Explicit Non-Abelian Gerbes with Connections

a]Dominik Rist 0000-0002-1817-3458
a]Christian Saemann 0000-0002-5273-3359
b]Martin Wolf 0009-0002-8192-3124

[a]Maxwell Institute for Mathematical Sciences, Department of Mathematics,
Heriot–Watt University, Edinburgh EH14 4AS, United Kingdom
[b]School of Mathematics and Physics,
University of Surrey, Guildford GU2 7XH, United Kingdom

We define the notion of adjustment for strict Lie 2-groups and provide the complete cocycle description for non-Abelian gerbes with connections whose structure 2-group is an adjusted 2-group. Most importantly, we depart from the common fake-flat connections and employ adjusted connections. This is an important generalisation that is needed for physical applications especially in the context of supergravity. We give a number of explicit examples; in particular, we lift the spin structure on $S^4$, corresponding to an instanton–anti-instanton pair, to a string structure, a 2-group bundle with connection. We also outline how categorified forms of Bogomolny monopoles known as self-dual strings can be obtained via a Penrose–Ward transform of string bundles over twistor space.

We thank Hyungrok Kim and Paul Skerritt for fruitful discussions. We are particularly grateful to David Roberts for his contribution to the `Workshop on Higher Gauge Theory and Higher Quantization' in 2014 and for a number of helpful discussions related to signs in the literature. The work of CS was partially supported by the Leverhulme Research Project Grant RPG-2018-329 `The Mathematics of M5-Branes'.

No additional research data beyond the data presented and cited in this work are needed to validate the research findings in this work. For the purpose of open access, the authors have applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising.

§ INTRODUCTION AND RESULTS

Higher forms arise as local connection forms or gauge potentials in a number of contexts within physics. The prime example is the Kalb–Ramond $B$-field of string theory, which is then found again in the low-energy supergravity limits. A whole family of higher form gauge potentials arises in the tensor hierarchies of gauged supergravity theories and, consequentially, double and exceptional field theory. Mathematically, these gauge potentials are connections on higher or categorified principal bundles also known as gerbes. This perspective is evidently crucial when supergravity theories are to be considered on topologically non-trivial spaces.

In the Abelian case, the theory of gerbes is well-established and used in many contexts. There are a number of equivalent descriptions, such as the geometrically appealing bundle gerbes <cit.> or the computationally useful Hitchin–Chatterjee gerbes <cit.>. Although the Kalb–Ramond $B$-field is locally an ordinary differential 2-form, Abelian gerbes are not fully sufficient for its description. In the presence of additional gauge bundles, the connections on these mix non-trivially with the connection on the Abelian gerbe. This is the case, e.g., in gauged supergravity, and it was in this context that what are now called connections on string bundles had been observed for the first time <cit.> in their local, infinitesimal form. Similarly, in the context of T-duality, one often wants to describe string theory backgrounds which correspond to Abelian gerbes on the total space of principal torus bundles. These can be conveniently captured by non-Abelian generalisations of gerbes at the topological level <cit.>. A differential refinement, leading to the full Buscher rules is also possible <cit.>. Moreover, there is reason to believe that aspects of the dynamics of multiple M5-branes can be captured by non-Abelian gerbes <cit.>; see also <cit.> for recent evidence that the string group $\sfString(3)$ naturally emerges on a single M5-brane.[See also <cit.> for a general review on higher structures in M-theory.] Finally, several kinds of non-Abelian gerbes have been used in a number of string-theory inspired twistor constructions <cit.>.

Non-Abelian gerbes have been introduced in various forms and all familiar definitions of principal bundles have found higher generalisations, cf. <cit.> and references therein. One of the earliest forms is perhaps the differential cocycle description given in <cit.> and <cit.>, and because of its computational usefulness, we shall focus mostly on this picture in the following. This form of non-Abelian gerbes generalises the notion of a principal fibre bundle by lifting the cocycle relations for transition functions and local connection forms to hold up to homotopies, which are then encoded in higher components of the cocycle.

Whilst the topological description of non-Abelian gerbes is relatively straightforward, its differential refinement by a connection is subtle. An explicit cocycle description for such a differential refinement was obtained in <cit.>, but these cocycles contain an additional local 2-form datum[In <cit.>, this 2-form is denoted by $\delta_{ij}$.] compared to the expected $L_\infty$-algebra-valued connections <cit.>. A detailed study of the evident notion of higher parallel transport <cit.> then suggested that the additional datum is, in a sense, spurious, and that one should work instead with reduced cocycles <cit.>, which have subsequently been used throughout much of the literature. This, however, led to a second problem. Consistency of the corresponding parallel transport requires the 2-form part $F$ of the total curvature of a non-Abelian gerbe with connection, which is also known as its fake curvature, to vanish <cit.>. The fake-flatness condition, however, is problematic for two reasons: firstly, we know that interesting string backgrounds do not satisfy this condition, and secondly, it can be shown that locally, a fake-flat non-Abelian gerbe with connection is always isomorphic to an Abelian gerbe with connection <cit.>, rendering it essentially useless for all but topological applications in physics. Another undesirable consequence is that after adding fake flatness to the cocycle conditions of non-Abelian gerbes with connection, ordinary principal bundles with non-flat connections are no longer trivially higher bundles. This is clearly undesirable from a category theoretical viewpoint.

The solution to these problems is provided by a third possible type of cocycles which are more general than the reduced ones of <cit.>, but do not contain the additional 2-form datum of <cit.>. In these adjusted cocycles the 3-form parts of the curvature as well as the gluing relations and gauge transformations for the 2-form part of the connection are deformed by a term proportional to the fake curvature.

At a very explicit, local and infinitesimal level and in one particular case, this had been observed already in <cit.>. A more general mathematical perspective was then given in <cit.>. In this picture, connections are locally morphisms from the Weil algebra of the (higher) gauge algebra to the de Rham complex on the relevant coordinate patch. The deformation to adjusted connection is then a coordinate change on the relevant Weil algebra. The motivation for the deformation here was a match with the Bianchi identities describing the Green–Schwarz anomaly cancellation derived previously from a physical perspective.

It was then found that adjusted connections have other desirable properties, too. In particular, the BRST complex, the Chevalley–Eilenberg complex dual to the higher gauge Lie algebroid, closes, i.e. the differential is indeed nilquadratic without further conditions on the curvature forms, contrary to the reduced cocycles of <cit.>. This property was then turned into the definition of adjusted connections in <cit.>, where also the adjustments for a number of more general higher Lie algebras were computed. Shortly after, it was then shown that a consistent adjusted higher parallel transport that does not rely on fake flatness is indeed possible <cit.>.[The discussion in <cit.> makes it clear why the adjustment is only visible at the level of connections or parallel transport. Topologically, higher principal bundles are functors from the Čech groupoid to the delooping of the higher structure group, but the extension of the Čech groupoid to a higher groupoid is trivial in the sense that all higher morphisms are identities. Parallel transport, however, is locally defined as a functor from a higher path groupoid to the delooping of the higher structure group. In the former, the higher morphisms are non-trivial.] Moreover, the origin of the adjustment was traced back to a generalisation of the notion of higher gauge algebra in <cit.>, where also adjustments for large classes of higher bundles were derived that are relevant in the context of gauged supergravity.

The explicit finite form of the adjusted cocycles for non-Abelian gerbes with connection has never been provided. This may be due to the fact that adjustments have to be discussed individually for each higher gauge group. In particular, certain higher gauge groups do not admit an adjustment, and an example can be found in <cit.>. It is the primary goal of this paper to fill this gap and to give explicit cocycle relations for adjusted connections with a focus on string bundles. A secondary goal is to construct an explicit example.

An evident higher gauge group to use in a such an example is a 2-group model of the string group. Recall that a spin structure over an $n$-dimensional oriented Riemannian manifold $X$, is a lift of the frame bundle, a principal $\sfSO(n)$-bundle, to a principal $\sfSpin(n)$-bundle. The obstruction to the existence of such a lift is the second Stiefel–Whitney class of $X$. One can now consider spin bundles over loop spaces, as suggested by string theory, and it turns out that the loop space $LX$ of a manifold $X$ carries a spin bundle if and only if[The theorem assumes $X$ simply connected and $n\geq5$. The `only if' part holds further requiring that $X$ be 2-connected.] spin bundles over $X$ can be lifted to principal $\sfString(n)$-bundles <cit.>. Here, $\sfString(n)$ is a topological group, defined only up to a large class of equivalence, which is a 3-connected cover of $\sfSpin(n)$, but otherwise shares the same homotopy groups as $\sfSpin(n)$.

It turns out that the group $\sfString(n)$ is conveniently described in terms of a 2-group model, cf. <cit.>, and from this perspective, string structures are indeed non-Abelian gerbes. This is also the form in which string structures most naturally arise in supergravity and string theory. There are now essentially two extreme string 2-group models. Firstly, there is an infinite-dimensional one based on Kac–Moody central extensions of loop groups, which was developed in <cit.>. On the other hand, there is the finite-dimensional, but much more complicated and less explicit model <cit.>.[See <cit.> for a description of (unadjusted) non-Abelian gerbes using this group.] For convenience, we shall be focusing on the infinite-dimensional model because it admits fully explicit formulas for all cocycles.

Explicitly, we have the following results to report:

* In <ref>, we define the category of adjusted crossed module of Lie groups. This is the algebraic datum underlying the following result.

* Our first main result is the full cocycle description of higher principal bundles[For us, `higher principal bundle' is synonymous with `principal 2-bundle'.] with adjusted structure 2-group as well as adjusted (and hence, generalised notion of) connection in <ref>.

* An instructive example is found when replacing a Lie group $\sfG$ by the equivalent crossed module $\caL\sfG$ of path and loop groups of $\sfG$, which induces an adjustment datum as we explain in <ref>. We then show that any principal bundle with connection can equivalently be described as a higher principal bundle with adjusted connection in <ref>.

* Concretely, we present the explicit cocycles for the spin structure on $S^4$, which can be regarded as the principal $\sfSpin(4)$-bundle $\sfSpin(5)\rightarrow S^4\cong\sfSpin(5)/\sfSpin(4)$, as an $\caL\sfSpin(4)$-bundle with adjusted connection in <ref>.

* We show that the quotients of certain string 2-groups reduce to quotients of ordinary groups and hence lead to ordinary manifolds regarded as Lie groupoids in <ref>. As a simple example, we consider the space $\sfString(3)/\sfString(2)\cong S^3$, which implies the categorified Hopf fibration of $\sfString(3)$ over $S^3$.

* Our second major result is the concrete example of a string 2-bundle constructed in <ref>. Here, we lift the principal $\caL\sfSpin(4)$-bundle equivalent to the spin structure on $S^4$ to an essentially unique $\sfString(4)$-bundle.[The case $\sfString(4)$ is particularly interesting from an M-theory perspective, because it contains the Lie group $\sfSpin(4)\cong\sfSU(2)\times\sfSU(2)$, which is important in the context of M2-brane models <cit.>.]

* Our formulas give the explicit finite gauge transformations for the non-Abelian self-dual strings introduced in <cit.>, closing a significant gap. This is explained in <ref>.

* Finally, in <ref>, we comment on an iterated Penrose–Ward transform that allows for a construction of solutions to the non-Abelian self-dual string equations of <cit.>.

An additional minor result is our clarification of the relation between two group cocycles used in the literature to describe the Kac–Moody central extension of the loop group of a group at the beginning of <ref>.

We have tried to be fairly self-contained and detailed in our presentation, mostly for two reasons. Firstly, we had to fix a number of sign inaccuracies in the literature we used, and we therefore wanted to present our computations in a verifiable way. Secondly, we would have found the detailed review part in this paper invaluable when we started this research project.

Throughout this paper, all manifolds $X,Y,\ldots$ are assumed to be smooth so we drop the adjective `smooth'. Furthermore, $\Omega^k(X)$ are the smooth differential $k$-forms on $X$. We set $\Omega^0(X)=\scC^\infty(X)$.

If not stated otherwise, all Lie groups $\sfG,\sfH,\ldots$ are assumed to be compact, simply-connected, and semi-simple. We denote the unit element of a (Lie) group by $\unit$. For simplicity, we will mostly use matrix Lie group/algebra notation. For example if $g$ is an element of a Lie group and $X$ an element of the corresponding Lie algebra, then $g^{-1}Xg$ is another element of the Lie algebra.

§ PRINCIPAL 2-BUNDLES WITH ADJUSTED CONNECTIONS

In this section, we first recap crossed modules of Lie groups and Lie algebras, their morphisms, and the most widely used notion of higher principal bundles with connections based on these, see e.g. <cit.>. This definition includes a partial flatness condition, known as fake flatness, on the higher connection, which greatly constrains these higher principal bundles and makes them less useful in many context. To resolve this problem, we need to globalise the adjusted curvatures of <cit.>, see also <cit.>, and introduce the notion of higher principal bundles based on adjusted crossed modules of Lie groups.

§.§ Crossed modules and their morphisms

Crossed modules of Lie groups.
Recall that a crossed module of Lie groups $\caG\coloneqq(\sfH\overset{\sft}{\longrightarrow}\sfG,\acton)$ consists of two Lie groups $\sfG$ and $\sfH$, an automorphism action $\acton$ of $\sfG$ on $\sfH$, and a morphism of Lie groups $\sft:\sfH\rightarrow\sfG$ such that the conditions

\begin{equation}\label{eq:crossedModuleConditions}
\sft(g\acton h_1)\ =\ g\sft(h_1)g^{-1}
\eand
\sft(h_1)\acton h_2\ =\ h_1h_2h_1^{-1}
\end{equation}

hold for all $g\in\sfG$ and for all $h_{1,2}\in\sfH$; the second condition is called the Peiffer identity. Crossed modules of Lie groups can be used to describe strict Lie 2-groups, cf. <cit.>.

Given two crossed modules of Lie groups, $\caG_{1,2}=(\sfH_{1,2}\overset{\sft_{1,2}}{\longrightarrow}\sfG_{1,2},\acton_{1,2})$, a strict morphism $\phi:\caG_1\rightarrow \caG_2$ is a pair of group homomorphisms $\phi_\sfG:\sfG_1\rightarrow \sfG_2$ and $\phi_\sfH:\sfH_1\rightarrow \sfH_2$ such that

\begin{equation}
\begin{aligned}
(\sft_2\circ \phi_\sfH)(h)=(\phi_\sfG\circ \sft_1)(h)
\eand
\phi_\sfH(g\acton_1 h)=\phi_\sfG(g)\acton_2\phi_\sfH(h)
\end{aligned}
\end{equation}

for all $g\in \sfG_1$ and $h\in \sfH_1$. A strict quasi-isomorphism is a strict morphism that preserves the subgroup $\ker(\sft_1)\subseteq\sfH_1$ as well as the quotient group $\coker(t_1)=\sfG_1/\im(\sft)$.

More generally, weak morphisms can be defined in terms of smooth flippable butterflies <cit.>. Specifically, given two crossed modules of Lie groups, $\caG_{1,2}=(\sfH_{1,2}\overset{\sft_{1,2}}{\longrightarrow}\sfG_{1,2},\acton_{1,2})$, a butterfly is a commutative diagram of Lie groups of the form

\begin{equation}
\begin{tikzcd}
\sfH_1 \arrow[dd,"\sft_1",swap] \arrow[dr,"\lambda_1"]& & \sfH_2 \arrow[dl,"\lambda_2",swap] \arrow[dd,"\sft_2"]
\\
& \sfE \arrow[dl,"\gamma_1",swap] \arrow[dr,"\gamma_2"]&
\\
\sfG_1 & & \sfG_2
\end{tikzcd}
\end{equation}

where $\sfE$ is a Lie group, $\lambda_{1,2}$ and $\gamma_{1,2}$ are morphisms of Lie groups, the NE–SW diagonal is a short exact sequence (i.e. a Lie group extension), and the NW–SE diagonal is a complex. In addtion, the actions need to compatible in the sense of

\begin{equation}
\lambda_{1,2}(\gamma_{1,2}(e)\acton_{1,2}h_{1,2})\ =\ e\lambda_{1,2}(h_{1,2})e^{-1}
\end{equation}

for all $e\in\sfE$ and for all $h_{1,2}\in\sfH_{1,2}$. The butterfly is called flippable whenever both diagonals are short exact sequences. Given such a flippable butterfly, we call $\caG_1$ and $\caG_2$ equivalent.

We also note that any butterfly between two crossed modules $\caG_{1,2}=(\sfH_{1,2}\overset{\sft}{\longrightarrow}\sfG_{1,2},\acton_{1,2})$ can be turned into a span of strict Lie 2-group morphisms <cit.> of the form

\begin{equation}
\begin{tikzcd}
& \caK \arrow[dl,"\phi"'] \arrow[dr,"\psi"]
\\
\caG_1 & & \caG_2
\end{tikzcd}
\end{equation}

where $\caK$ is a crossed module of the form $\sfH_1\times \sfH_2\rightarrow \sfE$, and $\phi$ and $\psi$ are strict morphisms of 2-groups with $\phi$ a quasi-isomorphism.

Crossed modules of Lie algebras.
Applying the Lie functor to a crossed module of Lie groups $\caG=(\sfH\overset{\sft}{\longrightarrow}\sfG,\acton)$, we obtain a crossed module of Lie algebras $\sfLie(\caG)\coloneqq(\frh\overset{\sft}{\longrightarrow}\frg,\acton)$ where $\frg$ and $\frh$ are the Lie algebras of $\sfG$ and $\sfH$, respectively.[We follow the literature and slightly abuse notation by denoting the linearisations of $\acton$ and $\sft$ again by the same symbols.] The linearisations of the conditions in (<ref>) read as

\begin{equation}\label{eq:crossedModuleLieAlgebras}
\sft(V\acton W_1)\ =\ [V,\sft(W_1)]
\eand
\sft(W_1)\acton W_2\ =\ [W_1,W_2]
\end{equation}

for all $V\in\frg$ and for all $W_{1,2}\in\frh$.

Both strict and weak morphisms (i.e. butterflies) between crossed modules of Lie groups differentiate straightforwardly to corresponding notions of morphisms of crossed modules of Lie algebras. We note that for a butterfly between crossed modules of Lie algebras $\sfLie(\caG_{1,2})=(\frh_{1,2}\xrightarrow{~t_{1,2}~}\frg_{1,2})$,

\begin{equation}
\begin{tikzcd}
\frh_1 \arrow[dd,"\sft_1",swap] \arrow[dr,"\lambda_1"]& & \frh_2 \arrow[dl,"\lambda_2",swap] \arrow[dd,"\sft_2"]
\\
& \fre \arrow[dl,"\gamma_1",swap] \arrow[dr,"\gamma_2"]&
\\
\frg_1 & & \frg_2
\end{tikzcd}
\end{equation}

with a section $\sigma:\frg_1\rightarrow \fre$ of $\gamma_1$ as morphism of vector spaces, we can construct a corresponding weak morphisms of $L_\infty$-algebras between $\sfLie(\caG_{1})$ and $\sfLie(\caG_{2})$, with both trivially regarded as strict 2-term $L_\infty$-algebras <cit.>. Explicitly, the splitting $\frg_1$ provides us with a second map $\tau:\fre\rightarrow \frh_2$, and we have maps

\begin{equation}
\begin{aligned}
\phi_\frh&:\frh_1\rightarrow \frh_2~,~~~&\phi_\frh(W)&=-\tau(\lambda_1(W))~,
\\
\phi_\frg&:\frg_1\rightarrow \frg_2~,~~~&\phi_\frg(V)&=\gamma_2(\sigma(V))~,
\\
\phi_2&:\frg_1\wedge \frg_1\rightarrow \frh_2~,~~~&\phi_2(V_1,V_2)&=-\tau([\sigma(V_1),\sigma(V_2)]_\fre)
\end{aligned}
\end{equation}

for all $V,V_{1,2}\in \frg_1$ and $W\in \frh_1$ defining a weak morphism of $L_\infty$-algebras between $\sfLie(\caG_{1})$ and $\sfLie(\caG_{2})$. That is,

\begin{equation}\label{eq:L_infty_alg}
\begin{aligned}
\sft_2(\phi_\frh(W))\ &=\ \phi_\frg (\sft_1(W))~,
\\
\phi_\frg([V_1,V_2]_{\frg_1})\ &=\ [\phi_\frg(V_1),\phi_\frg(V_2)]_{\frg_2}+\sft_2(\phi_2(V_1,V_2))~,
\\
\phi_\frh(V_1\acton_1 W)\ &=\ \phi_\frg(V_1)\acton_2 \phi_\frh(W)+\phi_2(V_1,\sft_1(W))~,
\\
0\ &=\ \phi_2(V_1,[V_2,V_3])+\phi_\frg(V_1)\acton_2\phi_2(V_2,V_3)+\mbox{cycl.}
\end{aligned}
\end{equation}

for all $V_{1,2}\in \frg$ and $W\in \frh$; see e.g. <cit.> for more details. Conversely, any weak morphism of $L_\infty$-algebras can be turned into a butterfly.

