#  Note on the group of vertical diffeomorphisms
of a principal bundle,
& its relation to the Frölicher-Nijenhuis bracket

J. François

###### Abstract

We consider the group of vertical diffeomorphisms of a principal bundle which
are not automorphisms, and the corresponding group of maps with value in the
structure group. The associated Lie algebra of general vertical vector fields
has a Lie bracket that is but an instance (in degree 0) of the Frölicher-
Nijenhuis bracket for vector-valued differential forms. The induces bracket on
the maps with value in the Lie algebra of the structure group happens to
reproduces an “extended” bracket introduced heuristically in physics to study
asymptotic symmetries of gravity. We thus correct the misconception that the
latter is a Lie algebroid bracket. We detail the elementary underlying
geometry, whose exposition is absent from most standard treatments of bundle
geometry, and we discuss its relevance to gauge field theory.

Department of Mathematics & Statistics, Masaryk University – MUNI.
Kotlářská 267/2, Veveří, Brno, Czech Republic.

Department of Philosophy – University of Graz.

Heinrichstraße 26/5, 8010 Graz, Austria.

Department of Physics, Mons University – UMONS.

Service _Physics of the Universe, Fields & Gravitation_.

20 Place du Parc, 7000 Mons, Belgium.

Keywords : Bundle geometry, Frölicher-Nijenhuis bracket, Nijenhuis-Lie
derivative, gauge field theory.

###### Contents

  1. 1 Introduction
  2. 2 Vertical diffeomorphisms of a principal bundle and the Frölicher-Nijenhuis bracket
    1. 2.1 Groups of vertical and gauge transformations
    2. 2.2 Linearisation and Lie algebraic structures
    3. 2.3 The Nijenhuis-Lie derivative and the Frölicher-Nijenhuis bracket on a principal bundle
  3. 3 General vertical transformations and the Nijenhuis-Lie derivative
    1. 3.1 Finite vertical and gauge transformations
    2. 3.2 Discussion
    3. 3.3 Infinitesimal vertical transformations
  4. 4 Local structure
    1. 4.1 Gluings and local active gauge transformations
    2. 4.2 Linear versions
    3. 4.3 Generalised local active gauge transformations
  5. 5 Conclusion

##  1 Introduction

In the some of the literature on gauge field theory and gravity, field-
dependent gauge parameters or diffeomorphisms are often considered. These are
needed, is it often argued, because of gauge-fixing or when boundary
conditions are imposed that needs to be preserved by gauge symmetries.
Consequently, a bracket extending the Lie algebra bracket of gauge parameters
(the bracket of vector fields in the case of diffeomorphisms) is introduced
heuristically, so as to take into account their field-dependence. Such a
bracket features often e.g. in the covariant phase space literature, for
example in investigations of spacetime asymptotic symmetries (BMS and
extensions). See e.g. [1; 2; 3; 4; 5] and references therein. Indeed this
literature often traces back the introduction of this bracket to a paper by
Barnich and Troessaert [6]. While these authors certainly deserve credit for
independently coming up with it, the extended bracket was introduced earlier
(maybe first) by Bergmann and Komar in [7], and then further used by Salisbury
and Sundermeyer in [8]. Its geometric origin remained unclear, few attempts
have been made to clarify its mathematical status. Barnich, shortly after re-
introducing it, proposed that it be understood as a Lie algebroid bracket [9].

In this note, we show that actually the extended bracket is the simplest
instance of the the Frölicher-Nijenhuis (FN) bracket of vector-valued
differential forms [10] on a principal bundle: the principal bundle in
question being the infinite dimensional field space $\Phi$ of a gauge theory,
whose structure group is the gauge group of the theory. The field space Lie
derivative along field-dependent vector fields is then simply the Nijenhuis-
Lie derivative.

This observation is quite natural, and can be articulated already for a finite
dimensional principal bundle $P$ with structure group $H$. Yet it doesn’t seem
to appear in most available standard treatments of the differential geometry
of principal bundles (probably explaining the confusion about the meaning of
such extended brackets). So, here we lay the case in some details: We first
describe the group $\operatorname{Diff}_{v}(P)$ of vertical diffeomorphisms of
$P$, and characterise the corresponding group of $H$-valued maps on $P$. Then,
we consider their respective Lie algebras and show that the bracket of
_general_ vertical vector fields is but an instance – in degree $0$ – of the
FN bracket, and that it corresponds to a natural _extended bracket_ on the
associated Lie algebra of Lie$H$-valued maps on $P$. Finally, we consider the
related issue of _general vertical transformations_ of forms on $P$, resulting
from the action of $\operatorname{Diff}_{v}(P)$, and we show that the
infinitesimal version of those is given by the Nijenhuis-Lie derivative along
general vertical vector fields.

As mentioned, this apply with very little adjustments (mainly notational) to
the infinite dimensional case of the field space $\Phi$ of gauge field
theories, whose structure group can either be the internal gauge group
$\mathcal{H}$ of the theory (i.e. of $P$) or $\operatorname{Diff}(M)$. We give
a detailed account of the later case in a separate work [11], for which this
paper can be considered a preparatory technical note.

##  2 Vertical diffeomorphisms of a principal bundle and the Frölicher-
Nijenhuis bracket

###  2.1 Groups of vertical and gauge transformations

Classical gauge gauge field theory is founded on the geometry of connections
on fiber bundles, whose central objects are principal bundles. Consider a
$H$-principal bundle $P$ over a base manifold $M$ (“spacetime” in gauge field
theory), $H$ its structure (Lie) group. As a manifold, $P$ has a group of
diffeomorphisms $\operatorname{Diff}(P)$, but as a bundle its maximal group of
transformations is the group of bundle automorphisms
$\operatorname{Aut}(P)\vcentcolon=\left\\{\psi\in\operatorname{Diff}(P)\,|\,\psi\circ
R_{h}=R_{h}\circ\psi\right\\}$. The latter, commuting with the right action of
$H$, preserve fibers, and thus project as diffeomorphisms of $M$.

The subgroup of _vertical_ diffeomorphisms of the $P$ is
$\operatorname{Diff}_{v}(P)\vcentcolon=\left\\{\psi\in\operatorname{Diff}(P)\,|\,\pi\circ\psi=\pi\right\\}$.
These are diffeos $\psi$ moving along fibers, therefore there is a unique
smooth map $\gamma:P\rightarrow H$ s.t. $\psi(p)=R_{\gamma(p)}p=p\gamma(p)$.
In other words, to $\psi\in\operatorname{Diff}_{v}(P)$ corresponds a unique
$\gamma\in C^{\infty}(P,H)$.

