varOmega ^3G}\). However, this action does not lift to the extension \({\widehat{{\varOmega ^3G}}}\), and there is no natural morphism \({\widehat{{\varOmega ^3G}}} \rightarrow {B^3_{\flat }G}\) which would satisfy the requirements of a crossed module. It is then clear that we must take two additional steps: first by extending the acting group \({B^3_{\flat }G}\) as

*SO*(4)-equivariance in this picture.

## Footnotes

## Notes

### Acknowledgements

Open access funding provided by University of Helsinki including Helsinki University Central Hospital. We wish to thank David M. Roberts for his comments and helpful criticism. This research was partially supported by the Emil Aaltonen Foundation (Grant No. 170185 N) and the Niilo Helander Foundation.

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