§.§ Unadjusted principal 2-bundles with connection

We now introduce categorified principal bundles with connections having crossed modules of Lie groups as their structure 2-groups in terms of their cocycle data.

Recall that a cover is a surjective submersion $\sigma:Y\rightarrow X$ of manifolds, not necessarily a local diffeomorphism, together with the fibre products

\begin{equation}\label{eq:fibreProducts}
Y^{[n]}\ \coloneqq\ \underbrace{Y\times_X\cdots\times_XY}_{\text{$n$ factors}}\ \coloneqq\ \{(y_1,\ldots,y_n)\in Y^n\,|\,\sigma(y_1)=\cdots=\sigma(y_n)\}
\end{equation}

for $n\in\IN$. We will sometimes make the additional assumption of working with a good cover by which we mean a cover in which all the $Y^{[n]}$ are disjoint unions of contractible spaces.

Descent data.
In the following, let $\caG\coloneqq(\sfH\overset{\sft}{\longrightarrow}\sfG,\acton)$ be a crossed module of Lie groups and $\sfLie(\caG)\coloneqq(\frh\overset{\sft}{\longrightarrow}\frg,\acton)$ the associated crossed module of Lie algebras, respectively.

The cocycles describing the descent data of a principal $\caG$-bundle with (unadjusted) connection over a manifold $X$ subordinate to a cover $Y\rightarrow X$ consist of

\begin{equation}
\begin{aligned}
h\ &\in\ \scC^\infty(Y^{[3]},\sfH)~,
\\
(g,\Lambda)\ &\in\ \scC^\infty(Y^{[2]},\sfG)\oplus\Omega^1(Y^{[2]},\frh)~,
\\
(A,B)\ &\in\ \Omega^1(Y^{[1]},\frg)\oplus\Omega^2(Y^{[1]},\frh)~,
\end{aligned}
\end{equation}

such that

\begin{equation}\label{eq:unadjustedCocycleConditionsB}
\begin{aligned}
h_{ikl}h_{ijk}\ &=\ h_{ijl}(g_{ij}\acton h_{jkl})~,
\\
g_{ik}\ &=\ \sft(h_{ijk})g_{ij}g_{jk}~,
\\
\Lambda_{ik}\ &=\ \Lambda_{jk}+g_{jk}^{-1}\acton \Lambda_{ij}-g_{ik}^{-1}\acton(h_{ijk}\nabla_i h_{ijk}^{-1})~,
\\
A_j\ &=\ g^{-1}_{ij}A_ig_{ij}+g^{-1}_{ij}\rmd g_{ij}-\sft(\Lambda_{ij})~,
\\
B_j\ &=\ g^{-1}_{ij}\acton B_i+\rmd\Lambda_{ij}+ A_j\acton\Lambda_{ij}+\tfrac12[\Lambda_{ij},\Lambda_{ij}]~,
\\
0\ &=\ \rmd A_i+\tfrac12[A_i,A_i]+\sft(B_i)
\end{aligned}
\end{equation}

for all appropriate $(i,j,\ldots)\in Y^{[n]}$; here $\nabla_i\coloneqq\rmd+A_i\acton$.

This definition is a special case of the original definitions in <cit.> and widely used in the literature.  This original definition also contained an additional map

\begin{equation}
\delta\ \in\ \Omega^2(Y^{[2]},\frh)
\end{equation}

giving a deformation of the above cocycle relations. This additional map is put to zero in most of the subsequent literature[see e.g. <cit.>] for a number of reasons. One of these is the fact that this additional map corresponds to an unexpected, additional higher gauge transformation which is absent in all natural local and infinitesimal pictures.[A closer look at the datum $\delta$ shows that it is akin to an (unwanted) non-trivial curvature for $\Lambda$.]

The data $(A,B,\Lambda)$ differentially refines the Čech cocycle of the principal $\caG$-bundle defined by the first two relations in (<ref>), and we shall simply refer to this data as a connection. The total curvature of this connection is given in terms of the 2-form fake curvature and the 3-form curvature

\begin{equation}\label{eq:unadjustedCurvatures}
F_i\ \coloneqq\ \rmd A_i+\tfrac12[A_i,A_i]+\sft(B_i)
\eand
H_i\ \coloneqq\ \rmd B_i+A_i\acton B_i
\end{equation}

for all $(i)\in Y^{[1]}$. The last condition in (<ref>) says that $F_i=0$, and this is known as the fake-flatness condition. Furthermore, these curvature forms satisfy the Bianchi identities

\begin{equation}\label{eq:unadjustedBianchiIdentities}
\nabla_i F_i\ =\ \sft(H_i)\ =\ 0
\eand
\nabla_i H_i\ =\ F_i\acton B_i\ =\ 0
\end{equation}

due to the fake-flatness condition.

Two principal $\caG$-bundles are called equivalent whenever the cocycles $(h,g,\Lambda,A,B)$ and $(\tilde h,\tilde g,\tilde\Lambda,\tilde A,\tilde B)$ encoding their descent data are linked by a coboundary consisting of the maps

\begin{equation}
\begin{aligned}
b\ &\in\ \scC^\infty(Y^{[2]},\sfH)~,
\\
(a,\lambda)\ &\in\ \scC^\infty(Y^{[1]},\sfG)\oplus\Omega^1(Y^{[1]},\frh)
\end{aligned}
\end{equation}

by means of

\begin{equation}
\begin{aligned}
\tilde h_{ijk}\ &=\ a_i^{-1}\acton(b_{ik}h_{ijk}(g_{ij}\acton b_{jk}^{-1})b_{ij}^{-1})~,
\\
\tilde g_{ij}\ &=\ a_i^{-1}\sft(b_{ij})g_{ij}a_j~,
\\
\tilde\Lambda_{ij}\ &=\ a^{-1}_j\acton\Lambda_{ij}+\lambda_j-\tilde{g}^{-1}_{ij}\acton\lambda_i+(a_j^{-1}g_{ij}^{-1})\acton(b_{ij}^{-1}\nabla_ib_{ij})~,
\\
\tilde A_i\ &=\ a_i^{-1}A_ia_i+a_i^{-1}\rmd a_i-\sft(\lambda_i)~,
\\
\tilde B_i\ &=\ a_i^{-1}\acton B_i+\rmd\lambda_i+\tilde{A}_i\acton\lambda_i+\tfrac12[\lambda_i,\lambda_i]
\end{aligned}
\end{equation}

for all appropriate $(i,j,\ldots)\in Y^{[n]}$. We call such coboundary transformations gauge transformations.

It is not too difficult to see that the curvature forms (<ref>) change under the gauge transformations (<ref>) as

\begin{equation}\label{eq:unadjustedCurvatureGaugeTransformation}
\tilde F_i\ =\ a_i^{-1}F_ia_i
\eand
\tilde H_i\ =\ a_i^{-1}\acton H_i+\tilde F_i\acton\lambda_i
\end{equation}

for all $(i)\in Y^{[1]}$.

Contrary to ordinary principal bundles, principal 2-bundles come with higher gauge transformations of bundle isomorphisms that link gauge transformations or bundle isomorphisms,

\begin{equation}
\begin{tikzcd}
\arrow[r,bend left=50,"{(a,b,\lambda)}"{name=U}]
\arrow[r,bend right=50,"{(\tilde a,\tilde b,\tilde\lambda)}"{name=D},swap]
\arrow[Rightarrow,"\,m", from=U, to=D,start anchor={[yshift=-1ex]},end anchor={[yshift=1ex]}]
& (\tilde h,\tilde g,\tilde\Lambda,\tilde A,\tilde B)~,
\end{tikzcd}
\end{equation}

Two gauge transformations $(a,b,\lambda)$ and $(\tilde a,\tilde b,\tilde\lambda)$ are called equivalent whenever they are linked by a higher coboundary consisting of maps

\begin{equation}
m\ \in\ \scC^\infty(Y^{[1]},\sfH)
\end{equation}

by means of

\begin{equation}\label{eq:unadjustedHigherGaugeTransformationsB}
\begin{aligned}
\tilde b_{ij}\ &=\ m_ib_{ij}(g_{ij}\acton m_j^{-1})~,
\\
\tilde a_i\ &=\ \sft(m_i)a_i~,
\\
\tilde \lambda_i\ &=\ \lambda_i+a_i^{-1}\acton(m_i^{-1}\nabla_i m_i)
\end{aligned}
\end{equation}

for all appropriate $(i,j,\ldots)\in Y^{[n]}$. We shall refer to them as higher gauge transformations.

Fake flatness.
The fake-flatness condition $F_i=0$ is part of our cocycle relations (<ref>), and it has been argued as vital for consistency of the higher parallel transport in <cit.>, cf. also the discussion in <cit.>. We can also see this condition emerge from the consistency of gauge transformations and, correspondingly, the cocycle relations. In particular, consider the gluing-together of two gauge transformations (<ref>) to a third one,

\begin{equation}\label{eq:tripleGaugeTransformations}
\begin{tikzcd}
& (A_2,B_2)\arrow[dr,"{(a_{23},\lambda_{23})}"]\arrow[d,Rightarrow,"{m_{123}}"]
\\
(A_1,B_1) \arrow[ur,"{(a_{12},\lambda_{12})}"] \arrow[rr,"{(a_{13},\lambda_{13})}",swap] & \phantom{a} & (A_3,B_3)
\end{tikzcd}
\end{equation}

Then, we have the following result.

The gauge transformations (<ref>) are related by a higher coboundary, parametrised by an $m_{123}\in\scC^\infty(Y,\sfH)$, in the following way[cf. (<ref>)]

\begin{equation}\label{eq:unadjustedCoboundaryComposition}
\begin{aligned}
a_{13}\ &=\ \sft(m_{123})a_{12}a_{23}~,
\\
\lambda_{13}\ &=\ \lambda_{23}+a_{23}^{-1}\acton\lambda_{12}-a_{13}^{-1}\acton(m_{123}\nabla_1m_{123}^{-1})~.
\end{aligned}
\end{equation}

The commutativity of the diagram (<ref>) amounts to

\begin{equation}\label{eq:fakeCurvatureConditionFromTriangle}
(a_{23}^{-1}a_{12}^{-1})\acton (m_{123}^{-1}(F_1\acton m_{123}))\ =\ 0~.
\end{equation}

The condition (<ref>) follows from the consistency of the gluing for $A$ and the condition (<ref>) for those of $B$, respectively. See <ref> for details.

This proposition makes it clear that consistent descent data satisfying (<ref>) except for fake flatness will usually be automatically fake flat, at least in the case of a faithful action $\acton$ in the crossed module. This is simply the infinitesimal analogue of the observation that fake flatness is required for a consistent parallel transport.

The ramifications of fake flatness, however, are rather severe. Most importantly, we have the following result, which impedes their application in many physical situations.

Fake-flat connections on higher principal bundles can be reduced locally to a connection on an Abelian gerbe.

We note that for the crossed module of Lie groups $(\sfG\overset{\sft}{\longrightarrow}\unit,\id)$, the group $\sfG=\ker(\sft)$ is necessarily Abelian. A principal $(\sfG\overset{\sft}{\longrightarrow}\unit,\id)$-bundle is then an Abelian gerbe with connection in the sense of Hitchin–Chatterjee <cit.>, and for these, the fake-flatness issue is absent. Recall that any of the more general bundle gerbes of <cit.> is stably isomorphic to a Hitchin–Chatterjee gerbe of <cit.> and thus also, albeit indirectly, captured by our approach.

§.§ Adjusted crossed modules and their morphisms

Local connection forms and their infinitesimal gauge transformations are described by generalised gauge potentials in an $L_\infty$-algebra. At this level, it is known that, at least for certain $L_\infty$-algebras, there is a modification of the curvature (implying a modification of the gauge transformations) that reproduces the appropriate Bianchi identities expected from string theory <cit.>.[A prominent example here is $\rmd H=\tfrac12 p_1$, where $H$ is the 3-form part of the curvature and $p_1$ the first Pontryagin class.] A more general form of this modification, still at the level of $L_\infty$-algebras was dubbed an adjustment in <cit.>. For our purposes, we shall need a finite version of this adjustment at the level of Lie 2-groups. As observed e.g. in <cit.>, the adjustment is an algebraic datum on the higher gauge Lie algebra in the infinitesimal case. In particular, the adjustment is necessary for the consistent definition of invariant polynomials in an $L_\infty$-algebra, as explained in <cit.>.

Adjustment for crossed modules of Lie groups.
The appropriate notion of an adjustment for a Lie 2-group in the form of a crossed module of Lie groups is the following.

An adjusted crossed module of Lie groups $\caG_\kappa=(\sfH\overset{\sft}{\longrightarrow}\sfG,\acton,\kappa)$ is a crossed module of Lie groups $\caG=(\sfH\overset{\sft}{\longrightarrow}\sfG,\acton)$ together with a map

\begin{equation}\label{eq:general_adjustment}
\kappa\,:\,\sfG\times\frg\ \rightarrow\ \frh~,
\end{equation}

called the adjustment, which is linear in $\frg$ and satisfies

\begin{align}
\kappa(\sft(h),V)\ &=\ h(V\acton h^{-1})~,\label{eq:alternativeAdjustmentCondition_a}
\\
\kappa(g_2g_1,V)\ &=\ g_2\acton\kappa(g_1,V)+\kappa\big(g_2,g_1Vg^{-1}_1-\sft(\kappa(g_1,V))\big)\label{eq:alternativeAdjustmentCondition_b}
\end{align}

for all $g_{1,2}\in\sfG$, $h\in\sfH$, and $V\in\frg$.

The conditions (<ref>) and (<ref>) are equivalent to

\begin{equation}\label{eq:adjustmentNormalisation}
\kappa(\unit,V)\ =\ 0~,
\end{equation}

\begin{equation}\label{eq:adjustmentCondition}
\begin{aligned}
&(g_2g_1)\acton(h(V\acton h^{-1}))+g_2\acton\kappa(g_1,V)
\\
&\kern1cm+\kappa(g_2,g_1Vg^{-1}_1-\sft(\kappa(g_1,V)))-\kappa(g_2g_1\sft(h),V)\ =\ 0
\end{aligned}
\end{equation}

for all $g_{1,2}\in\sfG$, for all $h\in\sfH$, and for all $V\in\frg$.

It is easy to see that upon choosing $g_1=g_2=\unit$ in (<ref>) and using (<ref>), we obtain (<ref>). Likewise, (<ref>) is obtained from (<ref>) by choosing $h=\unit$.

Conversely, upon choosing $g_2=\tilde g_2\tilde g_1$ and $g_1=\sft(h)$ in (<ref>) for $\tilde g_{1,2}\in\sfG$ and $h\in\sfH$ and using (<ref>) with $\sft(\kappa(\sft(h),V))=\sft(h(V\acton h^{-1}))=\sft(h)V\sft(h^{-1})-V$, we recover (<ref>). Setting $h=\unit$ in (<ref>), we get (<ref>).

Special adjustments.
The data of a crossed module of Lie groups $(\sfH\overset{\sft}{\longrightarrow}\sfG,\acton)$ that is preserved by quasi-isomorphisms are the kernel and cokernel of $\sft$. Whilst the adjustment datum on the remaining part is not canonical, it can often be determined by a simple map, as we now show.

Consider a crossed module of Lie groups $\caG\coloneqq(\sfH\overset{\sft}{\longrightarrow}\sfG,\acton)$ together with its crossed module of Lie algebras $\sfLie(\caG)\coloneqq(\frh\overset{\sft}{\longrightarrow}\frg,\acton)$. Let

\begin{equation}
P\,:\,\frg\ \rightarrow\ \frh
\end{equation}

be a morphism of vector spaces such that

\begin{equation}\label{eq:P_rels_1}
\sft\circ P\circ\sft\ =\ \sft~,
\end{equation}

and[Here, we use the shorthand notation $g[V,g^{-1}]\coloneqq gVg^{-1}-V$.]

\begin{equation}\label{eq:P_rels_2}
\sft(P(g_2\sft(P(g_1[V,g_1^{-1}]))g_2^{-1}))\ =\ g_2\sft(P(g_1[V,g_1^{-1}]))g_2^{-1}
\end{equation}

for all $g_{1,2}\in\sfG$ and for all $V\in\frg$. Then

\begin{equation}
\kappa_0(g,V)\ =\ P(g[V,g^{-1}])
\end{equation}

satisfies (<ref>) after applying $\sft$ to both sides. We will call an adjustment $\kappa$ such that $\sft\circ \kappa=\sft\circ \kappa_0$ for some $P$ a special adjustment.

We have

\begin{equation}
\begin{aligned}
\sft(\kappa_0(\sft(h),V))\ &=\ \sft(P(\sft(h)[V,\sft(h)^{-1}]))
\\
\ &=\ \sft(P(\sft(h(V\acton h^{-1}))))
\\
\ &=\ \sft(h(V\acton h^{-1}))
\end{aligned}
\end{equation}

for all $h\in\sfH$ and for all $V\in\frg$, as well as

\begin{equation}
\begin{aligned}
\sft(\kappa_0(g_2g_1,V))\ &=\ \sft(P(g_2g_1[V,g_1^{-1}g_2^{-1}]))
\\
\ &=\ g_2\sft(P(g_1[V,g_1^{-1}]))g_2^{-1}+\sft\Big(P(g_2[g_1Vg^{-1}_1,g^{-1}_2])\Big)
\\
\\
\ &=\ \sft\Big(g_2\acton P(g_1[V,g_1^{-1}])+P(g_2[g_1Vg^{-1}_1-\sft(P(g_1[V,g_1^{-1}])),g^{-1}_2])\Big)
\\
\ &=\ \sft\Big( g_2\acton\kappa_0(g_1,V)+\kappa\big(g_2,g_1Vg^{-1}_1-\sft(\kappa_0(g_1,V))\big)\Big)~.
\end{aligned}
\end{equation}

The intuition behind the map $P$ is that $\sft\circ P$ defines a projector onto $\im(\sft)\subseteq\frg$, and $\kappa_0$, whose definition is induced by $P$, captures the adjustment on this `redundant' part of the crossed module.

Path and loop groups.
Recall that the based (parametrised) path and loop groups of a Lie group are given by

\begin{equation}\label{eq:basedPathLoopGroups}
P_0\sfG\ \coloneqq\ \{p\in\scC^\infty([0,1],\sfG)\,|\,p(0)=\unit\}
\eand
L_0\sfG\ \coloneqq\ \{p \in P_0\sfG\,|\,p(1)=\unit\}~,
\end{equation}

respectively.[In our conventions for based path and loop groups, we follow <cit.>, and we refer to this paper for further details. We shall ignore technicalities such as introducing sitting instants in our loop and path parametrisations, as these can be trivially incorporated into our constructions. For a detailed discussion of these issues, see e.g. <cit.>. The important point to keep in mind is that the concatenation of two elements in our path space with the same endpoint always forms an element in our loop space.] Both $P_0\sfG$ and $L_0\sfG$ are Fréchet–Lie groups with the evident pointwise products. Furthermore, the endpoint evaluation map

\begin{equation}\label{eq:endpointEvaluation}
\begin{aligned}
\boundary\,:\,P_0\sfG\ &\rightarrow\ \sfG~,
\\
p\ &\mapsto\ p(1)
\end{aligned}
\end{equation}

is a morphism of Fréchet–Lie groups. Similarly, we define the based, parametrised path and loop Lie algebras of the Lie algebra $\frg$

\begin{equation}
P_0\frg\ \coloneqq\ \{\gamma\in\scC^\infty([0,1],\frg)\,|\,\gamma(0)=0\}\eand
L_0\frg\ \coloneqq\ \{\gamma\in P_0\frg\,|\,\gamma(1)=0\}
\end{equation}

with pointwise Lie brackets.