This generalises the subgroup of vertical automorphisms
$\operatorname{Aut}_{v}(P)\vcentcolon=\left\\{\psi\in\operatorname{Aut}(P)\,|\,\pi\circ\psi=\pi\right\\}$,
isomorphic to the gauge group
$\mathcal{H}\vcentcolon=\big{\\{}\gamma:P\rightarrow
H\,|\,R^{*}\gamma=h^{-1}\gamma h\big{\\}}$ via $\psi(p)=R_{\gamma(p)}\,p$: The
constraint of $H$-equivariance of $\psi$ defining the group of bundle
automorphisms $\operatorname{Aut}(P)$ is responsible for the specific
equivariance required on elements $\gamma$ of the gauge group. Obviously then,
$\operatorname{Diff}_{v}(P)\cap\operatorname{Aut}(P)=\operatorname{Aut}_{v}(P)$.

We may also notice that bundle automorphisms belong to the normaliser of
vertical diffeomorphisms: Indeed, for $\varphi\in\operatorname{Aut}(P)$ and
$\psi\in\operatorname{Diff}_{v}(P)\sim\gamma\in C^{\infty}(P,H)$ we have,

$\displaystyle\big{(}\varphi^{-1}\circ\psi\circ\varphi\big{)}(p)=\varphi^{-1}\circ\psi\big{(}\varphi(p)\big{)}=\varphi^{-1}\big{(}R_{\gamma\left(\varphi(p)\right)}\,\varphi(p)\big{)}=R_{\gamma\left(\varphi(p)\right)}\,\varphi^{-1}\circ\varphi(p)=R_{\gamma\left(\varphi(p)\right)}\,p.$

Therefore, $\pi\circ\big{(}\varphi^{-1}\circ\psi\circ\varphi\big{)}=\pi$, i.e.
$\varphi^{-1}\circ\psi\circ\varphi\in\operatorname{Diff}_{v}(P)$. Since
naturally a group is a subgroup of its normaliser, we have that,

$\displaystyle
N_{\operatorname{Diff}(P)}\big{(}\\!\operatorname{Diff}_{v}(P)\big{)}=\operatorname{Diff}_{v}(P)\cup\operatorname{Aut}(P).$
(1)

As a special case, we have
$N_{\operatorname{Diff}(P)}\big{(}\\!\operatorname{Aut}_{v}(P)\big{)}=\operatorname{Aut}(P)$,
i.e. the well-known fact that
$\operatorname{Aut}_{v}(P)\triangleleft\operatorname{Aut}(P)$, which in turn
gives us the short exact sequence (SES) of groups characteristic of a
principal bundle:

$\displaystyle\text{id}\rightarrow\operatorname{Aut}_{v}(P)\simeq\mathcal{H}\xrightarrow{\iota}\operatorname{Aut}(P)\xrightarrow{\pi}\operatorname{Diff}(M)\rightarrow\text{id}$
(2)

The linearised version is the SES of Lie algebras, a.k.a. the Atiyah Lie
algebroid associated to a principal bundle $P$.

$\displaystyle
0\rightarrow\mathfrak{aut}_{v}(P)=\Gamma_{H}(VP)\simeq\text{Lie}\mathcal{H}\xrightarrow{\iota}\mathfrak{aut}(P)=\Gamma_{H}(TP)\xrightarrow{\pi_{*}}\mathfrak{diff}(M)\simeq\Gamma(TM)\rightarrow
0.$ (3)

Here
$\Gamma_{H}(TP)\vcentcolon=\big{\\{}\boldsymbol{X}\in\Gamma(TP)\,|\,R_{h*}\boldsymbol{X}_{{}_{p}}=\boldsymbol{X}_{|ph}\big{\\}}$
are the right-invariant vector fields of $P\,$: Indeed, for
$\boldsymbol{X}\vcentcolon=\tfrac{d}{d\tau}\,\psi_{\tau}\,\big{|}_{\tau=0}$
with flow $\psi_{\tau}\in\operatorname{Aut}(P)$, we have

$\displaystyle
R_{h*}\boldsymbol{X}_{|p}\vcentcolon=\tfrac{d}{d\tau}\,R_{h}\big{(}\psi_{\tau}(p)\big{)}\,\big{|}_{\tau=0}=\tfrac{d}{d\tau}\,\psi_{\tau}\big{(}R_{h}\,p\big{)}\,\big{|}_{\tau=0}=\vcentcolon\boldsymbol{X}_{ph}.$
(4)

Naturally,
$\Gamma_{H}(VP)\vcentcolon=\big{\\{}\boldsymbol{X}\in\Gamma_{H}(TP)\,|\,\pi_{*}\boldsymbol{X}\equiv
0\big{\\}}$, while
Lie$\mathcal{H}\vcentcolon=\big{\\{}X:P\rightarrow\text{Lie}H\,|\,R^{*}X=\text{Ad}_{h^{-1}}X\big{\\}}$.
A $H$-invariant vertical vector field is written as
$X^{v}\vcentcolon=\tfrac{d}{d\tau}\,\psi_{\tau}\,\big{|}_{\tau=0}$ with flow
$\psi_{\tau}(p)=R_{\gamma_{\tau}(p)}\,p\in\operatorname{Aut}_{v}(P)$ and
$X\vcentcolon=\tfrac{d}{d\tau}\,\gamma_{\tau}\,\big{|}_{\tau=0}$. In other
words, elements of Lie$\mathcal{H}$ generate $H$-invariant fundamental vector
fields.