We can now combined $P_0\sfG$ and $L_0\sfG$ into the crossed module of Lie groups

\begin{equation}\label{eq:definitionLoopLieCrossed}
\caL\sfG\ \coloneqq\ (L_0\sfG\hookrightarrow P_0\sfG,\Ad)~,
\end{equation}

where $\Ad$ is the pointwise adjoint action of $L_0\sfG$ on $P_0\sfG$. The corresponding crossed module of Lie algebras $\sfLie(\caL\sfG)$ is then

\begin{equation}\label{eq:crossedModuleLg}
\caL\frg\ \coloneqq\ (L_0\frg\hookrightarrow P_0\frg,\ad)~.
\end{equation}

The crossed module $\caL\sfG$ is equivalent to $\sfG$, trivially regarded as the crossed module of Lie groups[`cm' refers to crossed module]

\begin{equation}\label{eq:definitionLieGroupLieCrossed}
\sfG_{\rm cm}\ \coloneqq\ (\unit\hookrightarrow\sfG,\id)~,
\end{equation}

and the evident flippable butterfly between $\caL \sfG$ and $\sfG_{\rm cm}$ is[Note that $L_0\sfG$ is, in fact, a normal subgroup of $P_0\sfG$, and we have

\begin{equation}\label{eq:identification}
\begin{tikzcd}[ampersand replacement=\&]
P_0\sfG \arrow[r,->>] \arrow[rr,bend right=30,"\boundary"] \& P_0\sfG/L_0\sfG \arrow[r,"\cong"] \& \sfG~.
\end{tikzcd}
\end{equation}

The equivalence between the crossed modules $\caL\sfG$ and $\sfG_{\rm cm}$ is a lift of the isomorphism between the quotient group $P_0\sfG/L_0\sfG$ and $\sfG$.]

\begin{equation}\label{eq:loopGroupButterfly}
\begin{tikzcd}
\unit \arrow[dd] \arrow[dr]& & L_0\sfG \arrow[dl] \arrow[dd]
\\
& P_0\sfG \arrow[dl,"\boundary",swap] \arrow[dr,"\id"]&
\\
\sfG & & P_0\sfG
\end{tikzcd}
\end{equation}

Consider some fixed map $\wp\in\scC^\infty([0,1],\IR)$ with $\wp(0)=0$ and $\wp(1)=1$. A special adjustment for $\caL\sfG$ is then given by the map

\begin{equation}
\begin{aligned}
\kappa\,:\,P_0\sfG\times P_0\frg\ &\rightarrow\ L_0\frg~,
\\
(g,V)\ &\mapsto\ P(g[V,g^{-1}])
\end{aligned}
\end{equation}

for all $g\in P_0\sfG$ and for all $V\in P_0\frg$ with

\begin{equation}\label{eq:Pmap}
P\ \coloneqq\ \id-\wp\cdot\boundary~,
\end{equation}

where $\id$ is the identity map.

We note that $\sft$ is injective, so it suffices to check the conditions for a special adjustment from <ref>. The relevant map here is evidently $P=\id-\wp\cdot\boundary$, and $\sft\circ P\circ\sft=\sft$ follows immediately since $\boundary\circ\sft=0$. Furthermore, we have for all $g\in P_0\sfG$ and for all $W\in P_0\frh$

\begin{equation}
P(g\sft(W)g^{-1})\ =\ P(\sft(g\acton W))\ =\ \sft(g(\acton W))~,
\end{equation}

which follows again from $\boundary\circ\sft=0$.

We note that the map $P$ yields a choice of embedding[There are certainly much more general choices.] of $\frg$ into $P_0\frg$, which leads to the decomposition

\begin{equation}
\begin{aligned}
P_0\frg\ &\rightarrow\ \frg\oplus L_0\frg~,
\\
V\ &\mapsto\ (P-\sfid)(V)+P(V)\ =\ \wp\cdot\boundary(V)+(\sfid-\wp\cdot\boundary)(V)
\end{aligned}
\end{equation}

for all $V\in P_0\frg$.

Note that the crossed module $\caL\frg$ is quasi-isomorphic to the Lie algebra $\frg$, regarded as the trivial crossed module $\frg_{\rm cm}\coloneqq(0\hookrightarrow\frg,\sfid)$, with the quasi-isomorphism the projection by endpoint evaluation. The embedding defined by $P$ then yields an inverse quasi-isomorphism,

\begin{equation}\label{eq:quasi-iso_Lie_algebras}
\frg_{\rm cm}\ \xrightarrow{~\cong~}\ \caL\frg~,
\end{equation}

see e.g. <cit.> for details.

The definition of strict morphisms of an adjusted crossed module of Lie groups is straightforward.

A strict morphism $\phi$ of crossed modules $\caG_{1,2}$ with adjustments $\kappa_{1,2}$ is a strict morphism of crossed modules $\phi:\caG_1\rightarrow\caG_2$ such that

\begin{equation}
\phi_\frh(\kappa_1(g,V))\ =\ \kappa_2(\phi_\sfG(g),\phi_\frg(V))
\end{equation}

for all $g\in\sfG$ and for all $V\in\frg$, where $\phi_\frg$ and $\phi_\frg$ denote the components of the strict morphism $\phi:\sfLie(\caG_1)\rightarrow\sfLie(\caG_2)$ induced by $\phi$. Strict quasi-isomorphisms of adjusted crossed modules are strict morphism of adjusted crossed modules that become strict quasi-isomorphisms after forgetting the adjustment.

To define weak morphisms of adjusted crossed modules, it is easiest to rely not on butterflies but the induced spans of strict Lie 2-group morphisms.

A (weak) morphism $\phi$ of crossed modules $\caG_{1,2}$ with adjustments $\kappa_{1,2}$ is a span of the form

\begin{equation}
\begin{tikzcd}
& \caK\arrow[dl,"\phi"'] \arrow[dr,"\psi"]
\\
\caG_1 & & \caG_2
\end{tikzcd}
\end{equation}

where $\caK$ is a third adjusted crossed modules and $\phi$ and $\psi$ are strict morphisms of adjusted crossed modules with $\phi$ a quasi-isomorphism.

As a simple example, consider the butterfly (<ref>) witnessing the equivalence between the crossed modules $\sfG_{\rm cm}$ from (<ref>), with the unique and trivial adjustment, and $\caL\sfG$ from (<ref>) with adjustment (<ref>). We have the following span of crossed modules

\begin{equation}
\begin{tikzcd}
& \caL\sfG \arrow[dl,"\phi"'] \arrow[dr,"\psi"]
\\
\sfG_{\rm cm} & & \caL\sfG
\end{tikzcd}
\end{equation}

where $\phi$ is the endpoint evaluation map and $\psi$ is the identity. Recall that $\phi$ is a quasi-isomorphism, and $\phi$ is trivially a strict morphism of adjusted crossed modules. We thus see that the adjustment on $\caL\sfG$ (up to choice of the map $P$) is induced by the trivial adjustment on $\sfG_{\rm cm}$.

Adjustment for crossed modules of Lie algebras.
It is useful to develop also the infinitesimal notion of adjustments. The Lie functor linearises an adjusted crossed module of Lie groups to an adjusted crossed module of Lie algebras in a similar fashion as in the unadjusted setting. As before, we slightly abuse notation and denote the linearisation (in the first entry) of the adjustment $\kappa$ again by $\kappa$, that is, the expression

\begin{equation}\label{eq:linearisedKappa}
\kappa(V_1,V_2)\ \coloneqq\ \lim_{\eps\to0}\tfrac1\eps\kappa(\exp(\eps V_1),V_2)
\end{equation}

for all $V_{1,2}\in\frg$.

We compute the following.

Given an adjusted crossed module of Lie groups $\caG$, the adjustment datum $\kappa:\frg\times \frg\rightarrow \frh$ in the corresponding crossed module of Lie algebras $\sfLie(\caG)\coloneqq(\frh\overset{\sft}{\longrightarrow}\frg,\acton)$ satisfies

\begin{equation}
\begin{aligned}
\kappa(\sft(W),V)\ &=\ -V\acton W~,
\\
\kappa([V_1,V_2],V_3)\ &=\ V_1\acton \kappa(V_2,V_3)-V_2\acton \kappa(V_1,V_3)+\kappa(V_1,[V_2,V_3])-\kappa(V_2,[V_1,V_3])
\\
\end{aligned}
\end{equation}

for all $V_{1,2}\in\frg$ and for all $W\in\frh$. Given a crossed module of Lie algebras with such a map $\kappa$, we call it an adjusted crossed module of Lie algebras.

This is a short, direct computation.

The special adjustment (<ref>) induces an adjustment datum on the corresponding crossed module of Lie algebras, which reads as

\begin{equation}\label{eq:adjustmentPathLoopAlgebras}
\begin{aligned}
\kappa\,:\,P_0\frg\times P_0\frg\ &\rightarrow\ L_0\frg~,
\\
(V_1,V_2)\ &\mapsto\ P([V_1,V_2])
\end{aligned}
\end{equation}

for all $V_{1,2}\in P_0\frg$ as follows by a direct computation.

Let us also briefly consider the form of morphisms of adjusted crossed modules of Lie algebras. Consider adjusted crossed modules $\sfLie(\caG_{1,2})\coloneqq(\frh_{1,2}\overset{\sft_{1,2}}{\longrightarrow}\frg_{1,2},\acton_{1,2},\kappa_{1,2})$ together with a weak morphism $\phi:\sfLie(\caG_{1})\rightarrow\sfLie(\caG_2)$ in the sense of 2-term $L_\infty$-algebra morphisms and satisfying (<ref>). As explained in <cit.>, the morphism $\phi$ can be turned into a butterfly

\begin{equation}
\begin{tikzcd}
\frh_1 \arrow[dd,"\sft_1",swap] \arrow[dr,"\lambda_1"]& & \frh_2 \arrow[dl,"\lambda_2",swap] \arrow[dd,"\sft_2"]
\\
& \frh_2\oplus \frg_1 \arrow[dl,"\gamma_1",swap] \arrow[dr,"\gamma_2"]&
\\
\frg_1 & & \frg_2
\end{tikzcd}
\end{equation}

with all maps evident except for

\begin{equation}
\gamma_2(W_2,V_1)\ =\ \sft_2(W_2)+\phi_\frg(V_1)
\end{equation}

for all $W_2\in\frh_2$ and for all $V_1\in\frg_1$. According to <cit.>, this butterfly can be turned into the span of Lie crossed modules

\begin{equation}
\begin{tikzcd}
& \frh_1\oplus\frh_2\rightarrow \frh_2\oplus \frg_1 \arrow[dl,"{\pr_1\rightarrow \pr_2}"'] \arrow[dr,"{\pr_2\rightarrow \gamma_2}"]
\\
\frh_1\rightarrow \frg_1 & & \frh_2\rightarrow \frg_2
\end{tikzcd}
\end{equation}

Lifting this to a span of adjusted crossed modules of Lie algebras with adjustments $\kappa_{1,2}$ and $\hat\kappa$ for $\frh_{1,2}\rightarrow\frg_{1,2}$ and $\frh_1\oplus\frh_2\rightarrow\frh_2\oplus\frg_1$, respectively, amounts to imposing

\begin{equation}
\begin{aligned}
\pr_1(\hat\kappa(W_2,V_1;W_2',V_1'))\ &=\ \kappa_1(V_1,V_1')~,
\\
\pr_2(\hat\kappa(W_2,V_1;W_2',V_1'))\ &=\ \kappa_2(\sft(W_2)+\phi_\frg(V_1),\sft(W'_2)+\phi_\frg(V'_1))
\end{aligned}
\end{equation}

for all $W_1,W'_1\in\frh_1$ and $W_2,W'_2\in\frh_2$ and for all $V_1,V'_1\in\frg_1$ and $V_2,V'_2\in\frg_2$. We hence see that the adjustment $\hat\kappa$ is fully fixed by the adjustments $\kappa_{1,2}$.

§.§ Adjusted descent data for higher principal bundles

Adjusted crossed modules of Lie groups now allow us to define higher principal 2-bundles with modified descend data. We will refer to them as adjusted, or having an adjusted connection, since the adjustment will only modify the differential refinement. The previous cocycle and coboundary relations (<ref>) and (<ref>), will then be called unadjusted.

Adjusted descent data.
In the following, let $\caG\coloneqq(\sfH\overset{\sft}{\longrightarrow}\sfG,\acton,\kappa)$ be an adjusted crossed module of Lie groups and $\sfLie(\caG)\coloneqq(\frh\overset{\sft}{\longrightarrow}\frg,\acton,\kappa)$ the associated adjusted crossed module of Lie algebras[Note, again, that we denote the adjustment of both $\caG$ and $\sfLie(\caG)$ by the same symbol.], respectively. Our first main result is the following definition, with all of its implications.

The cocycles describing the descent data of a principal $\caG$-bundle with adjusted connection over a manifold $X$ subordinate to a cover $Y\rightarrow X$ consist of

\begin{equation}
\begin{aligned}
h\ &\in\ \scC^\infty(Y^{[3]},\sfH)~,
\\
(g,\Lambda)\ &\in\ \scC^\infty(Y^{[2]},\sfG)\oplus\Omega^1(Y^{[2]},\frh)~,
\\
(A,B)\ &\in\ \Omega^1(Y^{[1]},\frg)\oplus\Omega^2(Y^{[1]},\frh)~,
\end{aligned}
\end{equation}

such that

\begin{equation}\label{eq:adjustedCocycleConditionsB}
\begin{aligned}
h_{ikl}h_{ijk}\ &=\ h_{ijl}(g_{ij}\acton h_{jkl})~,
\\
g_{ik}\ &=\ \sft(h_{ijk})g_{ij}g_{jk}~,
\\
\Lambda_{ik}\ &=\ \Lambda_{jk}+g_{jk}^{-1}\acton \Lambda_{ij}-g_{ik}^{-1}\acton(h_{ijk}\nabla_i h_{ijk}^{-1})~,
\\
A_j\ &=\ g^{-1}_{ij}A_ig_{ij}+g^{-1}_{ij}\rmd g_{ij}-\sft(\Lambda_{ij})~,
\\
B_j\ &=\ g^{-1}_{ij}\acton B_i+\rmd\Lambda_{ij}+ A_j\acton\Lambda_{ij}+\tfrac12[\Lambda_{ij},\Lambda_{ij}]-\kappa\big(g_{ij}^{-1},\rmd A_i+\tfrac12[A_i,A_i]+\sft(B_i)\big)\,,
\end{aligned}
\end{equation}

for all appropriate $(i,j,\ldots)\in Y^{[n]}$; here $\nabla_i\coloneqq\rmd+A_i\acton$.

The adjusted curvature forms are defined as

\begin{equation}\label{eq:adjustedCurvatures}
\begin{aligned}
F_i\ &\coloneqq\ \rmd A_i+\tfrac12[A_i,A_i]+\sft(B_i)\ \in\ \Omega^2(Y^{[1]},\frg)~,
\\
H_i\ &\coloneqq\ \rmd B_i+A_i\acton B_i-\kappa(A_i,F_i)\ \in\ \Omega^3(Y^{[1]},\frh)~,
\end{aligned}
\end{equation}

for all $(i)\in Y^{[1]}$, and the Bianchi identities change as follows.

We note that the fake curvature condition is absent in (<ref>), but for $F=0$, we recover the unadjusted cocycle relations (<ref>).

The adjusted curvatures (<ref>) satisfy the adjusted Bianchi identities

\begin{equation}\label{eq:adjustedBianchiIdentities}
\nabla_i F_i\ =\ \sft(H_i+\kappa(A_i,F_i))
\eand
\nabla_i H_i\ =\ \kappa(A_i,\sft(H_i))-\kappa(F_i,F_i)~.
\end{equation}

The first identity in (<ref>) follows from straightforward differentiation. From (<ref>) and (<ref>), we find

\begin{equation}
\kappa(\sft(B_i),F_i)\ =\ -F_i\acton B_i
\end{equation}

\begin{equation}
A_i\acton\kappa(A_i,F_i)\ =\ \kappa\big(\tfrac12[A_i,A_i],F_i\big)-\kappa\big(A_i,[A_i,F_i]-\sft(\kappa(A_i,F_i))\big)\,,
\end{equation}

respectively. Upon using these equations and the adjusted Bianchi identity for $F_i$, it is not too difficult to see that applying $\rmd$ to the definition of $H_i$ yields the second identity in (<ref>).

Two principal $\caG$-bundles with adjusted connections are said to be equivalent whenever the cocycles $(h,g,\Lambda,A,B)$ and $(\tilde h,\tilde g,\tilde\Lambda,\tilde A,\tilde B)$ encoding their descent data are linked by a coboundary consisting of maps

\begin{equation}
\begin{aligned}
b\ &\in\ \scC^\infty(Y^{[2]},\sfH)~,
\\
(a,\lambda)\ &\in\ \scC^\infty(Y^{[1]},\sfG)\oplus\Omega^1(Y^{[1]},\frh)
\end{aligned}
\end{equation}

by means of

\begin{equation}\label{eq:adjustedCoboundaryConditionsB}
\begin{aligned}
\tilde h_{ijk}\ &=\ a_i^{-1}\acton(b_{ik}h_{ijk}(g_{ij}\acton b_{jk}^{-1})b_{ij}^{-1})~,
\\
\tilde g_{ij}\ &=\ a_i^{-1}\sft(b_{ij})g_{ij}a_j~,
\\
\tilde\Lambda_{ij}\ &=\ a^{-1}_j\acton\Lambda_{ij}+\lambda_j-\tilde{g}^{-1}_{ij}\acton\lambda_i+(a_j^{-1}g_{ij}^{-1})\acton(b_{ij}^{-1}\nabla_ib_{ij})~,
\\
\tilde A_i\ &=\ a_i^{-1}A_ia_i+a_i^{-1}\rmd a_i-\sft(\lambda_i)~,
\\
\tilde B_i\ &=\ a_i^{-1}\acton B_i+\rmd\lambda_i+\tilde{A}_i\acton\lambda_i+\tfrac12[\lambda_i,\lambda_i]-\kappa\big(a_i^{-1},F_i\big)
\end{aligned}
\end{equation}

for all appropriate $(i,j,\ldots)\in Y^{[n]}$. We call such coboundary transformations adjusted gauge transformations.

The gauge transformations (<ref>) of the curvature forms change as follows.

The adjusted curvatures (<ref>) change under the adjusted gauge transformations (<ref>) as

\begin{equation}\label{eq:adjustedCurvatureGaugeTransformation}
\tilde F_i\ =\ a_i^{-1}F_ia_i-\sft(\kappa(a_i^{-1},F_i))
\eand
\tilde H_i\ =\ a_i^{-1}\acton H_i-\kappa(a_i^{-1},\sft(H_i))
\end{equation}

for all $(i)\in Y^{[1]}$. For a special adjustment (<ref>), we also have

\begin{equation}
\sft(\tilde H_i)\ =\ \sft(H_i)
\end{equation}

for all $(i)\in Y^{[1]}$.

Both results follow directly by using the formulas (<ref>) for the gauge-transformed gauge potentials $\tilde A_i$ and $\tilde B_i$ in the definition of adjusted curvatures in (<ref>), and simplifying the resulting expressions using identities from <ref>. The derivation for $\tilde H_i$ is more involved. One first shows that

\begin{equation}
\tilde H_i+\kappa(\tilde A_i, \tilde F_i)\ =\ a^{-1}_i\acton(H_i+\kappa(A_i,F_i))+(a_i^{-1}F_i a_i)\acton\lambda_i-\tilde\nabla_i\kappa(a_i^{-1},F_i)~,
\end{equation}

and one then expands the terms with $\kappa$ using (<ref>) and the Bianchi identity for $F_i$ from (<ref>) as well as (<ref>) and (<ref>). The case of a special adjustment follows from

\begin{equation}
\sft(\kappa(a_i^{-1},\sft(H_i))=\sft(P(\sft(a_i^{-1}[\sft(H_i),a_i])))=\sft(a_i^{-1}[\sft(H_i),a_i]))~.
\end{equation}

Finally, we also have the adjusted higher gauge transformations.

Two gauge transformations $(a,b,\lambda)$ and $(\tilde a,\tilde b,\tilde\lambda)$ are called equivalent whenever they are linked by a higher coboundary consisting of maps

\begin{equation}
m\ \in\ \scC^\infty(Y^{[1]},\sfH)
\end{equation}

by means of

\begin{equation}\label{eq:adjustedHigherGaugeTransformationsB}
\begin{aligned}
\tilde b_{ij}\ &=\ m_ib_{ij}(g_{ij}\acton m_j^{-1})~,
\\
\tilde a_i\ &=\ \sft(m_i)a_i~,
\\
\tilde \lambda_i\ &=\ \lambda_i+a_i^{-1}\acton(m_i^{-1}\nabla_i m_i)
\end{aligned}
\end{equation}

for all appropriate $(i,j,\ldots)\in Y^{[n]}$. We shall refer to them as adjusted higher gauge transformations.

Comparing the adjusted cocycle conditions (<ref>) with the unadjusted ones (<ref>) as well as the adjusted coboundary conditions (<ref>) with the unadjusted ones (<ref>) as well as the higher coboundary transformations (<ref>) and (<ref>), we realise that the only modification in the adjusted descent data arises in the patching and gauge transformations of the gauge potential $B$ as well as in the definition of the curvatures and the fake-flatness condition.