The composition law in $\operatorname{Diff}_{v}(P)$ induces a peculiar
composition law for the corresponding elements in $C^{\infty}(P,H)$. Indeed,
for $\psi,\varphi\in\operatorname{Diff}_{v}(P)$ to which are respectively
associated $\gamma,\eta\in C^{\infty}(P,H)$, we have
$\varphi\circ\psi\in\operatorname{Diff}_{v}(P)$ (since
$\pi\circ\varphi\circ\psi=\pi\circ\psi=\pi$) so one expects it has a
corresponding generating element in $C^{\infty}(P,H)$. The question is how the
latter is expressed in terms of $\eta$ and $\gamma$. We have,

$\displaystyle(\varphi\circ\psi)(p)$
$\displaystyle=\varphi\left(\psi(p)\right)=R_{\eta\left(\psi(p)\right)}\psi(p)=R_{\eta\left(R_{\gamma(p)}p\right)}R_{\gamma(p)}p,$
(5) $\displaystyle=R_{\gamma(p)\,\eta\left(R_{\gamma(p)}p\right)}\,p.$ (6)

So,

$\displaystyle\varphi\circ\psi\in\operatorname{Diff}_{v}(P)\quad\text{ is
associated to }\quad\gamma\,(\eta\circ R_{\gamma})\in C^{\infty}(P,H).$ (7)

Observe that for vertical automorphisms
$\psi,\varphi\in\operatorname{Aut}_{v}(P)$ to which correspond the gauge group
elements $\gamma,\eta\in\mathcal{H}$, we have $\eta\circ
R_{\gamma}=R^{*}_{\gamma}\eta=\gamma^{-1}\eta\gamma$. So $\gamma(\eta\circ
R_{\gamma})=\eta\gamma$, and (7) reduces to the standard result
$(\varphi\circ\psi)(p)=\big{(}R_{\eta\gamma}\big{)}p$.

For the inverse diffeomorphism $\psi^{-1}$,
$\psi^{-1}\circ\psi=\text{id}_{P}=\psi\circ\psi^{-1}$, we must have
$\tilde{\gamma}\in C^{\infty}(P,H)$ s.t.
$\psi^{-1}(p)=R_{\tilde{\gamma}(p)}p$. According to (7) we must then have that
on the one hand,

$\displaystyle\psi^{-1}\circ\psi=\text{id}_{P}\quad\text{ is associated to
}\quad\gamma(\tilde{\gamma}\circ R_{\gamma})=\text{id}_{H}.$ (8)

And on the other hand,

$\displaystyle\psi\circ\psi^{-1}=\text{id}_{P}\quad\text{ is associated to
}\quad\tilde{\gamma}(\gamma\circ R_{\tilde{\gamma}})=\text{id}_{H}.$ (9)

From (8) in particular, we get

$\displaystyle\tilde{\gamma}\circ R_{\gamma}=\gamma^{-1}_{R}.$ (10)

Observe that for $\gamma\in\mathcal{H}$, (9) is
$\gamma\tilde{\gamma}=\text{id}_{H}$, i.e. $\tilde{\gamma}=\gamma^{-1}_{R}$
for all $p\in P$. In particular for $ph$:
$\tilde{\gamma}(ph)=\gamma_{R}(ph)^{-1}=h^{-1}\gamma_{R}(p)^{-1}h=h^{-1}\tilde{\gamma}(p)h$,
i.e. $R^{*}_{h}\tilde{\gamma}=h^{-1}\tilde{\gamma}h$. So, (8) is
$\tilde{\gamma}\gamma=\text{id}_{H}$ and $\tilde{\gamma}=\gamma^{-1}_{L}$.
Thus, $\tilde{\gamma}=\gamma^{-1}$, as one would expect.

Following on (7), one shows that for
$\psi_{1},\psi_{2},\psi_{3}\in\operatorname{Diff}_{v}(P)$ associated
respectively to $\gamma_{1},\gamma_{2},\gamma_{3}\in C^{\infty}(P,H)$:

$\displaystyle\psi_{3}\circ\psi_{2}\circ\psi_{1}\quad\text{ is associated to
}\quad\gamma_{1}(\gamma_{2}\circ R_{\gamma_{1}})\big{(}\gamma_{3}\circ
R_{\gamma_{1}(\gamma_{2}\circ R_{\gamma_{1}})}\big{)}.$ (11)

The pattern is clear, for $k$ iterated compositions we have,

$\displaystyle\circ_{i=1}^{k}\psi_{i}\quad\text{ is associated to
}\quad\Pi_{i=1}^{k}t_{i},\ \ \text{ with }t_{i}\vcentcolon=\gamma_{i}\circ
R_{t_{i-1}}.$ (12)

We therefore see that the composition law of $\operatorname{Diff}_{v}(P)$
doesn’t give rise to a simple pointwise multiplicative group structure on the
corresponding generating elements in $C^{\infty}(P,H)$, but rather to a
recursive nested one involving the (yet unspecified) equivariance. Because of
this, when for simplicity of expression we will call $C^{\infty}(P,H)$
equipped with this composition law a “group”, one should keep in mind that we
are stretching the terminology.

###  2.2 Linearisation and Lie algebraic structures

We now turn to the question of linearisation, i.e. of describing the Lie
algebra $\mathfrak{diff}_{v}(P)$: Consider a 1-parameter element
$\psi_{\tau}\in\operatorname{Diff}_{v}(P)$ s.t.
$\psi_{\tau}(p)=R_{\gamma_{\tau}(p)}p$ with $\gamma_{\tau}\in
C^{\infty}(P,H)$. We have by definition,

$\displaystyle\tfrac{d}{d\tau}\gamma_{\tau}(p)\,\big{|}_{\tau=0}=\vcentcolon
X(p)\in\text{Lie}H,$ (13)

and $X\in C^{\infty}(P,\text{Lie}H)=\Omega^{0}(P,\text{Lie}H)$. By definition,
an element of $\mathfrak{diff}_{v}(P)$ is a _general_ vertical vector field:

$\displaystyle\tfrac{d}{d\tau}\psi_{\tau}(p)\,\big{|}_{\tau=0}$
$\displaystyle=\vcentcolon X(p)^{v}_{|p}\in V_{p}P,$ (14)
$\displaystyle=pX(p),$

and $X^{v}\in\mathfrak{diff}_{v}(P)\simeq\Omega^{0}(P,VP)$. Now,

$\displaystyle\tfrac{d}{d\tau}\,\psi_{\tau}^{-1}\circ\psi_{\tau}(p)\,\big{|}_{\tau=0}$
$\displaystyle=\tfrac{d}{d\tau}\,\psi_{\tau}(p)\,\tilde{\gamma}_{\tau}\big{(}\psi_{\tau}(p)\big{)}\,\big{|}_{\tau=0},$
$\displaystyle 0\ \ $
$\displaystyle=\tfrac{d}{d\tau}\,\psi_{\tau}(p)\,\big{|}_{\tau=0}\,\tilde{\gamma}_{0}\big{(}\psi_{0}(p)\big{)}+\psi_{0}(p)\,\tfrac{d}{d\tau}\,\tilde{\gamma}_{\tau}\big{(}\psi_{\tau}(p)\big{)}\,\big{|}_{\tau=0},$
$\displaystyle\Rightarrow\quad\tfrac{d}{d\tau}\,\tilde{\gamma}_{\tau}\big{(}\psi_{\tau}(p)\big{)}\,\big{|}_{\tau=0}$
$\displaystyle=-X(p)^{v}_{|p}.$ (15)

The bracket of vertical vector fields (14) must be a vertical vector field
itself. We want the expression of its generating element in
$\mathcal{C}^{\infty}(P,\text{Lie}H)$. Consider $X^{v}$ generated by
$\psi_{\tau}\in\operatorname{Diff}_{v}(P)$ with
$X(p)=\tfrac{d}{d\tau}\gamma_{\tau}(p)\,\big{|}_{\tau=0}\in
C^{\infty}(P,\text{Lie}H)$, and $Y^{v}$ generated by
$\varphi_{\tau}\in\operatorname{Diff}_{v}(P)$ with
$Y(p)=\tfrac{d}{d\tau}\eta_{\tau}(p)\,\big{|}_{\tau=0}\in
C^{\infty}(P,\text{Lie}H)$. Their bracket is:

$\displaystyle\big{[}X(p)^{v},Y(p)^{v}\big{]}_{|p}\vcentcolon=$
$\displaystyle\,\tfrac{d}{d\tau}\tfrac{d}{ds}\,\psi_{\tau}^{-1}\circ\varphi_{s}\circ\psi_{\tau}(p)\,\big{|}_{s=0}\big{|}_{\tau=0},$
$\displaystyle=$
$\displaystyle\,\tfrac{d}{d\tau}\tfrac{d}{ds}\,\psi_{\tau}(p)\cdot\eta_{s}\big{(}\psi_{\tau}(p)\big{)}\cdot\tilde{\gamma}_{\tau}\left(\varphi_{s}\big{(}\psi_{\tau}(p)\big{)}\right)\big{|}_{s=0}\big{|}_{\tau=0},\quad\text{by
\eqref{comp-3},}$ $\displaystyle=$
$\displaystyle\,\tfrac{d}{d\tau}\bigg{\\{}\psi_{\tau}(p)\,\underbrace{\tfrac{d}{ds}\,\eta_{s}\big{(}\psi_{\tau}(p)\big{)}\,\big{|}_{s=0}}_{Y\big{(}\psi_{\tau}(p)\big{)}}\cdot\underbrace{\tilde{\gamma}_{\tau}\bigg{(}\underbrace{\varphi_{0}\big{(}\psi_{\tau}(p)\big{)}}_{\psi_{\tau}(p)}\bigg{)}}_{\gamma_{\tau}(p)^{-1}\text{
by \eqref{inverse}}}\ +\
\psi_{\tau}(p)\cdot\underbrace{\eta_{0}\big{(}\psi_{\tau}(p)\big{)}}_{\text{id}_{H}}\,\underbrace{\tfrac{d}{ds}\,\tilde{\gamma}_{\tau}\left(\varphi_{s}\big{(}\psi_{\tau}(p)\big{)}\right)\big{|}_{s=0}}_{\big{[}Y^{v}(\tilde{\gamma}_{\tau})\big{]}\big{(}\psi_{\tau}(p)\big{)}}\bigg{\\}}\big{|}_{\tau=0},$
$\displaystyle=$
$\displaystyle\,\tfrac{d}{d\tau}\bigg{\\{}p\,\text{Ad}\big{(}\gamma_{\tau}(p)\big{)}Y\big{(}\psi_{\tau}(p)\big{)}+\psi_{\tau}(p)\cdot\big{[}Y^{v}(\tilde{\gamma}_{\tau})\big{]}\big{(}\psi_{\tau}(p)\big{)}\bigg{\\}}\big{|}_{\tau=0},$
$\displaystyle=$
$\displaystyle\,p\bigg{\\{}\tfrac{d}{d\tau}\,\text{Ad}\big{(}\gamma_{\tau}(p)\big{)}Y(p)\,\big{|}_{\tau=0}+\text{Ad}\big{(}\underbrace{\gamma_{0}(p)}_{\text{id}_{H}}\big{)}\tfrac{d}{d\tau}\,Y\big{(}\psi_{\tau}(p)\big{)}\,\big{|}_{\tau=0}\,+\,\tfrac{d}{d\tau}\,\big{[}Y^{v}(\tilde{\gamma}_{\tau})\big{]}\big{(}\psi_{\tau}(p)\big{)}\,\big{|}_{\tau=0}\bigg{\\}},$
$\displaystyle=$
$\displaystyle\,p\bigg{\\{}\text{ad}\big{(}X(p)\big{)}Y(p)+\big{[}X^{v}(Y)\big{]}(p)+\big{[}Y^{v}(-X)\big{]}(p)\bigg{\\}},$
$\displaystyle=$
$\displaystyle\\!:[X(p),Y(p)]^{v}_{|p}+\big{\\{}[X^{v}(Y)](p)\big{\\}}^{v}_{|p}-\big{\\{}[Y^{v}(X)](p)\big{\\}}^{v}_{|p}.$

Thus we obtain, for $X^{v},Y^{v}\in\Omega^{0}(P,VP)$ and $X,Y\in
C^{\infty}(P,\text{Lie}H)$:

$\displaystyle\big{[}X^{v},Y^{v}\big{]}=$ $\displaystyle\,\\{X,Y\\}^{v},$ (16)
$\displaystyle\text{with}\quad\\{X,Y\\}\vcentcolon=$
$\displaystyle\,[X,Y]_{\text{\tiny{Lie$H$}}}+X^{v}(Y)-Y^{v}(X).$

The sought after generating element of the bracket of general vertical vector
fields is then $\\{X,Y\\}\in C^{\infty}(P,\text{Lie}H)$. This  _extended_
bracket $\\{\ ,\,\\}$, reflecting the recursive nested structure (11)-(12) on
the elements of $C^{\infty}(P,H)$ generating $\operatorname{Diff}_{v}(P)$, is
manifestly antisymmetric in $X,Y$ and satisfies the Jacobi identity (as is
easily proven). Therefore, $C^{\infty}(P,\text{Lie}H)$ equiped with $\\{\
,\,\\}$ is a Lie algebra: We have thus the Lie algebra isomorphism
$\mathfrak{diff}_{v}(P)\simeq C^{\infty}(P,\text{Lie}H)$ (respective brackets
tacitly understood). It follows in particular that the Lie derivative an inner
product along $\mathfrak{diff}_{v}(P)$ satisfy:

$\displaystyle[L_{{X}^{v}},L_{{Y}^{v}}]$
$\displaystyle=L_{[{X}^{v},{Y}^{v}]}=L_{\\{{X},{Y}\\}^{v}},$ (17)
$\displaystyle[L_{{X}^{v}},\iota_{{Y}^{v}}]$
$\displaystyle=\iota_{[{X}^{v},{Y}^{v}]}=\iota_{\\{{X},{Y}\\}^{v}}.$

Notice that for $X,Y\in$ Lie$\mathcal{H}$, we have
$R^{*}_{h}Y=\text{Ad}_{h^{-1}}Y$, so $X^{v}(Y)=[Y,X]_{\text{\tiny{Lie$H$}}}$.
Therefore, the extended bracket reduces to
$\\{X,Y\\}=-[X,Y]_{\text{\tiny{Lie$H$}}}$ and
$[X^{v},Y^{v}]=(-[X,Y]_{\text{\tiny{Lie$H$}}})^{v}$ as it should. This is the
standard result that the “verticality map”
$|^{v}:\text{Lie}\mathcal{H}\rightarrow\mathfrak{aut}_{v}(P)\simeq\Gamma_{H}(VP)$,
$X\mapsto X^{v}$ is an anti-isomorphism.

We may also observe that given a connection $\omega$ on $P$, satisfying
$\omega(X^{v})=X$, its curvature is a tensorial 2-form that can be expressed
via Cartan’s structure equation
$\Omega=d\omega+\tfrac{1}{2}[\omega,\omega]_{\text{\tiny{Lie$H$}}}$. And
indeed one may check (in the very process of proving that equation) that, on
$X^{v},Y^{v}\in\mathfrak{diff}_{v}(P)$:

$\displaystyle\Omega(X^{v},Y^{v})$
$\displaystyle=d\omega(X^{v},Y^{v})+[\omega(X^{v}),\omega(Y^{v})]_{\text{\tiny{Lie$H$}}},$
$\displaystyle=X^{v}\cdot\omega(Y^{v})-Y^{v}\cdot\omega(X^{v})-\omega([X^{v},Y^{v}])\
+\ [X,Y]_{\text{\tiny{Lie$H$}}},$
$\displaystyle=X^{v}(Y)-Y^{v}(X)-\omega(\\{X,Y\\}^{v})\ +\
[X,Y]_{\text{\tiny{Lie$H$}}}\equiv 0.$ (18)

As we are now about to show, the extended bracket featuring in (16) is but the
degree 0 of the the Frölicher-Nijenhuis bracket of vector-valued forms on $P$,
while the Lie derivative along $\mathfrak{diff}_{v}(P)$ is but a case of the
Nijenhuis-Lie derivative.

###  2.3 The Nijenhuis-Lie derivative and the Frölicher-Nijenhuis bracket on a
principal bundle

We refer to [10] (chap. II, section 8) for a systematic presentation of the
following notions (on a generic smooth manifold) and for proofs of the
relations displayed.

As a manifold, $P$ has a space of differential forms $\Omega^{\bullet}(P)$.
Its space of derivations forms a graded Lie algebra
$\operatorname{Der}_{\bullet}\big{(}\Omega^{\bullet}(P)\big{)}=\bigoplus_{k}\operatorname{Der}_{k}\big{(}\Omega^{\bullet}(P)\big{)}$,
with graded bracket $[D_{k},D_{l}]=D_{k}\circ D_{l}-(-)^{kl}D_{l}\circ D_{k}$,
with $D_{i}\in\operatorname{Der}_{i}\big{(}\Omega^{\bullet}(P)\big{)}$.

The de Rham complex of $P$ is $\big{(}\Omega^{\bullet}(P);d\big{)}$ with
$d\in\operatorname{Der}_{1}$ the de Rham (exterior) derivative, which is
nilpotent – $d^{2}=0=\nicefrac{{1}}{{2}}[d,d]$ – and defined via the Koszul
formula. Given the exterior product $\wedge$ defined as usual on scalar-valued
forms, we have that $\big{(}\Omega^{\bullet}(P,\mathbb{K}),\wedge,d\big{)}$ is
a differential graded algebra.111The exterior product can also be defined on
the space $\Omega^{\bullet}(P,\sf A)$ of variational differential forms with
values in an algebra $(\sf A,\cdot)$, using the product in $\sf A$ instead of
the product in the field $\mathbb{K}$. So $\big{(}\Omega^{\bullet}(P,{\sf
A}),\wedge,d\big{)}$ is a again a differential graded algebra. On the other
hand, an exterior product cannot be defined on
$\Omega^{\bullet}(P,\boldsymbol{V})$ where $\boldsymbol{V}$ is merely a vector
space.

One may define the subset of vector-field valued differential forms
$\Omega^{\bullet}(P,TP)=\Omega^{\bullet}(P)\otimes TP$. Then, the subalgebra
of “ _algebraic_ ” derivations is defined as $D_{|\Omega^{0}(P)}=0$, they have
the form $\iota_{K}\in\operatorname{Der}_{k-1}$ for $K\in\Omega^{k}(P,TP)$,
with $\iota$ the inner product. For $\omega\otimes X\in\Omega^{\bullet}(P,TP)$
we have : $\iota_{K}(\omega\otimes X):=\iota_{K}\omega\otimes X=\omega\circ
K\otimes X$. On $\Omega^{\bullet}(P,TP)$, the _Nijenhuis-Richardson bracket_
(or _algebraic_ bracket) is defined by:

$\displaystyle[K,L]_{\text{\tiny{NR}}}:=\iota_{K}L-(-)^{(k-1)(l-1)}\,\iota_{L}K.$
(19)

It generalises the inner contraction of a form on a vector field, and it makes
the map:

$\displaystyle\iota:\Omega^{\bullet}(P,TP)$
$\displaystyle\rightarrow\operatorname{Der}_{\bullet}\big{(}\Omega^{\bullet}(P)\big{)}$
(20) $\displaystyle K$ $\displaystyle\mapsto\iota_{K}$

a graded Lie algebra morphism, since:

$\displaystyle[\iota_{K},\iota_{L}]=\iota_{[K,L]_{\text{\tiny{NR}}}}.$ (21)