Finally, let us consider the gluing-together of two gauge transformations (<ref>) to a third one,

\begin{equation}\label{eq:adjustedTripleGaugeTransformations}
\begin{tikzcd}
& (A_2,B_2)\arrow[dr,"{(a_{23},\lambda_{23})}"]\arrow[d,Rightarrow,"{m_{123}}"]
\\
(A_1,B_1) \arrow[ur,"{(a_{12},\lambda_{12})}"] \arrow[rr,"{(a_{13},\lambda_{13})}",swap] & \phantom{a} & (A_3,B_3)
\end{tikzcd}
\end{equation}

Then, <ref> changes as follows.

The gauge transformations (<ref>) are related by a higher coboundary, parametrised by an $ m_{123}\in\scC^\infty(Y,\sfH) $, in the following way

\begin{equation}\label{eq:adjustedCoboundaryComposition}
\begin{aligned}
a_{13}\ &=\ \sft(m_{123})a_{12}a_{23}~,
\\
\lambda_{13}\ &=\ \lambda_{23}+a_{23}^{-1}\acton\lambda_{12}-a_{13}^{-1}\acton(m_{123}\nabla_1m_{123}^{-1})~.
\end{aligned}
\end{equation}

The diagram (<ref>) is commutative if and only if

\begin{equation}\label{eq:adjustmentConditionFromTriangle}
(a_{23}^{-1}a_{12}^{-1})\acton (m_{123}^{-1}(F_1\acton m_{123}))\ =\ \kappa(a^{-1}_{13},F_1)-a_{23}^{-1}\acton\kappa(a^{-1}_{12},F_1)-\kappa(a^{-1}_{23},F_2)~,
\end{equation}

which, in turn, is equivalent to the adjustment condition (<ref>).

The conditions (<ref>) follow from the consistency of the gluing for $A$ and the condition (<ref>) for those of $B$, respectively. See <ref> for details.

Consequently, the fake-curvature condition has been fully subsumed into the adjustment, and there are no further conditions on $\kappa$ or $F$. This observation implies the following.

Over $Y$, the local connection forms and their infinitesimal gauge transformations form a closed BRST complex[i.e. the Chevalley–Eilenberg resolution of the quotient of connections by gauge transformations and higher gauge transformations], and hence the underlying Weil algebra is adjusted in the sense of <cit.>.

This is a simple and straightforward computation, but involves many technicalities which we did not introduce here, so we prefer to suppress the details.

Adjusted fake flatness.
It is natural to expect that equivalent 2-groups lead to equivalent principal 2-bundles with connections. A closer analysis of the situation, however, reveals that this is not necessarily the case, as e.g. the quasi-isomorphism of gauge $L_\infty$-algebras does not necessarily translate to an equivalence of the corresponding infinitesimal connection data.[As an example, consider the gauge crossed module of Lie algebras $\frg\xrightarrow{~\sfid~}\frg$, which is equivalent to the trivial crossed module $*\xrightarrow{~~}*$. The infinitesimal connection data is then obtained by tensoring these crossed modules, regarded as 2-term $L_\infty$-algebras, by the differential graded commutative algebra $\Omega^\bullet(Y)$, the de Rham complex on $Y$, and truncating degrees $\geq 2$. The two resulting $L_\infty$-algebras are clearly not quasi-isomorphic, as the first one contains non-trivial elements in the first cohomology group of the $L_\infty$-algebra differential.] Nevertheless, one can achieve this equivalence for special adjustments by imposing the following, additional condition.

Let $(\kappa,P)$ be a special adjustment (<ref>) of a crossed module of Lie groups. The adjusted fake-curvature condition reads as

\begin{equation}\label{eq:adjustedFakeCurvatureCondition}
P(F_i)\ =\ 0
\end{equation}

for all $i\in Y^{[1]}$.

To see the equivalence, note that for a special adjustment (<ref>) of a crossed module of Lie algebras $(\frh\xrightarrow{~\sft~}\frg)$, we have two decompositions

\begin{equation}
\frh\ \cong\ \ker(\sft)\oplus\im(P\circ\sft)
\eand
\frg\ \cong\ \coker(\sft)\oplus\im(\sft\circ P)~.
\end{equation}

Since the quasi-isomorphism class of the crossed module is defined by $\ker(\sft)$ and $\coker(\sft)$, we have to ensure that the connection 2-forms $B_i$ taking values in $\im(P\circ \sft)$ and the connection 1-form $A_i$ taking values in $\im(\sft\circ P)$ are fully determined by the remaining data. We first note that we can gauge-trivialise $(\sft\circ P)(A_i)$ using ordinary gauge transformations of $A_i$. The adjusted fake-curvature condition $P(F_i)=0$ then implies

\begin{equation}
(P\circ \sft)(B_i)\ =\ P(\rmd A_i+\tfrac12[A_i,A_i])~,
\end{equation}

and hence $(P\circ\sft)(B_i)$ is completely determined by $P((\id-\sft\circ P)(A_i))$.

§ EXAMPLES: PRINCIPAL BUNDLES AS ADJUSTED HIGHER PRINCIPAL BUNDLES

As an example, let us explain how ordinary principal $\sfG$-bundles with connection for $\sfG$ some Lie group can be seen as principal 2-bundles with adjusted connection and structure 2-group the adjusted crossed module of Lie groups $\caL\sfG\cong\sfG_{\rm cm}$, see (<ref>) and (<ref>).[Note that in the case of $\sfG_{\rm cm}$, condition (<ref>) is trivial, and there is no need to impose a fake-curvature condition.]

§.§ General setting

The descent data for a principal $(\caL\sfG,\kappa)$-bundle with $\kappa$ as in (<ref>) is given in <ref>. The adjusted curvatures (<ref>) reduce to

\begin{equation}\label{eq:curvature_LG_kompressed}
F_i\ =\ \rmd A_i+\tfrac12[A_i,A_i]+B_i
\eand
H_i\ =\ (\id-\wp\cdot\boundary)(\rmd F_i)
\end{equation}

for all $(i)\in Y^{[1]}$ which, in turn, allow us to rewrite the adjusted Bianchi identities (<ref>) as

\begin{equation}
\nabla_i F_i\ =\ H_i+\kappa(A_i,F_i)
\eand
\rmd H_i\ =\ 0
\end{equation}

for all $(i)\in Y^{[1]}$. Finally, the adjusted fake-curvature condition (<ref>) becomes

\begin{equation}\label{eq:adjustedFakeCurvatureConditionPathLoopGroups}
(\id-\wp\cdot\boundary)(F_i)\ =\ 0
\quad\Rightarrow\quad
H_i\ = 0
\end{equation}

for all $(i)\in Y^{[1]}$. We then expect the following statement.

The descent data of principal $\sfG$-bundles with connections and the descent data of principal $\caL\sfG$-bundles with adjusted connections satisfying the adjusted fake-curvature condition (<ref>) are equivalent.

We shall establish this claim in both directions in two steps. Firstly, we map the descent data of one type of bundle to descent data for the other type. We then show that isomorphism classes are indeed mapped to isomorphism classes.

Principal bundles $\Rightarrow$ higher principal bundles: Consider the descent data of a principal $\sfG$-bundle over $X$ subordinate to a good cover $Y\rightarrow X$,

\begin{equation}
g\ \in\ \scC^\infty(Y^{[2]},\sfG)
\eand
A\ \in\ \Omega^1(Y^{[1]},\frg)
\end{equation}

\begin{equation}
g_{ik}\ =\ g_{ij}g_{jk}
\eand
A_j\ =\ g_{ij}^{-1}A_i g_{ij}+g_{ij}^{-1}\rmd g_{ij}
\end{equation}

for all appropriate $(i,j,\ldots)\in Y^{[n]}$. Since the endpoint evaluation map $\boundary:P_0\sfG\rightarrow \sfG$ defined in (<ref>) is a surjective submersion, the pullbacks along $\boundary$ exist, and since also $Y^{[2]}$ is the disjoint union of contractible subsets of $X$ by assumption, there exists a lift of $g\in\scC^\infty(Y^{[2]},\sfG)$, i.e. a map $g^\circ\in\scC^\infty(Y^{[2]},P_0\sfG)$ such that

\begin{equation}
\begin{tikzcd}
& P_0\sfG \arrow[d,"\boundary"]
\\
Y^{[2]} \arrow[ru,dotted,"g^\circ"] \arrow[r,"g"] & \sfG
\end{tikzcd}
\end{equation}

is commutative.[Because $Y$ is a good cover, $Y^{[2]}$ is a disjoint union of contractible spaces, and hence the usual lifting lemma (e.g. <cit.>) works on each connected component of $Y^{[2]}$.]

The map $g^\circ$ only satisfies the cocycle condition for a principal $P_0\sfG$-bundle up to elements in $L_0\sfG$. To extend this to a cocycle for a principal $\caL\sfG$-bundle with adjusted connection, we hence introduce $h^\circ\in\scC^\infty(Y^{[3]},L_0\sfG)$ by

\begin{equation}
h^\circ_{ijk}\ \coloneqq\ g^\circ_{ik}(g^\circ_{jk})^{-1}(g^\circ_{ij})^{-1}
\eforall(i,j,k)\ \in\ Y^{[3]}~.
\end{equation}

This indeed defines a loop in $\sfG$, that is, $\boundary(h^\circ_{ijk})=\unit$, because of the gluing condition in (<ref>). It is possible and convenient (but not necessary) to define the lift $g^\circ$ in such a way that

\begin{equation}
g^\circ_{ii}\ =\ \unit
\eand
g^\circ_{ij}\ =\ (g^\circ_{ji})^{-1}
\end{equation}

with $\unit\in P_0\sfG$ the constant path, which also implies the normalisations

\begin{equation}
h^\circ_{iij}\ =\ h^\circ_{ijj}\ =\ h^\circ_{iji}\ =\ \unit~.
\end{equation}

For the connection forms, we generalise the usual prescription for transporting Maurer–Cartan elements in an $L_\infty$-algebra along quasi-isomorphisms, cf. <cit.>:

\begin{equation}\label{eq:connection_loop_embedded}
A^\circ_i\ \coloneqq\ \wp\cdot A_i
\eand
B^\circ_i\ \coloneqq\ \tfrac12(\wp-\wp^2)\cdot[A_i,A_i]
\eforall
(i)\ \in\ Y^{[1]}
\end{equation}

so that

\begin{equation}\label{eq:Lambda-lift}
\Lambda_{ij}^\circ\ \coloneqq\ (g_{ij}^\circ)^{-1}A_i^\circ g_{ij}^\circ+(g_{ij}^\circ)^{-1}\rmd g_{ij}^\circ-A_j^\circ
\eforall
(i,j)\ \in\ Y^{[2]}~.
\end{equation}

The right-hand side here is a loop in $\frg$ due to the gluing relation (<ref>) between $A_i$ and $A_j$. It is now not too difficult to verify that with all these definitions, all the cocycle conditions (<ref>) including the adjusted fake-flatness condition (<ref>) are satisfied. Altogether, we have obtained the descent data $(h^\circ,g^\circ,\Lambda^\circ,A^\circ,B^\circ)$ of a principal $\caL\sfG$-bundle with adjusted connection satisfying the adjusted fake-curvature condition.

Principal bundles $\Leftarrow$ higher principal bundles:
Conversely, we can apply the endpoint evaluation (<ref>) to project the descent data $(h^\circ,g^\circ,\Lambda^\circ,A^\circ,B^\circ)$ for a principal $\caL\sfG$-bundle over $X$ with adjusted connection and with the adjusted fake-flatness condition (<ref>) imposed to the data $(g,A)$ of a principal $\sfG$-bundle by means of

\begin{equation}\label{eq:recover_ordinary_cocycles}
g_{ij}\ \coloneqq\ \boundary(g_{ij}^\circ)
\eand
A_i\ \coloneqq\ \boundary(A_i^\circ)
\end{equation}

for all appropriate $(i,j,\ldots)\in Y^{[n]}$. Again, it follows that this data fulfils the required cocycle conditions (<ref>).

Gauge and higher gauge equivalence:
Above, we have established morphisms $\phi:(g,A)\mapsto(h^\circ,g^\circ,\Lambda^\circ,A^\circ,B^\circ)$ and $\psi:(h^\circ,g^\circ,\Lambda^\circ,A^\circ,B^\circ)\mapsto(g,A)$. Evidently, $\psi\circ\phi=\id$, and it remains to show that there is a bundle isomorphism $\phi\circ\psi\Rightarrow\id$. Consider some descent data $(h^\circ,g^\circ,\Lambda^\circ,A^\circ,B^\circ)$ and set

\begin{equation}
(\tilde h^\circ,\tilde g^\circ,\tilde\Lambda^\circ,\tilde A^\circ,\tilde B^\circ)\ \coloneqq\ (\phi\circ\psi)(h^\circ,g^\circ,\Lambda^\circ,A^\circ,B^\circ)~.
\end{equation}

We note that there is a coboundary $(a^\circ,b^\circ,\lambda^\circ)$ with

\begin{equation}\label{eq:coboundaryAlpha}
a^\circ_i\ \coloneqq\ \unit~,
\quad
b^\circ_{ij}\ \coloneqq\ \tilde g^\circ_{ij}(g^\circ_{ij})^{-1}~,
\eand
\lambda^\circ_i\ \coloneqq\ (\id-\wp\cdot\boundary)(A^\circ_i)
\end{equation}

for all appropriate $(i,j,\ldots)\in Y^{[n]}$ connecting $(h^\circ,g^\circ,\Lambda^\circ,A^\circ,B^\circ)$ with $(\tilde h^\circ,\tilde g^\circ,\tilde\Lambda^\circ,\tilde A^\circ,\tilde B^\circ)$. In particular, the difference between $g^\circ_{ij}$ and $\tilde g^\circ_{ij}$ is a loop and thus in the image of the bundle isomorphism with the given $b^\circ_{ij}$. Furthermore, the maps $\tilde h^\circ_{ijk}$ are fully determined by the maps $\tilde g^\circ_{ij}$ because $\sft$ is injective. Similarly, the difference between $A^\circ_i$ and $\tilde A^\circ_i$ is a loop and thus in the image of the bundle isomorphism with the given $\lambda^\circ_i$, which then fully determines the maps $\tilde \Lambda^\circ_{ij}$. The adjusted fake-flatness condition (<ref>) moreover fully determines $\tilde B^\circ_i$ in terms of $\tilde A^\circ_i$ again because $\sft$ is injective. Hence, for descent data satisfying the adjusted fake-flatness condition (<ref>), the coboundary data (<ref>) indeed provide a bundle isomorphism $\phi\circ\psi\Rightarrow\id$. In fact, the full gauge orbits are mapped to each other:

\begin{equation}
\begin{tikzcd}[row sep=2cm,column sep=3cm]
(g,A)\arrow[r,bend left=20,"\phi"]\arrow[d,"a"] & (h^\circ,g^\circ,\Lambda^\circ,A^\circ,B^\circ)\arrow[l,bend left=20,"\psi"{below}]\arrow[d,bend left=30,"{(\tilde a^\circ,\tilde b^\circ,\tilde\lambda^\circ)}"{name=U,right}]
\arrow[d,bend right=30,"{(a^\circ,b^\circ,\lambda^\circ)}"{name=D,left}]
\\
(\tilde g,\tilde A)\arrow[r,bend left=20,"\phi"] & (\tilde h^\circ,\tilde g^\circ,\tilde\Lambda^\circ,\tilde A^\circ,\tilde B^\circ)\arrow[l,bend left=20,"\psi"{below}]\arrow[Rightarrow,"m^\circ", from=D, to=U,start anchor={[xshift=1ex]},end anchor={[xshift=-1ex]}]
\end{tikzcd}
\end{equation}

Explicitly, $a^\circ$ and $\tilde a^\circ$ are lifts of $a$ such that

\begin{equation}\label{eq:boundary_kond_1}
\boundary(a^\circ_i)\ =\ a_i\ =\ \boundary(\tilde a_i^\circ)
\eforall
(i)\ \in\ Y^{[1]}~.
\end{equation}

The maps $b^\circ$ and $\tilde b^\circ$ are then determined by the explicit form of $\tilde g^\circ$,

\begin{equation}
b^\circ_{ij}\ \coloneqq\ a^\circ_i\tilde g^\circ_{ij}a^{\circ-1}_jg_{ij}^{\circ-1}
\eand
\tilde b^\circ_{ij}\ \coloneqq\ \tilde a^\circ_i\tilde g^\circ_{ij}\tilde a^{\circ-1}_j(g_{ij}^\circ)^{-1}
\eforall
(i,j)\ \in\ Y^{[2]}~,
\end{equation}

where the expressions on the right-hand sides are indeed loops due to (<ref>). Furthermore, comparing $A^\circ$ with $\tilde A^\circ$ fixes the maps $\lambda^\circ$ and $\tilde \lambda^\circ$,

\begin{equation}
\begin{aligned}
\lambda^\circ_i\ &\coloneqq\ (a_i^\circ)^{-1}A^\circ_ia^\circ_i+(a_i^\circ)^{-1}\rmd a^\circ_i-\tilde A^\circ_i~,
\\
\tilde \lambda^\circ_i\ &\coloneqq\ (\tilde a^\circ_i)^{-1} A^\circ_i\tilde a^\circ_i+(\tilde a^\circ_i)^{-1}\rmd\tilde a^\circ_i-\tilde A^\circ_i
\end{aligned}
\end{equation}

for all $(i)\in Y^{[1]}$. Having fixed the lifted gauge transformations, one can directly verify that the coboundary relations between the descent data $(h^\circ,g^\circ,\Lambda^\circ,A^\circ,B^\circ)$ and $(\tilde h^\circ,\tilde g^\circ,\tilde\Lambda^\circ,\tilde A^\circ,\tilde B^\circ)$ hold.

It remains to check that the two constructed gauge transformations are linked by a higher gauge transformation (<ref>). We put

\begin{equation}
m^\circ_i\ \coloneqq\ \tilde a^\circ_i(a^\circ_i)^{-1}
\eforall
(i)\ \in\ Y^{[1]}
\end{equation}

which ensures that $a^\circ$ and $\tilde a^\circ$ are related appropriately. The corresponding relations between $b^\circ$ and $\tilde b^\circ$ as well as $\lambda^\circ$ and $\tilde\lambda^\circ$ are then straightforwardly checked.

§.§ The instanton–anti-instanton pair as an LSpin(4)-bundle

Motivation. Let us also give a concrete and explicit example of how a principal $\sfG$-bundle can be equivalently described as a principal $\caL\sfG$-bundle. For physical applications, an interesting candidate is certainly the principal $\sfSU(2)$-bundle over $S^4$ underlying the elementary instanton solution on $S^4$. Recall that this bundle is the principal $\sfSU(2)$-bundle given by the quaternionic Hopf fibration $S^3\hookrightarrow S^7\rightarrow S^4$. This bundle has a non-vanishing second Chern class whose image in de Rham cohomology integrates to the unit instanton charge.

Because $\sfSpin(4)\cong \sfSU(2)\times \sfSU(2)$, it is very natural to extend this bundle to that of an instanton–anti-instanton pair, which is the $\sfSpin(4)\cong \sfSU(2)\times \sfSU(2)$-bundle

\begin{equation}\label{eq:Spin4-bundle}
\sfSU(2)\times\sfSU(2)\ \hookrightarrow\ P\ \coloneqq\ \sfSpin(5)\ \rightarrow\ \sfSpin(5)/\sfSpin(4)\ \cong\ \sfSO(5)/\sfSO(4)\ \cong\ S^4~.
\end{equation}

This gives the unique spin structure on $S^4$, cf. <cit.>.

The connections of both the elementary instanton and the instanton–anti-instanton pair can be obtained as the canonical connection on the corresponding reductive homogeneous spaces; see <ref> for a brief review of this construction. The resulting connection on the bundle (<ref>) turns out to consist of the sum of an $\sfSU(2)$-instanton and an $\sfSU(2)$-anti-instanton.

This extension is also natural from a physical perspective. As explained in <cit.>, self-dual gauge configurations are part of non-Abelian self-dual strings that naturally arise when M2-branes end on M5-branes. Moreover, the quaternionic Hopf fibration underlying the elementary instanton also features prominently in the so-called hypothesis H, cf. e.g. <cit.>, which states that the charge quantisation in M-theory happens in a particular cohomology theory.