The _Nijenhuis-Lie derivative_ is the map,

$\displaystyle L:=[\iota,d]:\Omega^{\bullet}(P,TP)$
$\displaystyle\rightarrow\operatorname{Der}_{\bullet}\big{(}\Omega^{\bullet}(P)\big{)}$
$\displaystyle K$ $\displaystyle\mapsto L_{K}:=\iota_{K}d-(-)^{k-1}d\iota_{K}$

We have $L_{K}\in\operatorname{Der}_{k}$ for $K\in\Omega^{k}(P,TP)$. It
generalises the Lie derivative along vector fields,
$L_{X}\in\operatorname{Der}_{0}$. It is such that $[L_{K},d]=0$, and it is a
morphism of graded Lie algebras:

$\displaystyle[L_{K},L_{J}]=L_{[K,J]_{\text{\tiny{FN}}}},$ (22)

where $[K,J]_{\text{\tiny{FN}}}$ is the _Frölicher-Nijenhuis bracket_.
Explicitely, for $K=\mathsf{K}\otimes X\in\Omega^{k}(P,TP)$ and
$J=\mathsf{J}\otimes Y\in\Omega^{l}(P,TP)$, it is:

$\displaystyle[K,J]_{\text{\tiny{FN}}}\vcentcolon=\mathsf{K}\wedge\mathsf{J}\otimes[X,Y]+\mathsf{K}\wedge
L_{X}\mathsf{J}\otimes Y-L_{Y}\mathsf{K}\wedge J\otimes
X+(-)^{k}\big{(}d\mathsf{K}\wedge\iota_{X}\mathsf{J}\otimes
Y+\iota_{Y}\mathsf{K}\wedge d\mathsf{J}\otimes X\big{)}.$ (23)

We further have the relations:

$\displaystyle[L_{K},\iota_{J}]$
$\displaystyle=\iota_{[K,J]_{\text{\tiny{FN}}}}-(-)^{k(l-1)}L_{(\iota_{K}J)},$
(24) $\displaystyle[\iota_{J},\boldsymbol{L}_{K}]$
$\displaystyle=L_{(\iota_{K}J)}+(-)^{k}\,\iota_{[J,K]_{\text{\tiny{FN}}}}.$

The FN bracket (23) reproduces as a special case the extended bracket (16).
Indeed, specialising first (19) in degree $0$, for $f=\mathsf{f}\otimes X$ and
$g=\mathsf{g}\otimes Y\in\Omega^{0}(P,TP)$, we have:

$\displaystyle[f,{dg}]_{\text{\tiny{NR}}}$
$\displaystyle=\iota_{f}{dg}-(-)^{0}\cancel{\iota_{{dg}}f}=\big{(}\mathsf{f}\wedge\iota_{X}d\mathsf{g}\big{)}\otimes
Y=\big{(}\mathsf{f}\wedge L_{X}\mathsf{g}\big{)}\otimes Y,$ (25)
$\displaystyle[{df},{g}]_{\text{\tiny{NR}}}$
$\displaystyle=\cancel{\iota_{{df}}{g}}-(-)^{0}\iota_{{g}}{df}=-[{g},{df}]_{\text{\tiny{NR}}}=-\big{(}\mathsf{g}\wedge
L_{Y}\mathsf{f}\big{)}\otimes X.$

So that (23) is:

$\displaystyle[f,g]_{\text{\tiny{FN}}}$
$\displaystyle=\mathsf{f}\wedge\mathsf{g}\otimes[X,Y]+f\wedge
L_{X}\mathsf{g}\otimes Y-L_{Y}f\wedge\mathsf{g}\otimes X,$
$\displaystyle=\mathsf{f}\wedge\mathsf{g}\otimes[X,Y]+[f,{dg}]_{\text{\tiny{NR}}}-[{g},{df}]_{\text{\tiny{NR}}},$
(26)

and (22)-(24) reduces to:

$\displaystyle[L_{f},L_{g}]$ $\displaystyle=L_{[f,g]_{\text{\tiny{FN}}}},$
(27) $\displaystyle[L_{f},\iota_{g}]$
$\displaystyle=\iota_{[f,g]_{\text{\tiny{FN}}}}.$

Now, the map
$|^{v}:C^{\infty}\big{(}P,\text{Lie}H\big{)}\rightarrow\Gamma(VP)$,
${X}\mapsto{X}^{v}$, allows to think of ${X}^{v}\in\mathfrak{diff}_{v}(P)$ as
a (vertical) vector-valued 0-form on $P$, i.e.
${X}^{v}\in\Omega^{0}(P,VP)\subset\Omega^{\bullet}(P,TP)$. Therefore, the
Nihenhuis-Richardson and Frölicher-Nijenhuis brackets naturally apply: in
particular, specialising further eq.(25) we have, for
${X}^{v},{Y}^{v}\in\Omega^{0}(P,VP)$,

$\begin{split}[{X}^{v},d{Y}^{v}]_{\text{\tiny{NR}}}&=\\{\iota_{{X}^{v}}{d}{Y}\\}^{v}=\\{{X}^{v}({Y})\\}^{v},\\\
[d{X}^{v},{Y}^{v}]_{\text{\tiny{NR}}}&=-[{Y}^{v},d{X}^{v}]_{\text{\tiny{NR}}}=-\\{\iota_{{Y}^{v}}{d}{X}\\}^{v}=-\\{{Y}^{v}({X})\\}^{v},\end{split}$
(28)

so that the FN bracket for 0-forms (2.3) specialises to:

$\displaystyle[{X}^{v},{Y}^{v}]_{\text{\tiny{FN}}}$
$\displaystyle=\big{(}[{X},{Y}]_{\text{{\tiny
Lie$H$}}}\big{)}^{v}+[{X}^{v},d{Y}^{v}]_{\text{\tiny{NR}}}-[{Y}^{v},d\boldsymbol{X}^{v}]_{\text{\tiny{NR}}},$
$\displaystyle=\big{(}[{X},{Y}]_{\text{{\tiny
Lie$H$}}}+{X}^{v}({Y})-{Y}^{v}({X})\,\big{)}^{v}=\\{{X},{Y}\\}^{v}.$ (29)

Which is indeed the extend bracket (16) on $C^{\infty}(P,\text{Lie}H)$, as
claimed. Then of course we have the following special cases of (21),
(22)-(24)/(27), among derivations in
$\operatorname{Der}^{\bullet}\big{(}\Omega^{\bullet}(P)\big{)}$:

$\displaystyle=\iota_{[{X}^{v},d{Y}^{v}]_{\text{\tiny{NR}}}}=\iota_{\\{\iota_{{X}^{v}}{d}{Y}\\}^{v}},$
(30) $\displaystyle[L_{\boldsymbol{X}^{v}},\iota_{{Y}^{v}}]$
$\displaystyle=\iota_{[{X}^{v},{Y}^{v}]_{\text{\tiny{FN}}}}=\iota_{\\{{X},{Y}\\}^{v}},$
$\displaystyle[L_{\boldsymbol{X}^{v}},L_{{Y}^{v}}]$
$\displaystyle=L_{[{X}^{v},{Y}^{v}]_{\text{\tiny{FN}}}}=L_{\\{{X},{Y}\\}^{v}}.$

As one should expect, (30) reproduces (17).