$S^4$ as the coset $\sfSpin(5)/\sfSpin(4)$.
The explicit description of the fibration (<ref>) is found e.g. in <cit.>, and we briefly review this description in the following. Recall that

\begin{equation}\label{eq:def_Sp2}
\sfSpin(5)\ \cong\ \sfSp(2)\ =\ \sfU(2,\IH)\ \coloneqq\ \left\{g\in\sfMat(2,\IH)\,|\,g^\dagger g=gg^\dagger=\unit_2\right\},
\end{equation}

and thus, the elements of $\sfSpin(5)$ can be identified with matrices

\begin{equation}\label{eq:Spin5-elements}
g\ =\ \begin{pmatrix}
a & b
\\
c & d
\end{pmatrix},
\quad
\quad
\begin{array}{ll}
|a|^2+|b|^2\ =\ 1~,&a\bar c+b\bar d\ =\ 0~,
\\
|c|^2+|d|^2\ =\ 1~,&c\bar a+d\bar b\ =\ 0~,
\\
|a|^2+|c|^2\ =\ 1~,&\bar a b+\bar c d\ =\ 0~,
\\
|b|^2+|d|^2\ =\ 1~,&\bar b a+\bar d c\ =\ 0~.
\end{array}
\end{equation}

Note that the above eight relations contain six independent conditions, so that $\sfSpin(5)$ is ten-dimensional. We further identify $\sfSU(2)$ with the unit quaternions $\sfSp(1)\cong\sfU(1,\IH)$, and consider the embedding

\begin{equation}
\begin{aligned}
\sfSU(2)\times\sfSU(2)\ &\hookrightarrow\ \sfSpin(5)~,
\\
\left(\frac{e}{|e|},\frac{f}{|f|}\right)\ &\mapsto\
\begin{pmatrix}
\frac{e}{|e|} & 0
\\
0 & \frac{f}{|f|}
\end{pmatrix}.
\end{aligned}
\end{equation}

We then have a canonical projection $\sfSpin(5)\rightarrow\sfSpin(5)/\sfSpin(4)$, where we identify

\begin{equation}\label{eq:Spin4Action}
\begin{pmatrix}
a & b
\\
c & d
\end{pmatrix}\ \sim\
\begin{pmatrix}
a & b
\\
c & d
\end{pmatrix}
\begin{pmatrix}
\frac{e}{|e|} & 0
\\
0 & \frac{f}{|f|}
\end{pmatrix}
\eforall
e,f\ \in\ \IH~.
\end{equation}

Given an element $g\in\sfSpin(5)$ of the form (<ref>), we note that the product

\begin{equation}
\begin{pmatrix}
a & b
\\
c & d
\end{pmatrix}
\begin{pmatrix}
1 & 0
\\
0 & -1
\end{pmatrix}
\begin{pmatrix}
a & b
\\
c & d
\end{pmatrix}^\dagger
\ = \
\begin{pmatrix}
2|a|^2-1 & 2a\bar c
\\
2 c\bar a & 2|a|^2-1
\end{pmatrix}
\end{equation}

is invariant under the right-action (<ref>) of $\sfSpin(4)$ on $g$. Moreover, we can identify

\begin{equation}
x^1+\rmi x^2+\rmj x^3+\rmk x^4\ \coloneqq\ 2 c\bar a~\eand
x^5\ \coloneqq\ 2|a|^2-1~,
\end{equation}

where $\rmi$, $\rmj$, and $\rmk$ are the standard quaternionic units and $x^1,\ldots,x^5\in\IR$. Because of the constraints in (<ref>), we obtain $\sum_{i=1}^5(x^i)^2=1$, that is, $S^4\hookrightarrow\IR^5$. This yields a projection

\begin{equation}
\pi\,:\,\sfSpin(5)\ \rightarrow\ S^4~,
\end{equation}

which we shall use below.

Similarly, we have a projection $\sfSpin(5)\rightarrow S^7\cong\sfSpin(5)/\sfSU(2)$ that is given by

\begin{equation}
\begin{pmatrix}
a & b
\\
c & d
\end{pmatrix}
\ \mapsto\ (a,c)~,
\end{equation}

where we regard $S^7$ as the unit vectors in $\IH^2\cong \IR^8$ and the quotient group $\sfSU(2)$ is identified with the second factor in $\sfSpin(4)\cong \sfSU(2)\times \sfSU(2)$.

Instantons on $S^4$.
We write elements of $\frspin(5)$ as quaternionic matrices

\begin{equation}
X\ =\
\begin{pmatrix}
x & z
\\
-\bar z & y
\end{pmatrix}
\eforall
x,y,z\ \in\ \IH
\ewith
\Re(x)\ =\ \Re(y)\ =\ 0~.
\end{equation}

Here, $\frspin(4)\cong\frsu(2)\oplus\frsu(2)$ is identified with the two imaginary quaternions $x$ and $y$, which yields the decomposition

\begin{equation}\label{eq:split_spin5}
\frspin(5)\ \cong\ \frspin(4)\oplus\frm~,
\quad
\begin{pmatrix}
x & z
\\
-\bar z & y
\end{pmatrix}
\ =\
\begin{pmatrix}
x & 0
\\
0 & y
\end{pmatrix}
\begin{pmatrix}
0 & z
\\
-\bar z & 0
\end{pmatrix}
\end{equation}

with the evident relations

\begin{equation}\label{eq:algebraic structure}
[\frspin(4),\frspin(4)]\ \subseteq\ \frspin(4)~,
\quad
[\frspin(4),\frm]\ \subseteq\ \frm~,
\eand
[\frm,\frm]\ \subseteq\ \frspin(4)~.
\end{equation}

Our discussion in <ref> then yields the canonical connection on the bundle $\sfSpin(5)\rightarrow S^4$. As we shall see below, this connection is the direct sum of a fundamental instanton and a fundamental anti-instanton.

Clutching construction.
Following our discussion in <ref>, we can describe the principal $\sfSpin(4)$-bundle $\sfSpin(5)\rightarrow S^4$ subordinate to the cover given by the bundle itself with the transition function

\begin{equation}\label{eq:transition_function_1}
\begin{aligned}
g\,:\,(\sfSpin(5))^{[2]}\ &\rightarrow\ \sfSU(2)\times\sfSU(2)~,
\\
(g_1,g_2)\ &\mapsto\ g_1^{-1}g_2~,
\end{aligned}
\end{equation}

where we have again used the notation (<ref>).

In order to make contact with the usual clutching construction of principal bundles over $S^4$, we embed $S^4$ into $\IR^5$, set $U_\pm\coloneqq S^4\setminus\{x^5=\pm1\}$, and derive a cover $U_+\sqcup U_-\to S^4$ from the stereographic projections

\begin{equation}
\begin{aligned}
\pi_\pm\,:\,U_\pm\ &\rightarrow\ \pi_\pm(U_\pm)\ \cong \IH~,
\\
(x^1,\ldots,x^5)\ &\mapsto\ q_\pm\ \coloneqq\ \frac{1}{1\mp x^5}(x^1\pm\rmi x^2\pm\rmj x^3\pm\rmk x^4)~,
\end{aligned}
\end{equation}

where, as before, $\rmi$, $\rmj$, and $\rmk$ are the standard quaternionic units. Consequently, $q_+q_-=1$ on $\pi_+(U_+\cap U_-)\cong\pi_+(U_+\cap U_-)\cong\IH\setminus\{0\}$.

We then consider the commutative diagram

\begin{equation}
\begin{tikzcd}
U_+\sqcup U_-\arrow[rr,"\phi"]\arrow[rd] & &\sfSpin(5)\arrow[ld]
\\
& S^4
\end{tikzcd}
\end{equation}

where $\phi$ reads in local coordinates as

\begin{equation}\label{eq:comp1}
\begin{gathered}
\pi_\pm(U_\pm)\ \rightarrow\ \sfSpin(5)~,
\\
q_+\ \mapsto\ \frac{1}{\sqrt{1+|q_+|^2}}
\begin{pmatrix}
\bar q_+ & 1
\\
1 & -q_+
\end{pmatrix}
\eand
q_-\ \mapsto\ \frac{1}{\sqrt{1+|q_-|^2}}
\begin{pmatrix}
1 & q_-
\\
\bar q_- & -1
\end{pmatrix}.
\end{gathered}
\end{equation}

The pullback $g_{+-}\coloneqq\phi^*g$ of the transition function (<ref>) to $(U_+\sqcup U_-)^{[2]}$ then reads in local coordinates as

\begin{equation}
\begin{aligned}
(\pi_\pm^{-1})^*g_{+-}|_{\pi_\pm(U_+\cap U_-)}\,:\,\pi_\pm(U_+\cap U_-)\ &\rightarrow\ \sfSU(2)\times\sfSU(2)~,
\\
q_\pm\ &\mapsto\ \left(\frac{q_+}{|q_+|},\frac{\bar q_+}{|q_+|}\right)\ =\ \left(\frac{\bar q_-}{|q_-|},\frac{q_-}{|q_-|}\right).
\end{aligned}
\end{equation}

This principal bundle can be endowed with a connection given by the gauge potentials $A_\pm$ on $U_\pm$ with

\begin{equation}
A_-\ =\ g_{+-}^{-1}A_+g_{+-}+g_{+-}^{-1}\rmd g_{+-}
\eon
U_+\cap U_-~.
\end{equation}

This connection describes a pair of an elementary instanton with an elementary anti-instanton, which, in quaternionic notation, is given by

\begin{equation}
(\pi_\pm^{-1})^*A_\pm\ \coloneqq\ \frac12\left(\frac{q_\pm\rmd\bar q_\pm-\rmd q_\pm\bar q_\pm}{1+|q_\pm|^2},\frac{\bar q_\pm\rmd q_\pm-\rmd\bar q_\pm q_\pm}{1+|q_\pm|^2}\right)
\eon
\pi_\pm(U_\pm)~,
\end{equation}

cf. <cit.>. Consequently, the curvature is

\begin{equation}
(\pi_\pm^{-1})^*F_\pm\ =\ \left(\frac{\rmd q_\pm\wedge\rmd\bar q_\pm}{(1+|q_\pm|^2)^2},\frac{\rmd\bar q_\pm\wedge\rmd q_\pm}{(1+|q_\pm|^2)^2}\right)
\eon
\pi_\pm(U_\pm)
\end{equation}

with the first component describing an instanton with self-dual field strength and the second component describing an anti-instanton with anti-self-dual field strength. The total first Pontryagin class (as is evident from its image in de Rham cohomology) vanishes, and this will become important later.

Choice of cover for a lift. In order to lift the cocycle data (<ref>) of the above defined $\sfSpin(4)$-bundle $P$ to that of a principal $\caL\sfSpin(4)$-bundle, we have to choose a different cover.[For a recent application of based loop groups as gauge groups in Yang–Mills theory, see also <cit.>.]
The cover $U_+\sqcup U_-$ introduced above is not suitable for a lift of $P$ to a principal $\caL\sfSpin(4)$-bundle, because the transition function $g_{+-}$ in (<ref>) does not factor through $P_0\sfSpin(4)$. This is easy to see: $U_+\sqcup U_-\cong S^3\times \IR$, and the equivalence classes of transition functions are thus given by $\pi_3(\sfSpin(3)\times \sfSpin(3))\cong \IZ\times \IZ$, but the path space $P_0\sfSpin(4)$ is contractible, and hence its homotopy groups and in particular $\pi_3$ vanishes. Therefore, the topological information has to be encoded in the $L_0\sfSpin(4)$-valued function. Since we only have a 2-patch cover and since $U_+\cap U_-\cong S^3\times\IR$, the homotopy classes of this map are $\pi_3(L_0\sfSpin(4))\cong\pi_4(\sfSpin(4))\cong\pi_4(S^3\times S^3)\cong\IZ_2\oplus\IZ_2$, which, evidently is not large enough.[The identification $\pi_{n}(\sfG)\cong\pi_{n-1}(L_0\sfG)$ is as follows. Let $X$ be a topological space and $p_0\in X$ fixed. The reduced suspension of $(X,p_0)$ is the quotient $\Sigma X\coloneqq(X\times[0,1])/\!\!\sim$ with the equivalence relation given by $(p,0)\sim(p,1)\sim(p_0,t)$ for all $p\in X$ and for all $t\in[0,1]$. Then, for $Y$ another topological space, there is the natural identification $\scC_0(\Sigma X,Y)\cong\scC_0(X,L_0Y)$ of continuous maps that preserve the base point. In the context of spheres, we have a homeomorphism between $\Sigma S^{n-1}$ and $S^n$. Indeed, we mark poles $x_0\in S^{n-1}$ and $y_0\in S^n$ and identify $x_0$ as well as all points $S^{n-1}\times\{0,1\}$ with $y_0$. We then identify the remaining points $(0,1)$ with $\IR$ and map $S^{n-1}\setminus\{x_0\}$ to $\IR^{n-1}$ by stereographic projection at $x_0$. The Cartesian product $\IR^{n-1}\times\IR$ is then identified with $S^n\setminus\{y_0\}$ by inverse stereographic projection at $y_0$.]

We can, however, construct a suitable 3-patch replacement cover of the 2-patch cover $U_+\sqcup U_-$ rather directly by hand. First of all, we note that $\pi_3(\sfSpin(4))\cong\pi_2(L_0\sfSpin(4))$ so that we would like our triple overlaps to be homeomorphic to $S^2\times\IR^2$. We can reduce $\pi_+(U_+\cap U_-)\cong\pi_-(U_+\cap U_-) \cong\IH\setminus\{0\}$ to such a space by removing a straight line through the origin from $\IH$ since, topologically, $\IH\setminus\IR\cong S^2\times\IR^2$. In particular, let

\begin{equation}
\begin{aligned}
L_+\ &\coloneqq\ \{q\in\IH\,|\,\Re(q)>0\text{ and }\Im(q)=0\}~,
\\
L_-\ &\coloneqq\ \{q\in\IH\,|\,\Re(q)<0\text{ and }\Im(q)=0\}~,
\end{aligned}
\end{equation}

and set $L\coloneqq L_-\cup\{0\}\cup L_+$; evidently $L\cong\IR$. Next, we define the 3-patch cover $U_1\sqcup U_2\sqcup U_3\rightarrow S^4$ by[Note that this is not an open cover. We can refine this cover to an open cover; however, to keep the formulas simple, we shall work with this 3-patch cover.]

\begin{equation}\label{eq:3patchCover}
\begin{gathered}
\ewith
U_1\ \coloneqq\ \pi_+^{-1}(\underbrace{\pi_+(U_+)\setminus L_+}_{\cong\,\IH\setminus L_+})
\eand
\pi_1\ \coloneqq\ \pi_+|_{U_1}~,
\\
\ewith
U_2\ \coloneqq\ \pi_+^{-1}(\underbrace{\pi_+(U_+)\setminus L_-}_{\cong\,\IH\setminus L_-})
\eand
\pi_2\ \coloneqq\ \pi_+|_{U_2}~,
\\
\ewith
U_3\ \coloneqq\ U_-
\eand
\pi_3\ \coloneqq\ \pi_-~.
\end{gathered}
\end{equation}

Note that $U_1\cup U_2=U_+$ and $U_1\cap U_2\cap U_3\cong S^2\times\IR^2$. With respect to this cover, the bundle $\sfSpin(5)\rightarrow S^4$ is described by transition functions

\begin{equation}
g_{12}\ \coloneqq\ -\unit~,
\quad
g_{13}\ \coloneqq\ -g_{+-}~,
\eand
g_{23}\ \coloneqq\ g_{+-}~,
\end{equation}

where we inserted a constant change of frame between patches $U_1$ and $U_2$. This does not affect the gauge potential, and we have

\begin{equation}
\begin{aligned}
(\pi_{1,2}^{-1})^*A_{1,2}\ &\coloneqq\ \frac12\left(\frac{q_{1,2}\rmd\bar q_{1,2}-\rmd q_{1,2}\bar q_{1,2}}{1+|q_{1,2}|^2},\frac{\bar q_{1,2}\rmd q_{1,2}-\rmd\bar q_{1,2} q_{1,2}}{1+|q_{1,2}|^2}\right),
\\
(\pi_3^{-1})^*A_3\ &\coloneqq\ \frac12\left(\frac{q_3\rmd\bar q_3-\rmd q_3\bar q_3}{1+|q_3|^2},\frac{\bar q_3\rmd q_3-\rmd\bar q_3 q_3}{1+|q_3|^2}\right),
\end{aligned}
\end{equation}

with $q_{1,2,3}\in\pi_{1,2,3}(U_{1,2,3})$, where $q_{1,2}$ are restrictions of $q_+$ and $q_3=q_-$.

Lifted cocycle data.
The cocycle (<ref>) can now be straightforwardly lifted, following the prescription in <ref>. Relative to the cover (<ref>), we can define the following lift of the cocycles:

\begin{equation}
g^\circ_{12}\ \coloneqq\ \big(t\mapsto(\rme^{\rmi\pi \wp(t)},\rme^{-\rmi\pi \wp(t)})\big)
\eand
g^\circ_{i3}\ \coloneqq\ \big(t\mapsto(u_i(t),\bar u_i(t))\big)
\end{equation}

for all $i=1,2$ with

\begin{equation}\label{eq:cocycles_LSpin_bundle_gui}
u_i(t)\ \coloneqq\ \left(q_i \mapsto \frac{1-\wp(t)+(-1)^i\wp(t)q_i}{|1-\wp(t)+(-1)^i\wp(t)q_i|}\right)
\end{equation}

and for all $t\in[0,1]$, which induces

\begin{equation}
h^\circ_{123}\ \coloneqq\ g_{13}^\circ(g_{23}^\circ)^{-1}(g_{12}^\circ)^{-1}~.
\end{equation}

Furthermore, the lift of the connection reads as

\begin{equation}
\begin{gathered}
A^\circ_1\ \coloneqq\ \wp\cdot A_1~,
\quad
A^\circ_2\ \coloneqq\ \wp\cdot A_2~,
\quad
A^\circ_3\ \coloneqq\ \wp\cdot A_3~,
\\
B^\circ_1\ \coloneqq\ \tfrac12(\wp-\wp^2)\cdot[A_1,A_1]~,
\quad
B^\circ_2\ \coloneqq\ \tfrac12(\wp-\wp^2)\cdot[A_2,A_2]~,
\\
B^\circ_3\ \coloneqq\ \tfrac12(\wp-\wp^2)\cdot[A_3,A_3]~,
\end{gathered}
\end{equation}

\begin{equation}
\begin{aligned}
\Lambda_{12}^\circ\ &\coloneqq\ \bigg(t\mapsto\wp(t)\cdot(\rme^{-\rmi\pi \wp(t)},\rme^{\rmi\pi \wp(t)}) \left[A_+,(\rme^{\rmi\pi \wp(t)},\rme^{-\rmi\pi \wp(t)})\right]\bigg)\,,
\\
\Lambda_{i3}^\circ\ &\coloneqq\ \bigg(t\mapsto\frac{1}{1+|q_+|^2}\bigg(\wp(t)|q_+|^2\bar u_i(t)u_i(1)\rmd\big(\bar u_i(1)u_i(t)\big)+|q_+|^2(1-\wp(t))\bar u_i(t)\rmd u_i(t)
\\
&\kern4.5cm+\bar u_i(t)\rmd u_i(t)-\wp(t)\bar u_i(1)\rmd u_i(1),u_i(t)\leftrightarrow\bar u_i(t)\bigg)\bigg)
\end{aligned}
\end{equation}

for all $i=1,2$ with $u_i$ as given in (<ref>) and for all $t\in[0,1]$. Here, $u_i(t)\leftrightarrow\bar u_i(t)$ denotes the same expression but with $u_i(t)$ and $\bar u_i(t)$ interchanged.

This is the complete cocycle data of a non-trivial and non-Abelian gerbe, albeit one which is equivalent to an ordinary principal bundle. The original cocycles are recovered by endpoint evaluation of the lifted cocycles as detailed in (<ref>).