All of the above applies, mutatis mutandis, to the the infinite dimensional
field space $P=\Phi$ of a gauge field theory, which can be seen (under some
adequate mild restrictions) as a principal bundle with structure group
$H=\mathcal{H}$, the internal gauge group a theory, or
$H=\operatorname{Diff}(M)$.

To the best of my knowledge, the bracket (16)-(2.3) has been first introduced
in physics by Bergmann & Komar [7], for the field space
$\Phi=\\{g_{\mu\nu}\\}$, and Salisbury & Sundermeyer [8] in their
investigations of the largest possible covariance group of General Relativity
– see eq.(3.1)-(3.2) in [7] and eq.(2.1) in [8]. It was latter reintroduced by
Barnich & Troessaert in their study [6] of asymptotic symmetries of gravity in
the flat limit at null infinity. This Bergmann-Komar-Salisbury-Sundermeyer
(BKSS) bracket for _field-dependent vector fields_ $\xi,\zeta\in
C^{\infty}\big{(}\Phi,\mathfrak{diff}(M)\big{)}$,
$\\{\xi,\zeta\\}_{\text{\tiny{BKSS}}}=-\\{\xi,\zeta\\}$, appears in eq.(8) in
[6] and more recently e.g. in eq.(3.3) in [2], eq.(2.12) in [3], eq.(2.21) in
[1], eq.(2.3) in [5], eq.(1.1) in [4]. As this references show, this bracket
is commonly used in the covariant phase space literature concerned with the
covariant symplectic structure of gravity and with the analysis of its
asymptotic symmetries (BMS and extensions). This literature derives, sometimes
at quite a computational cost, the above displayed identities satisfied (in
degree 0) by the NL-derivative, the FN and RN brackets. For example, (30)
reproduces e.g. eq.(3.1) in [2], eq.(2.13) in [3], or eq.(A.9) in [5]. We show
here that they flow naturally from well-understood general geometric
structures.

If some have tried to interpret this extended bracket for field-dependent
vector fields (or gauge parameters) as a Lie algebroid bracket (see [9] eq.
(28)), we established that it is actually just the natural bracket on
$C^{\infty}\big{(}\Phi,\mathfrak{diff}(M)\big{)}$ – or
$C^{\infty}\big{(}\Phi,\text{Lie}\mathcal{H}\big{)}$ – arising from
$\boldsymbol{\mathfrak{diff}}_{v}(\Phi)$, and is but the FN-bracket in degree
$0$ on $\Phi$.

##  3 General vertical transformations and the Nijenhuis-Lie derivative

Standard gauge transformations of a form $\beta\in\Omega^{\bullet}(P)$ are
defined as _vertical transformations given by the action of
$\operatorname{Aut}_{v}(P)$ via pullback_, expressible in terms of the
associated elements in the gauge group $\mathcal{H}$ of $P$ :
$\beta^{\gamma}\\!\vcentcolon=\psi^{*}\beta$. Geometrically, this is computed
by relying on the duality pullback/pushforward,
${\beta^{\gamma}}_{|p}(\boldsymbol{X}_{|p})\vcentcolon=\psi^{*}\beta_{|\psi(p)}(\boldsymbol{X}_{|p})=\beta_{|\psi(p)}(\psi_{*}\boldsymbol{X}_{|p})$,
and one then only needs to find the result of the pushforward
$\psi_{*}\boldsymbol{X}$ of any vector field $\boldsymbol{X}\in\Gamma(TP)$ by
a $\psi\in\operatorname{Aut}_{v}(P)$. This is a standard computation, which
turns out not to depend on the $H$-equivariance of $\psi$, so the result holds
as well for $\psi\in\operatorname{Diff}_{v}(P)$. This will thus allow to
define _general vertical transformations_ of $\beta$ as the action of
$\operatorname{Diff}_{v}(P)$ by pullback, expressible in terms of the
associated elements of $C^{\infty}(P,H)$. The linearisation of these is
treated next, and seen to be exactly given by the Nijenhuis-Lie derivative.

###  3.1 Finite vertical and gauge transformations

As a useful lemma, let us first derive the pushforward by the right $H$-action
of $X^{v}\in\mathfrak{diff}_{v}(P)$ generated by
$X=\tfrac{d}{d\tau}\,\gamma_{\tau}\,\big{|}_{\tau=0}\in
C^{\infty}(P,\text{Lie}H)$, for $\gamma_{\tau}\in C^{\infty}(P,H)$:

$\displaystyle R_{h*}X(p)^{v}_{|p}$
$\displaystyle=R_{h*}\tfrac{d}{d\tau}\,R_{\gamma_{\tau}(p)}\,p\,\big{|}_{\tau=0}=\tfrac{d}{d\tau}\,R_{h}R_{\gamma_{\tau}(p)}\,p\,\big{|}_{\tau=0}=\tfrac{d}{d\tau}\,R_{\gamma_{\tau}(p)h}\,p\,\big{|}_{\tau=0}=\tfrac{d}{d\tau}\,R_{{\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}h^{-1}}\gamma_{\tau}(p)h}\,R_{{\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}h}}\,p\,\big{|}_{\tau=0},$
(31) $\displaystyle=\vcentcolon\big{(}\text{Ad}_{h^{-1}}X(p)\big{)}^{v}_{|ph}$

Remark that, for $X\in$ Lie$\mathcal{H}$, we have
$R^{*}_{h}X=\text{Ad}_{h^{-1}}X$, so $R_{h*}X(p)^{v}_{|p}=X(ph)^{v}_{|ph}$.
That is, $X^{v}$ is a right-invariant vertical vector field, those forming (as
already observed) the Lie algebra of vertical automorphisms
$\Gamma_{H}(VP)\simeq\mathfrak{aut}_{v}(P)$.