Alternative lift.
The above lift has the disadvantage of being computationally rather involved. With a view towards our later discussion, we therefore also introduce a simplifying lift. To this end, we replace the 3-patch cover $U_1\sqcup U_2\sqcup U_3$ by the 2-patch cover $\bar U_1\sqcup \bar U_2\rightarrow S^4$ with

\begin{equation}\label{eq:2patchCover}
\begin{gathered}
(\bar U_1,\bar\pi_1)
\ewith
\bar U_1\ \coloneqq\ \pi_+^{-1}(\pi_+(U_+)\setminus(L_+\cup L_-))
\eand
\bar\pi_1\ \coloneqq\ \pi_+|_{\bar U_1}
\\
(\bar U_2,\bar\pi_2)
\ewith
\bar U_2\ \coloneqq\ U_-
\eand
\bar\pi_2\ \coloneqq\ \pi_-~.
\end{gathered}
\end{equation}

Note that $\bar U_1\cap\bar U_2\cong S^2\times\IR^2$. With respect to this cover, the bundle $\sfSpin(5)\rightarrow S^4$  is described by the transition function

\begin{equation}
g_{12}\ \coloneqq\ g_{+-}\big|_{(\bar U_1\sqcup\bar U_2)^{[2]}}~.
\end{equation}

The local connection 1-form reads as

\begin{equation}
(\pi_{1,2}^{-1})^*A_{1,2}\ \coloneqq\ \left(\frac{\Im{(q_{1,2}\rmd\bar q_{1,2})}}{1+|q_{1,2}|^2},\frac{\Im{(\bar q_{1,2}\rmd q_{1,2})}}{1+|q_{1,2}|^2}\right),
\end{equation}

with $q_{1,2}\in\bar\pi_{1,2}(\bar U_{1,2})$, where $q_1$ is a restriction of $q_+$ and $q_2=q_-$.

The cocycle (<ref>) can then be straightforwardly lifted, following the prescription in <ref>. Relative to the cover (<ref>), we can define

\begin{equation}
g^\circ_{12}\ \coloneqq\ (u_{q},u_{\bar q})
\eand
g^\circ_{21}\ \coloneqq\ (v_{q},v_{\bar q})~,
\end{equation}

where $ q=q_+ $ and for all $ t \in [0,1] $

\begin{equation}
\begin{aligned}
u_q(t)\ &\coloneqq\ \cos(\theta_q\wp(t))+\frac{\Im(q)}{|\Im(q)|}\sin(\theta_q \wp(t))~,
\\
v_q(t)\ &\coloneqq\ \cos(\theta'_q\wp(t))+\frac{\Im(q)}{|\Im(q)|}\sin(\theta'_q \wp(t))
\end{aligned}
\end{equation}

\begin{equation}\label{eq:cocycles_LSpin_bundle_theta}
\theta_q\ \coloneqq\ \frac{\pi}{2}-\arctan\left(\frac{\Re(q)}{|\Im(q)|}\right) \eand \theta'_q\ \coloneqq\ 2\pi - \theta_q ~.
\end{equation}

This induces

\begin{equation}
\begin{aligned}
h^\circ_{121}\ &\coloneqq\ (g_{12}^\circ g_{21}^\circ)^{-1}
\\
&\kern2.5pt=\ \left(t\mapsto \left(\cos(2\pi \wp(t))+\frac{\Im(\bar q)}{|\Im(q)|}\sin(2\pi \wp(t)),\cos(2\pi \wp(t))+\frac{\Im(q)}{|\Im(q)|}\sin(2\pi \wp(t))\right)\right),
\\
h^\circ_{212}\ &\coloneqq\ (g_{21}^\circ g_{12}^\circ)^{-1}\ =\ h^\circ_{121}~,
\end{aligned}
\end{equation}

where the last equality follows because $g^\circ_{12}$ and $g^\circ_{21}$ commute. We note that the above two expressions only depend on $\frac{\Im(q)}{|\Im(q)|}$, which describes the embeddings $S^2\hookrightarrow\IR^3\hookrightarrow\IR^4$, as expected from the abstract discussion involving reduced suspension mentioned previously.

The lift of the connection reads as

\begin{equation}
\begin{gathered}\label{eq:cocycles_LSpin_bundle_AB2}
A^\circ_1\ \coloneqq\ \wp\cdot A_1~,
\quad
A^\circ_2\ \coloneqq\ \wp\cdot A_2~,
\\
B^\circ_1\ \coloneqq\ \tfrac12(\wp-\wp^2)\cdot[A_1,A_1]~,
\quad
B^\circ_2\ \coloneqq\ \tfrac12(\wp-\wp^2)\cdot[A_2,A_2]~,
\end{gathered}
\end{equation}

and, for all $ t \in [0,1] $,

\begin{equation}
\begin{aligned}
\Lambda_{12}^\circ\ &\coloneqq\ \bigg(t\mapsto\frac{1}{1+|q|^2}\bigg(|q|^2\wp(t)\bar u_q(t)u_q(1)\rmd\big(\bar u_q(1)u_q(t)\big)+|q|^2(1-\wp(t))\bar u_q(t)\rmd u_q(t)
\\
&\kern4.5cm+\bar u_q(t)\rmd u_q(t)-\wp(t)\bar u_q(1)\rmd u_q(1),u_q(t)\leftrightarrow\bar u_q(t)\bigg)\bigg)~.
\end{aligned}
\end{equation}

\begin{equation}
\begin{aligned}
\Lambda_{12}^\circ(t)\ &=\ \frac{1}{1+|q|^2}\left(Q_q(t) \frac{\Im\big(\Im(q)\rmd \Im(\bar q)\big)}{2|\Im(q)|^2},Q_{\bar q}(t)\frac{\Im\big(\Im(\bar q)\rmd \Im(q)\big)}{2|\Im(q)|^2}\right)
\\
&=\ \big(Q_q(t),Q_{\bar q}(t)\big)\frac{\Im\big(\Im(q)\rmd \Im(q)\big)}{2|\Im(q)|^2(1+|q|^2)}~,
\end{aligned}
\end{equation}

\begin{equation}
Q_q(t)\ \coloneqq\ \big(1+|q|^2+\wp(t)(q^2-|q|^2)\big)\bar u_q^2(t)-\left(1+|q|^2-\wp(t)\left(1-\frac{\bar q^2}{|q|^2}\right)\right).
\end{equation}

Similarly, we find

\begin{equation}
\begin{aligned}
\Lambda_{21}^\circ\ &\coloneqq\ \bigg(t\mapsto\frac{1}{1+|q|^2}\bigg(\wp(t)\bar{v}_{q}(t)v_q(1)\rmd\big(\bar{v}_{q}(1)v_q(t)\big)+(1-\wp(t))\bar{v}_{q}(t)\rmd v_q(t)
\\
&\kern4.5cm+|q|^2(\bar{v}_{q}(t)\rmd v_q(t)-\wp(t)\bar{v}_{q}(1)\rmd v_q(1)),~v_q(t)\leftrightarrow\bar{v}_q(t)\bigg)\bigg)
\\
&\kern2.5pt=\ \bigg(t\mapsto\bigg(Q'_q(t),Q'_{\bar q}(t)\bigg)\frac{\Im\big(\Im(q)\rmd\Im(q)\big)}{2|\Im(q)|^2(1+|q|^2)}\bigg)\,,
\end{aligned}
\end{equation}

\begin{equation}
Q'_q(t)\ \coloneqq\ \left(1+|q|^2-\wp(t)\left(1-\frac{\bar q^2}{|q|^2}\right)\right)\bar{v}_{q}^2(t)-\big(1+|q|^2+\wp(t)(q^2-|q|^2)\big)\,.
\end{equation}

The form of the lift of the gauge potentials $A$ and $B$ given in (<ref>) was convenient from the local, infinitesimal perspective, and it leads to the general lifting formulas (<ref>).

We note that using a suitable path space as a cover may lead to much more natural description of the lift of the spin structure on $S^4$. This, however, may seem too esoteric to physicists, and we therefore prefer to work in this slightly more complicated setting.

§ EXAMPLES: STRING BUNDLES WITH CONNECTIONS

We now turn to truly higher principal bundles with connection that are not mere reformulations of ordinary principal bundles with connections. We will focus on principal 2-bundles with structure 2-group a 2-group model of the string group which encode string structures, higher generalisations of spin structures; see <ref> for a concise review. We will then show how to endow these bundles with adjusted connections. As an explicit example, we shall lift the $\sfSpin(4)$-structure on $S^4$ constructed in <ref> to a string structure, and endow the resulting principal 2-bundle with an adjusted connection.

Lifting a generically non-flat spin bundle to a string bundle, we obtain a generically non-fake-flat principal 2-bundle, and hence we have to use adjusted connections. We note that at the local and infinitesimal level, such connections had first been identified in the context of supergravity <cit.> and before the invention of string structures. They were then put into the context of higher gauge theory in <cit.> either at the local level, or in an abstract integrated form, or using an alternative perspective on string structures. The latter arises from the observation that isomorphism classes of string structures with connections on a principal bundle $P$ are in bijection with isomorphism classes of trivialisations of the Chern–Simons 2-gerbe with connection associated to $P$ <cit.>, and this set is a torsor for the Deligne cohomology group $H^3_{\rm D}(X,\IZ)$. For very recent work in this direction, see also <cit.>, where our adjusted connections also emerged in this context.

§.§ Strict Lie 2-group model

As mentioned in <ref>, the string group is defined up to $A_\infty$-equivalence, and various models exist. Here, we focus on the strict Lie 2-group model constructed in <cit.>, see also <cit.> as well as <cit.> and <cit.> for discussions of the involved central extension. Recall that a string 2-group model for a Lie group $\sfG$ in the sense of <cit.> is a Lie 2-group $\caG=(G_1\multirightarrow{2}G_0)$ with a 3-connected cover $\pi:\caG\rightarrow\sfG$ such that the group of isomorphism classes of objects is isomorphic to $\sfG$ and the group of automorphisms of $\unit\in G_0$ is isomorphic to $\sfU(1)$.

String Lie 2-group as 2-group extension.
Consider the based path and loop groups defined in (<ref>). Given the close relationship between string structures and spin structures on loop spaces and the fact that the spin group is a central extension of another group, it is perhaps not surprising that a 2-group model $\sfString(\sfG)$ for a Lie group $\sfG$ can be built as the central extension of 2-groups

\begin{equation}\label{eq:centralExtension2Groups}
\unit\ \longrightarrow\ \sfB\sfU(1)\ \longrightarrow\ \sfString(\sfG)\ \longrightarrow\ \caL\sfG\ \longrightarrow\ \unit~,
\end{equation}

where $\sfB\sfU(1)$ is the crossed module $(\sfU(1)\longrightarrow*,\id)$ and the 2-group $\caL\sfG$ is defined in (<ref>). Let us now explain the construction of $\sfString(\sfG)$ in detail, reviewing and slightly expanding the discussion in <cit.>, which, at the time of writing, contained some sign errors.

Kac–Moody extension as principal circle bundle.
Our starting point is the Kac–Moody central extension[See <ref> for a brief review of central extensions.]

\begin{equation}\label{eq:Kac-Moody-group-extension}
\unit\ \longrightarrow\ \sfU(1)\ \overset{\iota}{\longrightarrow}\ \widehat{L_0\sfG}\ \overset{\pi}{\longrightarrow}\ L_0\sfG\ \longrightarrow\ \unit
\end{equation}

of $L_0\sfG$. To construct a group product on $\widehat{L_0\sfG}$, we follow <cit.>, see also <cit.>. In particular, it is convenient to regard the trivialisation of the principal $\sfU(1)$-bundle $\widehat{L_0\sfG}$ over the path space[All our definitions of based, parametrised path and loop spaces are the same as in <ref>.] $P_0L_0\sfG$. Here, we identify $L_0G\cong P_0L_0\sfG/L_0L_0\sfG$, generalising (<ref>), and $\widehat{L_0\sfG}\cong \big(P_0L_0\sfG\times\sfU(1)\big)/\sfN$, with $N$ defined in the following proposition:

The subset $\sfN\subseteq L_0L_0\sfG\times\sfU(1)$ with

\begin{equation}\label{eq:subgroupN}
\sfN\ \coloneqq\ \left\{(f,z)\in L_0L_0\sfG\times\sfU(1)\,\middle|\,z=\sfhol^{-1}(f)=\exp\left(\int_{D_f}\omega\right)\right\},
\end{equation}

where $D_f$ is a disc in $L_0\sfG$ with boundary $f$, is a normal subgroup of $P_0L_0\sfG\times\sfU(1)$.

The subgroup property follows because of

\begin{equation}
\left(f_1,\exp\left(\int_{D_{f_1}}\omega\right)\right)\left(f_2,\exp\left(\int_{D_{f_2}}\omega\right)\right)\ =\ \left(f_1f_2,\exp\left(\int_{D_{f_1f_2}}\omega\right)\right)
\end{equation}

due to

\begin{equation}
c(f_1,f_2)\ =\ \exp\left(-\int_{D_{f_1}}\omega-\int_{D_{f_2}}\omega+\int_{D_{f_1f_2}}\omega\right),
\end{equation}

which, in turn, is a direct consequence of the definition (<ref>) specialised to loops $f_{1,2}\in L_0L_0\sfG$.

Moreover, the normality of $\sfN$ follows because of

\begin{equation}
\begin{aligned}
(f_1,z_1)(f_2,\sfhol^{-1}(f_2))(f_1,z_1)^{-1}\ &=\ (f_1f_2 f_1^{-1},\sfhol^{-1}(f_2)\,c(f_1,f_2)\,c(f_1f_2,f_1^{-1})\,c^{-1}(f_1,f_1^{-1}))
\\
&=\ (f_1f_2 f_1^{-1},\sfhol^{-1}(f_1f_2 f_1^{-1}))
\end{aligned}
\end{equation}

for all $f_1\in P_0L_0\sfG$, for all $f_2\in L_0L_0\sfG$, and for all $z_1\in\sfU(1)$. Here, the last equality follows from combining the holonomy $\sfhol^{-1}(f_2)$ with the holonomies in the cocycles, comparing the boundaries and invoking Stokes' theorem, and using (<ref>) once.

Altogether, we obtain the following commutative diagram

\begin{equation}\label{eq:KacMoodyExtensionLoopGroup}
\begin{tikzcd}
P_0L_0\sfG\times\sfU(1)\arrow[r,"\hat \boundary"] & \boundary^*\widehat{L_0\sfG}\arrow[r]\arrow[d] & \big(P_0L_0\sfG\times\sfU(1)\big)/\sfN\arrow[d]\arrow[r,"\cong"] & \widehat{L_0\sfG}\arrow[d,"\pi"]
\\
& P_0L_0\sfG\arrow[r]\arrow[rr,bend right=20,"\boundary"] & P_0L_0\sfG/L_0L_0\sfG\arrow[r,"\cong"] & L_0\sfG
\end{tikzcd}
\end{equation}

where the pullback bundle $\boundary^*\widehat{L_0\sfG}$ along the endpoint evaluation map $\boundary$ is given by the fibre product

\begin{equation}
\boundary^*\widehat{L_0\sfG}\ \coloneqq \ P_0L_0\sfG~\times_{L_0\sfG}\widehat{L_0\sfG}~,
\end{equation}

and the isomorphism $\hat \boundary$ reads as

\begin{equation}
\hat \boundary (f,z)\ \coloneqq\ (f,\hat f(1)z)
\end{equation}

for all $(f,z)\in P_0L_0\sfG\times\sfU(1)$, where $\hat f$ is the horizontal lift[with respect to the connection constructed in <ref>] of $f\in P_0L_0\sfG$ to $\widehat{L_0\sfG}$ with $\hat f(0)=\unit\in \widehat{L_0\sfG}$.

Multiplication on $\widehat{L_0\sfG}$.
In order to construct the multiplication on $\widehat{L_0\sfG}$, we require a suitable group cocycle $c$, cf. <ref>. Here, $c$ is a map

\begin{equation}\label{eq:group_cocycle}
c\,:\,P_0L_0\sfG\times P_0L_0\sfG\ \rightarrow\ \sfU(1)
\ewith
c(f_1,f_2)c(f_1f_2,f_3)\ =\ c(f_1,f_2f_3)c(f_2,f_3)
\end{equation}

for all $f_{1,2,3}\in P_0L_0\sfG$, which results in the associative, unital product

\begin{equation}\label{eq:productOnP0L0G}
(f_1,z_1)(f_2,z_2)\ \coloneqq\ \big(f_1f_2,z_1z_2c(f_1,f_2)\big)
\end{equation}

for all $(f_{1,2},z_{1,2})\in P_0L_0\sfG\times\sfU(1)$. We would like the group cocycle $c$ to arise as the integrated form of the Kac–Moody 2-cocycle on the Lie algebra $L_0\frg$ with $\frg$ the Lie algebra of $\sfG$.

\begin{equation}\label{eq:Lie_algebra_cocycle}
\begin{aligned}
\omega\,:\,L_0\frg\times L_0\frg\ &\rightarrow\ \fru(1)~,
\\
(\beta_1,\beta_2)\ &\mapsto\ \frac{\rmi}{2\pi}\int_0^1 \rmd r\,\innerLarge{\beta_1(r)}{\parder[\beta_2(r)]{r}}.
\end{aligned}
\end{equation}

Here, $\inner{-}{-}$ is the inner product on $\frg$ which is normalised in a standard fashion <cit.>.[Explicitly, $\inner{h_\alpha}{h_\alpha}=2$ when $h_\alpha$ is the co-root corresponding to the highest root. In particular, we have $\inner{U}{V}=-\tr(UV)$ for $\fru(n)$ with anti-Hermitian generators and $\inner{U}{V}=-\tfrac12\tr(UV)$ for $\frso(2n)$ with antisymmetric generators.] Note that the cocycle (<ref>) also gives rise to the standard expression for the left-invariant 2-form curvature $\omega\in\Omega^2(L_0\sfG,\fru(1))$ of the bundle $\widehat{L_0\sfG}\rightarrow L_0\sfG$ <cit.>,

\begin{equation}
\omega_g(g\beta_1,g\beta_2)\ \coloneqq\ \omega(\beta_1,\beta_2)
\end{equation}

for all $g\in L_0\sfG$ and for all $g\beta_{1,2}\in T_gL_0\sfG$, or

\begin{equation}\label{eq:2FormCurvatureKacMoody}
\omega_g \ =\ \frac{\rmi}{4\pi}\int_0^1\rmd r\,\innerLarge{\theta_{g(r)}}{\parder[\theta_{g(r)}]{r}},
\end{equation}

where $\theta$ is the left-invariant Maurer–Cartan form on $L_0\sfG$. This is a closed 2-form. The integrated cocycle is then obtained from the holonomy of $\omega$. Explicitly, given paths $f_{1,2}\in P_0L_0\sfG$, we construct the loop given by the triangle

\begin{equation}\label{eq:triangle}
\ell(f_1,f_2)\ \coloneqq\ \overline{f_1f_2}\circ f_1(1)f_2\circ f_1\ \in\ L_0L_0\sfG~,
\end{equation}

where $\overline{f}$ denotes the path $f\in P_0L_0\sfG$ with reversed orientation, that is, $\overline{f}(t,r)\coloneqq f(1-t,r)$. Then, we have the following result.

Let $f_{1,2}\in P_0L_0\sfG$ and $D_{\ell(f_1,f_2)}$ be an arbitrary disc in $L_0\sfG$ with boundary $\ell(f_1,f_2)$ given by (<ref>). Then,

\begin{equation}\label{eq:cocyclePLGU}
\begin{aligned}
c(f_1,f_2)\ &\coloneqq\ \sfhol(\ell(f_1,f_2))
\\
&\coloneqq\ \exp\left(-\int_{D_{\ell(f_1,f_2)}}\omega\right)
\\
&\,=\ \exp\left(-\frac{\rmi}{2\pi}\int_0^1\rmd r\int_0^1\rmd s\int_0^s \rmd t\,\innerLarge{f_1^{-1}\parder[f_1]{s}}{f_2\left\{\parder{r}\left(f_2^{-1}\parder[f_2]{t}\right)\right\}f_2^{-1}}\right)
\end{aligned}
\end{equation}

for all $f_{1,2}\in P_0L_0\sfG$ which we parametrised as $f_{1,2}=f_{1,2}(s,r)$ with $s,t$ the path parameters and $r$ the loop parameter.

Note that using the identity

\begin{equation}\label{eq:helpful_identity}
\parder{t}\left(\parder[f_2(t,r)]{r}f_2^{-1}(t,r)\right)\ =\ f_2(t,r)\left\{\parder{r}\left(f_2^{-1}(t,r)\parder[f_2(t,r)]{t}\right)\right\}f_2^{-1}(t,r)~,
\end{equation}

we can simplify the expression (<ref>) to <cit.>

\begin{equation}\label{eq:cocyclePLGUBaez}
c(f_1,f_2)\ =\ \exp\left(-\frac{\rmi}{2\pi}\int_0^1\rmd r\int_0^1\rmd s\,\innerLarge{f_1^{-1}(s,r)\parder[f_1(s,r)]{s}}{\parder[f_2(s,r)]{r}f_2^{-1}(s,r)}\right),
\end{equation}

where, again, $s$ and $r$ are the path and loop parameters, respectively. The group cocycle condition (<ref>) for $c$ is then straightforwardly verified using the form (<ref>).

We note that the left-invariance of the curvature $\omega$ implies that

\begin{equation}\label{eq:holonomy_shift}
\sfhol(\ell(f_1,f_2))\ =\ \sfhol(g\ell(f_1,f_2))
\end{equation}

for all $g\in L_0\sfG$. The cocycle condition (<ref>) then amounts to the integral of the curvature $\omega$ over a tetrahedron vanishing, and the sides of this tetrahedron are given by the triangles

\begin{equation}
\begin{aligned}
\\
\end{aligned}
\end{equation}

cf. <cit.>.

Altogether, by means of (<ref>) and (<ref>), we have constructed a group product on $P_0L_0\sfG\times\sfU(1)$ and thus on $\big(P_0L_0\sfG\times\sfU(1)\big)/\sfN$. Furthermore, we obtain a group homomorphism

\begin{equation}
\sft\,:\,P_0L_0\sfG\times\sfU(1)\big)/\sfN\ \rightarrow\ P_0\sfG
\end{equation}

via the composition of the evident projection $\big(P_0L_0\sfG\times\sfU(1)\big)/\sfN\rightarrow P_0L_0\sfG/L_0L_0\sfG\rightarrow L_0\sfG$ with the embedding $L_0\sfG\hookrightarrow P_0\sfG$. Explicitly, for all $[(f,z)]\in\big(P_0L_0\sfG\times\sfU(1)\big)/\sfN$ we have

\begin{equation}
\sft\big([(f,z)]\big)\ =\ \boundary(f)\ \in\ L_0\sfG~,
\end{equation}

which is indeed well-defined.

Different group cocycles. Let us briefly leave the main thread of our presentation in order to answer a question raised in <cit.>.

The starting point of our discussion around (<ref>) is the group cocycle[Note that in the following, we fix some sign errors appearing in the original literature.]

\begin{equation}
c(f_1,f_2)\ =\ \exp\left(-\frac{\rmi}{2\pi}\int_0^1\rmd r\int_0^1\rmd s\int_0^s \rmd t\,\innerLarge{f_1^{-1}\parder[f_1]{s}}{f_2\left\{\parder{r}\left(f_2^{-1}\parder[f_2]{t}\right)\right\}f_2^{-1}}\right)
\end{equation}

given by Murray <cit.> for all $f_{1,2}\in P_0L_0\sfG$ parametrised as $f_1=f_1(s,r)$ and $f_2=f_2(t,r)$ with $s,t$ the path parameters and $r$ the loop parameter. Furthermore, with the help of the identity (<ref>), Murray's group cocycle becomes

\begin{equation}\label{eq:MurrayBaez}
c(f_1,f_2)\ =\ \exp\left(-\frac{\rmi}{2\pi}\int_0^1\rmd r\int_0^1\rmd s\,\innerLarge{f_1^{-1}(s,r)\parder[f_1(s,r)]{s}}{\parder[f_2(s,r)]{r}f_2^{-1}(s,r)}\right),
\end{equation}

and this is the form of the group cocycle given in Baez–Stevenson–Crans–Schreiber <cit.>.

Next, Mickelsson provides another group cocycle given by <cit.>

\begin{equation}
\begin{aligned}
\tilde c(f_1,f_2)\ &\coloneqq\ \exp\left(-\frac{\rmi}{4\pi}\int_0^1\rmd r\int_0^1\rmd s\left\{\innerLarge{f_1^{-1}(s,r)\parder[f_1(s,r)]{s}}{\parder[f_2(s,r)]{r}f_2^{-1}(s,r)}\right.\right.
\\
\end{aligned}
\end{equation}

The group cocycles $c$ and $\tilde c$ are related by a coboundary transformation

\begin{equation}\label{eq:coboundaryDifferentGroupCocycles}
\tilde c(f_1,f_2)\ =\ c(f_1,f_2)(d(f_1f_2))^{-1}d(f_1)d(f_2)~,
\end{equation}

cf. (<ref>), with

\begin{equation}
\begin{aligned}
d\,:\,P_0L_0\sfG\ &\rightarrow\ \sfU(1)~,
\\
f\ &\mapsto\ \exp\left(-\frac{\rmi}{4\pi}\int_{0}^{1}\rmd r\int_0^1\rmd s\,\innerLarge{f^{-1}(s,r)\parder[f(s,r)]{s}}{f^{-1}(s,r)\parder[f(s,r)]{r}}\right).
\end{aligned}
\end{equation}

This answers the question raised in <cit.> as to what the relation between $c$ and $\tilde c$ is.

Lift of the action in $\caL\sfG$.
Returning to the main discussion, we finally have to lift the action of $P_0\sfG$ on $L_0\sfG$ in $\caL\sfG$ to an automorphism action of $P_0\sfG$ on $P_0L_0\sfG\times\sfU(1)\big)/\sfN$. Consider the left-invariant 1-form $\xi_g\in\Omega^1(L_0\sfG,\fru(1))$ given by (cf. <cit.> and <cit.>)

\begin{equation}\label{eq:Baez1Form}
(\xi_g)_\ell\ \coloneqq\ \frac{\rmi}{2\pi}\int_0^{1}\rmd r\,\innerLarge{\theta_{\ell(r)}}{g(r)^{-1}\parder[g(r)]{r}}
\end{equation}

for all $g\in P_0\sfG$ and for all $\ell\in L_0\sfG$. We have $(\Ad_g)^*\omega=\omega+\rmd \xi_g$, see <ref>, which implies that for loops $h\in L_0L_0\sfG$,

\begin{equation}\label{eq:simp_loops}
\exp\left(-\int_0^1\rmd s\,\xi_g\left(h^{-1}(s)\parder[h(s)]{s}\right)\right)\ =\ \sfhol(ghg^{-1})\sfhol^{-1}(h)~.
\end{equation}

Here, $h(s)\in L_0\sfG$ for all $s\in[0,1]$.

The group action <cit.>

\begin{equation}\label{eq:actionOfP0G}
\begin{aligned}
g\acton(f,z)\ &\coloneqq\ \left(gfg^{-1},z\exp\left(\int_0^1\rmd s\,\xi_g\left(f^{-1}(s)\parder[f(s)]{s}\right)\right)\right)
\\
&\,=\ \left(gfg^{-1},z\exp\left(\frac{\rmi}{2\pi}\int_0^1\rmd r\int_0^1\rmd s\,\innerLarge{f^{-1}(s,r)\parder[f(s,r)]{s}}{g^{-1}(r)\parder[g(r)]{r}}\right)\right)
\end{aligned}
\end{equation}

for all $g\in P_0\sfG$ and for all $(f,z)\in P_0L_0\sfG\times\sfU(1)$ is compatible with the group product on $P_0L_0\sfG\times\sfU(1)$. It also closes on the subgroup $\sfN$ defined in (<ref>) and so, it descents to an action of $P_0\sfG$ on $\big(P_0L_0\sfG\times\sfU(1)\big)/\sfN$. The group homomorphism (<ref>) and the automorphism action (<ref>) satisfy the crossed module conditions (<ref>).

Compatibility with the group product follows from a direct calculation using the explicit expressions (<ref>) and (<ref>) of the group cocycle $c$ and of $\xi_g$, respectively. Furthermore, with (<ref>), it is also clear that the action closes on $\sfN$.

We also note that the action trivially satisfies $\sft(g\acton\hat h)=g\sft(\hat h)g^{-1}$ for all $g\in P_0\sfG$ and for all $\hat h\in P_0L_0\sfG\times\sfU(1)$. It only remains to show that $(\sft(\hat h_1)\acton\hat h_2)\hat h_1\hat h_2^{-1}\hat h_1^{-1}\in\sfN$ for all $\hat h_{1,2}\in P_0L_0\sfG\times\sfU(1)$ as this implies the Peiffer identity. The proof can be found in <ref>.

String 2-group model. In conclusion, our above discussion now yields the following definition.

The string 2-group model $\sfString(\sfG)$ is the crossed module of Lie groups

\begin{equation}\label{eq:string2GroupDefinition}
\sfString(\sfG)\ \coloneqq\ \big(\underbrace{\big(P_0L_0\sfG\times\sfU(1)\big)/\sfN}_{\cong\,\widehat{L_0\sfG}}\overset{\sft}{\longrightarrow}P_0\sfG,\acton\big)~.
\end{equation}

We shall also write $\sfString(n)$ as a shorthand notation for $\sfString(\sfSpin(n))$.

String Lie 2-algebra model.
In order to derive the corresponding crossed module of Lie algebras, we identify the Lie algebra of $P_0L_0\sfG\times\sfU(1)$ with $P_0L_0\frg\oplus\fru(1)$. The Lie bracket is obtained in the usual way from the group commutator using (<ref>),

\begin{equation}\label{eq:P0L0G+R_commutator}
\begin{aligned}
[(\gamma_1,q_1),(\gamma_2,q_2)]\ &=\ \bigg([\gamma_1,\gamma_2],-\frac{\rmi}{2\pi}\int_0^1\rmd r\int_0^1\rmd s\,\bigg\{\innerLarge{\parder[\gamma_1(s,r)]{s}}{\parder[\gamma_2(s,r)]{r}}
\\
&\kern6cm -\innerLarge{\parder[\gamma_2(s,r)]{s}}{\parder[\gamma_1(s,r)]{r}}\bigg\}\bigg)
\\
&=\ \left([\gamma_1,\gamma_2],-\frac{\rmi}{2\pi}\int_0^1\rmd    r\,\innerLarge{\boundary(\gamma_1(s,r))}{\parder[\,\boundary(\gamma_2(s,r))]{r}}\right),
\end{aligned}
\end{equation}

where we have integrated by parts in the second line. Note that $\boundary(\gamma_{1,2}(s,r))=\gamma_{1,2}(1,r)$. The extra minus sign in the above formula looks a bit odd, comparing it with the expression for the Lie algebra cocycle (<ref>), but it is expected on general grounds. This follows as the difference between the horizontal lift of a commutator of vector fields and the commutator of horizontal lifts of vector fields is given by the curvature.[More explicitly, given a $\frg$-valued connection 1-form $\mu$ on a $\sfG$-bundle whose curvature is $\omega$, we have for any vector fields $V_{1,2}$ on the base space, the formula $\omega(V_1^h,V_2^h)=-\mu([V_1^h,V_2^h])$, where ${}^h$ denotes the horizontal lift. Upon identifying the Lie algebra $\frg$ with the space of vertical vector fields it generates, we can write this as $\omega(V_1^h,V_2^h)=[V_1,V_2]^h-[V_1^h,V_2^h]$, where we have used that $[V_1^h,V_2^h]_H=[V_1,V_2]^h$, where ${}_H$ denotes the horizontal part of a vector (field). For the case of a central extension $\hat\sfK$ of the group $\sfK$ by $\sfG=\sfU(1)$, this formula at the identity gives the Lie algebra commutator $[(V_1,0),(V_2,0)]=([V_1,V_2],0)-(0,\omega(V_1^h,V_2^h))=([V_1,V_2],-\omega(V_1,V_2))$, where we have used the connection to split the Lie algebra $\hat\frk=\frk\oplus\fru(1)$ and construct the horizontal lifts. We emphasise this point here, as it had been the source of sign inconsistencies plaguing our calculations.]

Furthermore, because of

\begin{equation}\label{eq:derhol}
\dder{t}\bigg|_0\sfhol(\exp(t\gamma))\ =\ 0
\end{equation}

for all $\gamma\in L_0L_0\frg$, which we verify in <ref>, the Lie algebra $\frn$ of $\sfN$ is just

\begin{equation}
\frn\ =\ \{(\gamma,0)\in L_0L_0\frg\oplus\fru(1)\}\ \cong\ L_0L_0\frg~.
\end{equation}

From (<ref>) we immediately see that $ \frn $ is indeed an ideal. Consequently,

\begin{equation}
(P_0L_0\frg\oplus\fru(1))/\frn\ \cong\ L_0\frg\oplus\fru(1)~.
\end{equation}

In particular, since the second component in the Lie algebra commutator depends only on the endpoint, we recover the Kac–Moody 2-cocycle (<ref>) in the factor algebra. In addition, the linearisation of the action (<ref>) is

\begin{equation}
\alpha\,\acton\,(\gamma,q)\ =\ \left([\alpha,\gamma],\frac{\rmi}{2\pi}\int_0^1\rmd r\,\innerLarge{\parder[\alpha]{r}}{\gamma}\right)
\end{equation}

for all $\alpha\in P_0\frg$ and for all $(\gamma,q)\in L_0\frg\oplus\fru(1)$. In summary, we thus have obtained the crossed module of Lie algebras <cit.>

\begin{equation}\label{eq:strict_string_Lie_2_algebra}
\frstring(\frg)\ \coloneqq\ \big(L_0\frg\oplus\fru(1)\overset{\sft}{\longrightarrow}P_0\frg,\acton\big)
\end{equation}

as a strict model of the string Lie 2-algebra. Here, $\sft$ is just the projection onto $L_0 \frg$ followed by the embedding into $P_0\frg$.

Minimal model.
We recall that a minimal model of the string Lie 2-algebra is given by the 2-term $L_\infty$-algebra

\begin{equation}
\frstring^\circ(\frg)\ \coloneqq\ \fru(1)\oplus\frg~,
\end{equation}

where $\frstring^\circ(\frg)_{-1}\coloneqq\fru(1)$ and $\frstring^\circ(\frg)_0\coloneqq\frg$ with the non-trivial higher products

\begin{equation}
\begin{aligned}
\mu_2\,:\,\frg\times\frg\ &\rightarrow\ \frg~~,
\\
(V_1,V_2)\ &\mapsto\ [V_1,V_2]~,
\\
\mu_3\,:\,\frg\times\frg\times\frg\ &\rightarrow\ \fru(1)~,
\\
(V_1,V_2,V_3)\ &\mapsto\ \rmi\inner{V_1}{[V_2,V_3]}
\end{aligned}
\end{equation}

for all $V_{1,2,3}\in\frg$. We can now extend the quasi-isomorphism (<ref>) between the Lie algebra $\frg$, trivially regarded as a Lie 2-algebra, and the Lie 2-algebra $\caL\frg$ to a quasi-isomorphism

\begin{equation}
\frstring^\circ(\frg)\ \xrightarrow{~\phi~}\ \frstring(\frg)\ \xrightarrow{~\psi~}\ \frstring^\circ(\frg)~.
\end{equation}

Explicitly, we have the chain maps

\begin{equation}
\begin{gathered}
\begin{tikzcd}
\fru(1)\arrow[d,"0"]\arrow[r,"\phi_1"] & L_0\frg\oplus\fru(1)\arrow[d,"\sft"]\arrow[r,"\psi_1"] & \fru(1)\arrow[d,"0"]
\\
\frg \arrow[r,"\phi_1\coloneqq\cdot_\wp"] & P_0\frg\arrow[r,"\psi_1\coloneqq\boundary"]  & \frg
\end{tikzcd}
\end{gathered}
\end{equation}

where in the top row $\phi_1$ and $\psi_1$ are the evident embedding and projection maps. In this case, both $\phi_2$ and $\psi_2$ are non-trivial,

\begin{equation}
\begin{aligned}
\phi_2\,:\,\frg\times\frg\ &\rightarrow\ L_0\frg~,
\\
(V_1,V_2)\ &\mapsto\ (\wp-\wp^2)\cdot[V_1,V_2]
\end{aligned}
\end{equation}

for all $V_{1,2}\in\frg$ and

\begin{equation}
\begin{aligned}
\psi_2\,:\,P_0\frg\times P_0\frg\ &\rightarrow\ \fru(1)~,
\\
(V_1,V_2)\ &\mapsto\ -\rmi\int_0^1\rmd r\left(\innerLarge{\parder[V_1]{r}}{V_2}-\innerLarge{V_1}{\parder[V_2]{r}}\right)
\end{aligned}
\end{equation}

for all $V_{1,2}\in P_0\frg$.

We notice that the string 2-group model crucially depends on the correct normalisation of the defining cocycle. Usually, $\sfString(\sfG)$ is therefore defined only for simply-connected compact simple Lie groups $\sfG$. The normalisation of the cocycle translates to the correct Lie 2-group extension of $\caL\sfG$ by $\sfB\sfU(1)$, the Lie 2-group corresponding to the crossed module $(\sfU(1)\rightarrow *,\sfid)$. These extensions are characterised by $H^3(\sfG,\IZ)$, and we have $H^3(\sfG,\IZ)\cong \IZ$ for simply-connected compact simple Lie groups with preferred element the generator $+1$. This limits us to $\sfSpin(n)$ for $n=3$ and $n\geq 5$ for $\sfString(n)$. As we shall see, however, there are also two equivalent, preferred choices for $n=4$, where $H^3(\sfSpin(4),\IZ)\cong \IZ\times \IZ$, which are the generators $(\pm 1,\mp 1)$ of opposite signs, so that we can speak of $\sfString(4)$.

§.§ Adjusted descent data

Let us now discuss adjustment for the descent data based on $\sfString(\sfG)$ for a Lie group $\sfG$. We note that the local kinematical data for higher gauge theory with gauge Lie 2-algebra $\frstring(\frg)$ was already given in <cit.>.

Adjustment of $\sfString(\sfG)$.
Recall the adjustment $\kappa$ defined in (<ref>). We can now lift this adjustment datum from $\caL\sfG$ to $\sfString(\sfG)$ as follows.

The map

\begin{equation}\label{eq:adjustmentStringGroup}
\begin{aligned}
\hat\kappa\,:\,P_0\sfG\times P_0\frg\ &\rightarrow\ L_0\frg\oplus\fru(1)~,
\\
(g,V)\ &\mapsto\ \left(\kappa(g,V),\frac{\rmi}{2\pi}\int_0^1\rmd r\,\innerLarge{g^{-1}\parder[g]{r}}{V}\right).
\end{aligned}
\end{equation}

for all $g\in P\sfG$ and for all $V\in P_0\frg$ defines an adjustment on $\sfString(\sfG)$. It also induces an adjustment at the level of crossed modules of Lie algebras with

\begin{equation}\label{eq:adjustmentStringAlgebra}
\begin{aligned}
\hat\kappa\,:\,P_0\frg\times P_0\frg\ &\rightarrow L_0\frg\oplus\fru(1)~,
\\
(V_1,V_2)\ &\mapsto\ \left(\kappa(V_1,V_2),\frac{\rmi}{2\pi}\int_0^1\rmd r\,\innerLarge{\parder[V_1]{r}}{V_2}\right)
\end{aligned}
\end{equation}

for all $V_{1,2}\in P_0\frg$ where $\kappa(V_1,V_2)$ is the adjustment datum defined in (<ref>). This infinitesimal adjustment is the one derived in <cit.>.

This follows from using the results from <ref>. In particular, (<ref>) directly follows from (<ref>). Moreover, using (<ref>) and (<ref>), we ultimately obtain (<ref>). The infinitesimal adjustment is then derived by a short computation.

Let us now briefly list the adjusted descent data from <ref>, specialised to structure 2-group $\sfString(\sfG)$ with adjustment (<ref>), and simplified. Such descent data consists of maps

\begin{equation}
\begin{aligned}
\hat h=[(h,h')]\ &\in\ \scC^\infty\Big(Y^{[3]},\big(P_0L_0\sfG\times\sfU(1)\big)/\sfN\Big)~,
\\
(g,\hat\Lambda=(\Lambda,\Lambda'))\ &\in\ \scC^\infty(Y^{[2]},P_0\sfG)~\oplus~\Omega^1(Y^{[2]},L_0\frg\oplus\fru(1))~,
\\
(A,\hat B=(B,B'))\ &\in\ \Omega^1(Y^{[1]},P_0\frg)~\oplus~\Omega^2(Y^{[1]},L_0\frg\oplus\fru(1))~,
\end{aligned}
\end{equation}

such that for all appropriate $(i,j,\ldots)\in Y^{[n]}$, with $Y^{[n]}$ given in (<ref>), we have

\begin{equation}
\begin{gathered}
g_{ik}\ =\ \sft(h_{ijk})g_{ij}g_{jk}~,
\\
(h_{ikl}h_{ijk})^{-1}h_{ijl}(g_{ij}\acton h_{jkl})\ \in\ L_0L_0\sfG~,
\\
h'_{ikl}h'_{ijk}\ =\ h'_{ijl}h'_{jkl} \exp\left(\frac{\rmi}{2\pi}\int_0^1\rmd s\int_0^1\rmd r\,\innerLarge{g_{ij}^{-1}\parder[g_{ij}]{r}}{h_{jkl}^{-1}\parder[h_{jkl}]{s}}\right)\exp\left(-\int_\Box\omega\right),
\end{gathered}
\end{equation}

where $\Box$ denotes the loop $\overline{h_{ikl}}\circ\big(\boundary(h_{ikl})\overline{h_{ijk}}\big)\circ\big(\boundary(h_{ijl})(g_{ij}\acton h_{jkl})\big)\circ h_{ijl}$. The cocycle conditions for the differential refinement specialise to

\begin{equation}
\begin{aligned}
\Lambda_{ik}\ &=\ \Lambda_{jk}+g_{jk}^{-1}\acton\Lambda_{ij}-g_{ik}^{-1}\acton\big(\boundary(h_{ijk})\nabla_i\boundary(h_{ijk}^{-1})\big)\,,
\\
\Lambda'_{ik}\ &=\ \Lambda'_{jk}+\Lambda'_{ij}-h'_{ijk}\rmd h'^{-1}_{ijk}
\\
&\kern1cm-\frac{\rmi}{2\pi}\int_0^1\rmd r\,\left\{\innerLarge{\parder[g_{jk}]{r}g_{jk}^{-1}}{\Lambda_{ij}}-\innerLarge{\parder[g_{ik}]{r}g_{ik}^{-1}}{\boundary(h_{ijk})\nabla_i\boundary(h_{ijk}^{-1})}\right.
\\
&\kern2cm+\left.\innerLarge{\boundary(h_{ijk}^{-1})\parder[\,\boundary(h_{ijk})]{r}}{A_i}+\int_{0}^{1}\rmd s\,\innerLarge{\parder[h_{ijk}]{r}h_{ijk}^{-1}}{\parder[(h_{ijk}\rmd h_{ijk}^{-1})]{s}}\right\},\\
A_j\ &=\ g^{-1}_{ij}A_ig_{ij}+g^{-1}_{ij}\rmd g_{ij}- \Lambda_{ij}~,\\
B_j\ &=\ B_i+\rmd\Lambda_{ij}+A_j\acton \Lambda_{ij}+\tfrac12[\Lambda_{ij},\Lambda_{ij}]-\kappa(g^{-1}_{ij},F_i-B_i)~,\\
B'_j\ &=\ B'_i+\rmd\Lambda'_{ij} +\frac{\rmi}{2\pi}\int_0^1\rmd r\left\{\innerLarge{\parder[A_j]{r}}{\Lambda_{ij}}+\frac12\innerLarge{\parder[\Lambda_{ij}]{r}}{\Lambda_{ij}}+\innerLarge{\parder[g_{ij}]{r}g_{ij}^{-1}}{F_i-B_i}\right\}.
\end{aligned}
\end{equation}

Furthermore, using the linearisation (<ref>), the adjusted curvatures (<ref>) reduce to

\begin{equation}
\begin{aligned}
F_i\ &=\ \rmd A_i+\tfrac12[A_i,A_i]+\sft(\hat B_i)~,
\\
H_i\ &=\ \rmd B_i+A_i\acton B_i-\kappa(A_i,F_i)\ =\ (\id-\wp\cdot\boundary)(\rmd F_i)~,
\\
H'_i\ &=\ \rmd B'_i-\frac{\rmi}{2\pi}\int_0^1\rmd r\,\innerLarge{\parder[A_i]{r}}{F_i-\sft(B_i)}\ =\ \rmd\big(B'_i-\tfrac12\omega(A_i,A_i)\big)-\tfrac{\rmi}{4\pi}{\rm cs}(\boundary(A_i))
\end{aligned}
\end{equation}

for all $(i)\in Y^{[1]}$. Here, in analogy to writing $\hat B=(B,B')$ we did write for the 3-form curvature $\hat H=(H,H')$, and we have made us of the Chern–Simons 3-form

\begin{equation}\label{eq:chern-simons}
{\rm cs}(A_i)\ \coloneqq\ \inner{A_i}{\rmd A_i}+\tfrac13\inner{A_i}{[A_i,A_i]}~
\end{equation}

for all $(i)\in Y^{[1]}$.  The adjusted Bianchi identities (<ref>) thus become

\begin{equation}\label{eq:Bianchi_identities_extended}
\begin{aligned}
\nabla_iF_i\ &=\ \sft(H_i+\kappa(A_i,F_i))~,
\\
\rmd H_i\ &=\ 0~,
\\
\rmd H'_i\ &=\ -\tfrac{\rmi}{4\pi}\inner{\boundary(F_i-\sft(B_i))}{\boundary(F_i-\sft(B_i))}\ =\ -\tfrac{\rmi}{4\pi}\inner{\boundary(F_i)}{\boundary(F_i)}
\end{aligned}
\end{equation}

for all $(i)\in Y^{[1]}$.

§.§ The string 2-bundle over S4

Having a lift of the spin group $\sfSpin(n)$ to a 2-group $\sfString(n)$, it is natural to ask, if there is a lift of

\begin{equation}
\sfSpin(5)\ \rightarrow\ \sfSpin(5)/\sfSpin(4)~~~\mbox{to}~~~\sfString(5)\ \rightarrow\ \sfString(5)/\sfString(4)~.
\end{equation}

As we will show, such a lift exists, and we first learnt about this construction, albeit without connection, from <cit.>. It turns out that both quotient spaces can be identified with $S^4$, the latter, with the manifold $S^4$ trivially regarded as the discrete Lie groupoid $(S^4\multirightarrow{2}S^4)$, and we begin by commenting on this point.

Quotients of crossed modules.
Recall that the categories of crossed modules of Lie groups and of strict Lie 2-groups are equivalent, cf. <cit.>. Explicitly, given a crossed module of Lie groups $\caG=(\sfH\overset{\sft}{\longrightarrow}\sfG,\acton)$, we associated to $\caG$ a strict Lie 2-group, denoted by $\grp{\caG}$ and given by the data

\begin{equation}\label{eq:crossedModuleGroupoid}
\begin{gathered}
\grp{\caG}\ \coloneqq\ (\sfG\ltimes\sfH\multirightarrow{2}\sfG)
\ewith
\kern-.8cm
\begin{tikzcd}[column sep=2.0cm,row sep=large]
\phantom{\sft(h)}g & \sft(h^{-1})\,g\arrow[l,bend left,swap,out=-20,in=200]{}{(g,h)}
\end{tikzcd}\!,
\\
(g_1,h_1)\circ(\sft(h^{-1}_1)g_1,h_2)\ =\ (g_1,h_1h_2)~,
\\
(g_1,h_1)\otimes (g_2,h_2)\ \coloneqq\ (g_1g_2,(g_1\acton h_2)h_1)
\end{gathered}
\end{equation}

for all $g,g_{1,2}\in\sfG$ and for all $h,h_{1,2}\in\sfH$. Note that our conventions here are those of <cit.>, which are slightly different from <cit.>.

By an embedding of a crossed module $\caG_1\coloneqq(\sfH_1\overset{\sft_1}{\longrightarrow}\sfG_1,\acton_1)$ into a crossed module $\caG_2\coloneqq(\sfH_2\overset{\sft_2}{\longrightarrow}\sfG_2,\acton_2)$, we mean a (strict) injective morphism of crossed modules $\sfe:\caG_1\hookrightarrow\caG_2$ which is given by embeddings of groups $\sfe_0:\sfG_1\hookrightarrow\sfG_2$ and $\sfe_1:\sfH_1\hookrightarrow\sfH_2$ such that

\begin{equation}\label{eq:consistency_2_group_embedding}
\sfe_0(\sft_1(h_1))\ =\ \sft_2(\sfe_1(h_1))
\eand
\sfe_0(g_1)\acton_2\sfe_1(h_1)\ =\ \sfe_1(g_1\acton_1 h_1)~.
\end{equation}

for all $h_1\in\sfH_1$ and for all $g_1\in\sfG$. Such an embedding then induces a right action of the morphisms of $\grp{\caG_1}$ on the morphisms of $\grp{\caG_2}$ by means of

\begin{equation}
\caR_{(g_1,h_1)}(g_2,h_2)\ \coloneqq\ (g_2,h_2)\otimes (\sfe_0(g_1),\sfe_1(h_1))\ =\ \big(g_2 \sfe_0(g_1),(g_2\acton_2\sfe_1(h_1))h_2\big)
\end{equation}

for all $(g_1,h_1)\in\sfG_1\ltimes\sfH_1$ and for all $(g_2,h_2)\in\sfG_2\ltimes\sfH_2$. Upon identifying morphisms under this action, we obtain the quotient Lie groupoid[A more sophisticated definition in terms of quotient stacks is certainly possible, but not necessary here.]

\begin{equation}\label{eq:quotionLieGroupoid}
\caG_2/\caG_1\ \coloneqq\ \grp{\caG_2}/\grp{\caG_1}\ \coloneqq\ \Big(\big(\sfG_2\ltimes\sfH_2)/(\sfG_1\ltimes\sfH_1)\multirightarrow{2}\sfG_2/\sfG_1\Big)~,
\end{equation}

where, again, the equivalence relation is

\begin{equation}
g_2\ \sim\ g_2\sfe_0(g_1)
\eand
(g_2,h_2)\ \sim\ (g_2 \sfe_0(g_1),(g_2\acton_2\sfe_1(h_1))h_2)
\end{equation}

for all $(g_1,h_1)\in\sfG_1\ltimes\sfH_1$ and for all $(g_2,h_2)\in\sfG_2\ltimes\sfH_2$. The composition of morphisms in this groupoid is induced from the vertical composition $\circ$ in (<ref>).

A useful and simple example of the above construction for the purposes of our discussion is given by the crossed modules $\sfB\sfU(1)=(\sfU(1)\longrightarrow*,\id)$ and $\sfString(\sfG)$ as defined in (<ref>). We can embed $\sfB\sfU(1)$ into $\sfString(\sfG)$ as follows:

\begin{equation}
\begin{tikzcd}
\sfU(1)\arrow[r,"\sfe_1"]\arrow[d] & \widehat{L_0\sfG}\arrow[d]
\\
*\arrow[r,"\sfe_0"] & P_0\sfG
\end{tikzcd}
\quad
\begin{array}{c}
\\[-5pt]
\sfe_1\,:\,\sfU(1)\ \ni\ z\ \mapsto\ [(\unit,z)]\ \in\ \big(P_0L_0\sfG\times\sfU(1)\big)/\sfN\ \cong\ \widehat{L_0\sfG}~,
\\
\kern-4pt\sfe_0\,:\,*\ \ni\ \unit\ \mapsto\ \unit\ \in\ P_0\sfG~.
\end{array}
\end{equation}

The resulting quotient is

\begin{equation}\label{eq:cosetStringBU}
\sfString(\sfG)/\sfB\sfU(1)\ \cong\ \caL\sfG
\end{equation}

with $\caL\sfG$ the crossed module defined in (<ref>). This is, in fact, the essence of the short exact sequence (<ref>). Since $\sfG$ can be trivially identified with the crossed module $(*\hookrightarrow\sfG,\id)$ and because of the latter's identification with $\caL\sfG$ by virtue of (<ref>), any Lie group $\sfG$ can thus be viewed as the coset $\sfString(\sfG)/\sfB\sfU(1)$.

We note that the string 2-group $\sfString(1)$ can be identified with the group $\sfB\sfU(1)$, so that <ref> can be regarded as the quotient $\sfString(\sfG)/\sfString(1)$. We shall consider more general quotients of the string group later on.

Also, we note that a special case of <ref> is the categorification of the Hopf fibration $S^1\hookrightarrow S^3\rightarrow S^2$ to

\begin{equation}\label{eq:categorified_Hopf}
\sfString(1)\ \hookrightarrow\ \sfString(3)\ \rightarrow\ \caL\sfSpin(3) \ \cong\ \sfSpin(3)~.
\end{equation}

This fibration underlies the Abelian self-dual string, see e.g. <cit.> for more details.

Quotients of string 2-groups. Consider two string groups $\sfString(\sfH)$ and $\sfString(\sfG)$ of the form (<ref>) for some Lie groups $\sfH$ and $\sfG$ together with an embedding $\sfH\hookrightarrow\sfG$. Such an embedding induces an embedding of crossed modules $\sfString(\sfH)\hookrightarrow \sfString(\sfG)$, and we have resulting group homomorphisms

\begin{equation}
P_0\sfH\ \hookrightarrow\ P_0\sfG~,
\quad
P_0L_0\sfH\ \hookrightarrow\ P_0L_0\sfG~,
\quad
L_0L_0\sfH\ \hookrightarrow\ L_0L_0\sfG~.
\end{equation}

From the definition (<ref>), it is furthermore clear that we obtain an embedding of the normal subgroup $\sfN_\sfH$ into $\sfN_\sfG$. These embeddings are compatible with the structure maps, and we have

\begin{equation}
\begin{aligned}
\sfString(\sfG)/\sfString(\sfH)\ &= \ \left(\frac{P_0\sfG\ltimes((P_0L_0\sfG\times\sfU(1))/\sfN_\sfG)}{P_0\sfH\ltimes((P_0L_0\sfH\times \sfU(1))/\sfN_\sfH)}\multirightarrow{2}P_0\sfG/P_0\sfH\right)
\\
\ &\cong \ \left(\frac{P_0\sfG\ltimes L_0\sfG}{P_0\sfH\ltimes L_0\sfH}\multirightarrow{2}P_0\sfG/P_0\sfH\right)
\\
\ &\cong \ \caL\sfG/\caL\sfH
\\
\ &\cong \ \sfG/\sfH~.
\end{aligned}
\end{equation}

In the last equivalence, we have used the fact that our notion of quotient of Lie groups is compatible with the equivalences $\caL\sfG\cong\sfG_{\rm cm}$, see (<ref>) and (<ref>), which immediately follows from the fact that underlying this equivalence are surjective morphisms $\caL\sfG\rightarrow \sfG_{\rm cm}$ and $\caL\sfH\rightarrow\sfH_{\rm cm}$ which map orbits to orbits. The above discussion has the following corollary.

The quotient $\sfString(5)/\sfString(4)$ is a Lie groupoid, which is equivalent to $S^4$, regarded as the discrete Lie groupoid $(S^4\multirightarrow{2}S^4)$.

Total space description.
Before presenting explicit cocycles, let us briefly sketch the total space description of the string bundle over $S^4$, which is

\begin{equation}\label{eq:nonAbelian_coset_bundle}
\sfString(5)\ \rightarrow\ S^4\ \cong\ \sfString(5)/\sfString(4)~.
\end{equation}

Together with the groupoid equivalence

\begin{equation}
(S^4\multirightarrow{2}S^4)\ \cong\ \big(L_0S^4\multirightarrow{2}P_0S^4\big)\,,
\end{equation}

we then have the following diagram

\begin{equation}
\begin{tikzcd}
\widehat{L_0\sfSpin(5)} \arrow[d]&
\\
L_0\sfSpin(5)\arrow[r,shift left]\arrow[r,shift right] \arrow[d] & P_0\sfSpin(5)\arrow[d]
\\
L_0S^4\arrow[r,shift left]\arrow[r,shift right] & P_0S^4\arrow[d,"\boundary"]
\\
& S^4
\end{tikzcd}
\end{equation}

This diagram underlies the description of the principal $\sfString(4)$-bundle as a non-Abelian bundle gerbe, cf. <cit.> as well as <cit.>, but we refrain from giving further details.

Cocycle description.
The above total space description is not suitable for the direct extraction of a cocycle description: the bundle $\widehat{L_0\sfSpin(5)}$ is a non-trivial higher principal bundle over $L_0S^4$. We therefore need to switch to a different cover. We also note that the different lifts form a torsor for $H^3(S^4,\IZ)$. Because this group is trivial, the lift is unique up to isomorphism. This allows us to work with the same simple cover $\bar U_1\sqcup\bar U_2\rightarrow S^4$ defined in (<ref>), which proved to be sufficient in the description of the coset space $\caL\sfSpin(5)/\caL\sfSpin(4)\cong\sfSpin(5)/\sfSpin(4)\cong S^4$ in <ref>. The cocycles for this bundle will then evidently be an extension of the cocycles introduced in (<ref>).

Out second main result is then the following.

Subordinate to the cover $\bar U_1\sqcup\bar U_2$, the string bundle with adjusted connection describing the string structure of $S^4$ is uniquely, up to 2-bundle isomorphism, given by the following descend data

\begin{equation}
\begin{aligned}
\hat h_{ijk}\,:\,\bar Y^{[3]}\ &\rightarrow\ \big(P_0L_0\sfSpin(4)\times\sfU(1)\big)/\sfN~,
\\
q_+\ &\mapsto\ \big[\big((s,t)\mapsto h^\circ_{ijk}(q_+,st),h'_{ijk}\big)\big]\,,
\end{aligned}
\end{equation}

as well as

\begin{equation}
h'_{111}\ \coloneqq\ h'_{222}\ \coloneqq\ h'_{112}\ \coloneqq\ h'_{122}\ \coloneqq\ h'_{121}\ \coloneqq\ 1~,
\end{equation}

\begin{equation}
\Lambda'_{11}\ \coloneqq\ \Lambda'_{22}\ \coloneqq\ \Lambda'_{12}\ \coloneqq\ 0 \eand B'_{1}\ \coloneqq\ 0~.
\end{equation}

The cocycle relations (<ref>) then reduce to

\begin{equation}
\begin{aligned}
\hat h_{212}\ &=\ g_{21}\acton \hat h_{121}~,
\\
\hat \Lambda_{21}\ &=\ \hat h_{121}\nabla_1\hat h_{121}^{-1}-g_{21}^{-1}\acton \hat \Lambda_{12}~,
\\
\hat B_2\ &=\ \hat B_1+\rmd\hat \Lambda_{12}+A_2\acton \hat \Lambda_{12}+\tfrac12[\hat \Lambda_{12},\hat \Lambda_{12}]-\hat \kappa(g^{-1}_{12},F_1-B_1)
\end{aligned}
\end{equation}

thus providing the missing data. The additional components arising in the lift to a string bundle read as

\begin{equation}
\begin{aligned}
h_{212}\ &=\ h_{121}~,
\\
h'_{212}\ &=\ \exp\left(\frac{\rmi}{2\pi}\int_{0}^{1}\rmd t \int_{0}^{1}\rmd s\innerLarge{g_{21}^{\circ-1}(q,t)\parder[g^\circ_{21}(q,t)]{t}}{h^{\circ-1}_{121}(q,st)\parder[h^{\circ}_{121}(q,st)]{s}}\right)
\\
&=\ \exp(\rmi\theta_q)\ =\ \frac{\Re(q)}{|q|}+\rmi\frac{|\Im(q)|}{|q|}~,
\\
\Lambda'_{21}\ &=\ -\frac{\rmi}{2\pi}\int_0^1\rmd r\left\{\innerLarge{\parder[g^\circ_{12}]{r}g_{12}^{\circ-1}}{A^\circ_1}+\innerLarge{\parder[g^\circ_{21}]{r}g_{21}^{\circ-1}}{A^\circ_2}\right.
\\
&\kern2cm-\left.\innerLarge{\parder[g^\circ_{21}]{r}g_{21}^{\circ-1}}{g_{12}^{\circ-1} \rmd g^\circ_{12}}-\int_{0}^{1}\rmd s\,\innerLarge{\parder[h^\circ_{121}]{r}h_{121}^{\circ-1}}{\parder[(h^\circ_{121}\rmd h_{121}^{\circ-1})]{s}}\right\},
\\
B'_{2}\ &=\ \frac{\rmi}{4\pi}\left.\left(\innerLarge{g_{12}^{\circ-1}\rmd g^\circ_{12} + g_{12}^{\circ-1} A^\circ_1 g^\circ_{12}}{A^\circ_2} +\innerLarge{g_{12}^{\circ-1} A^\circ_1 g^\circ_{12}}{g_{12}^{\circ-1}\rmd g^\circ_{12}}\right)\right|_{r=1}
\\
&\kern1cm+\frac{\rmi}{2\pi}\rmd\int_0^1\rmd r\innerLarge{\parder[g^\circ_{12}]{r}g_{12}^{\circ-1}}{A^\circ_1}
\\
&\kern1cm+\frac{\rmi}{4\pi}\int_0^1\rmd r\left\{\innerLarge{\parder[A^\circ_1]{r}}{A^\circ_1}-\innerLarge{\parder[A^\circ_2]{r}}{A^\circ_2}+\innerLarge{\parder[g_{12}^{\circ-1}\rmd g^\circ_{12}]{r}}{g_{12}^{\circ-1}\rmd g^\circ_{12}}\right\}.
\end{aligned}
\end{equation}

The proof is simply a verification of the adjusted cocycle relations for $\sfString(4)$-bundles, which is a tedious but straigthforward exercise.

Finally, we note that <ref> also applies here, and much more suitable covers may exist, that are, however, less suitable for immediate physical applications.

§ OUTLOOK: APPLICATION TO SELF-DUAL STRINGS