Now, the classic computation: For $\boldsymbol{X}\in\Gamma(TP)$ with flow
$\phi_{\tau}$, and given $\psi\in\operatorname{Diff}_{v}(P)$ to which
corresponds $\gamma\in C^{\infty}(P,H)$, we have,

$\displaystyle\psi_{*}\boldsymbol{X}_{|p}\vcentcolon=\tfrac{d}{d\tau}\,\psi\big{(}\phi_{\tau}(p)\big{)}\,\big{|}_{\tau=0}$
$\displaystyle=\tfrac{d}{d\tau}\,R_{\gamma\big{(}\phi_{\tau}(p)\big{)}}\,\phi_{\tau}(p)\,\big{|}_{\tau=0},$
$\displaystyle=\tfrac{d}{d\tau}\,R_{\gamma\big{(}\phi_{\tau}(p)\big{)}}\,p\,\big{|}_{\tau=0}+\tfrac{d}{d\tau}\,R_{\gamma(p)}\,\phi_{\tau}(p)\,\big{|}_{\tau=0},$
$\displaystyle=\tfrac{d}{d\tau}\,R_{{\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}\gamma(p)\gamma(p)^{-1}}\gamma\big{(}\phi_{\tau}(p)\big{)}}\,p\,\big{|}_{\tau=0}+R_{\gamma(p)*}\boldsymbol{X}_{|p},$
$\displaystyle=\tfrac{d}{d\tau}\,R_{\gamma(p)^{-1}\gamma\big{(}\phi_{\tau}(p)\big{)}}\,\underbrace{R_{\gamma(p)}\,p}_{\psi(p)}\,\big{|}_{\tau=0}+R_{\gamma(p)*}\boldsymbol{X}_{|p}.$

The first term manifestly is a vertical vector field at $\psi(p)$, one needs
only to find a nice way to write the associated Lie algebra element
(corresponding to the curve $\gamma(p)^{-1}\gamma\big{(}\phi_{\tau}(p)\big{)}$
through $e=\text{id}_{H}$ in $H$). Let us first notice that, since
$\gamma:P\rightarrow H$, generically
$\tfrac{d}{d\tau}\,\gamma\big{(}\phi_{\tau}(p)\big{)}\,\big{|}_{\tau=0}=d\gamma_{|p}\big{(}\boldsymbol{X}_{|p}\big{)}=\gamma_{*}\big{(}\boldsymbol{X}_{|p}\big{)}\in
T_{\gamma(p)}H$. While the Maurer-Cartan form on $H$ is,

$\displaystyle\theta_{\text{\tiny{MC}},\,|h}\vcentcolon=L_{h^{-1}*}:T_{h}H$
$\displaystyle\rightarrow T_{e}H=\vcentcolon\text{Lie}H,$
$\displaystyle\upchi_{|h}$ $\displaystyle\mapsto L_{h^{-1}*}\,\upchi_{|h}.$
$\displaystyle\text{So,}\
\quad\big{[}d\gamma_{|p}\big{(}\boldsymbol{X}_{|p}\big{)}\big{]}_{|\gamma(p)}$
$\displaystyle\mapsto
L_{\gamma(p)^{-1}*}\big{[}d\gamma_{|p}\big{(}\boldsymbol{X}_{|p}\big{)}\big{]}_{|\gamma(p)}=\big{[}\gamma(p)^{-1}d\gamma_{|p}\big{(}\boldsymbol{X}_{|p}\big{)}\big{]}_{|e},$
$\displaystyle\hskip
17.07182ptL_{\gamma(p)^{-1}*}\tfrac{d}{d\tau}\,\gamma\big{(}\phi_{\tau}(p)\big{)}\,\big{|}_{\tau=0}=\tfrac{d}{d\tau}\,\gamma(p)^{-1}\gamma\big{(}\phi_{\tau}(p)\big{)}\,\big{|}_{\tau=0}.$

We thus finally obtain, using the equivariance for vertical vector fields
(31):

$\displaystyle\psi_{*}\boldsymbol{X}_{|p}$
$\displaystyle=R_{\gamma(p)*}\boldsymbol{X}_{|p}+\big{[}\gamma(p)^{-1}d\gamma_{|p}\big{(}\boldsymbol{X}_{|p}\big{)}\big{]}^{v}_{|\psi(p)},$
$\displaystyle=R_{\gamma(p)*}\left(\boldsymbol{X}_{|p}+\big{[}d\gamma_{|p}\big{(}\boldsymbol{X}_{|p}\big{)}\gamma(p)^{-1}\big{]}^{v}_{|p}\right).$
(32)

Obviously, iterations of generalised vertical transformations would rely on
the same formula: e.g. for $\psi^{*}\varphi^{*}=(\varphi\circ\psi)^{*}$ to
which corresponds $\gamma\,(\eta\circ R_{\gamma})$ by (7), one has,

$\displaystyle(\varphi\circ\psi)^{*}\boldsymbol{X}=R_{\gamma\,(\eta\circ
R_{\gamma})\,*}\left(\boldsymbol{X}+\big{[}d[\gamma\,(\eta\circ
R_{\gamma})]\big{(}\boldsymbol{X}\big{)}\,[\gamma\,(\eta\circ
R_{\gamma})]^{-1}\big{]}^{v}\right).$ (33)

The case of $k$-iterations is obvious, though complicated, relying on (12).

To illustrate, consider the general vertical transformations of a (principal)
connection $\omega\in\mathcal{C}$ and of a tensorial form
$\alpha\in\Omega^{\bullet}_{tens}(P,\rho)$ – for $\rho:H\rightarrow GL(V)$ a
representation of $H$. By definition of a connection,
$R^{*}_{h}\omega_{|ph}=\text{Ad}_{h^{-1}}\omega_{|p}$ and
$\omega_{|p}(X^{v}_{|p})=X\in$ Lie$H$, so one has:

$\displaystyle{\omega^{\gamma}}_{|p}(\boldsymbol{X}_{|p})\vcentcolon=\psi^{*}\omega_{|\psi(p)}(\boldsymbol{X}_{|p})$
$\displaystyle=\omega_{|\psi(p)}(\psi_{*}\boldsymbol{X}_{|p}),$
$\displaystyle=\omega_{|\psi(p)}\left(R_{\gamma(p)*}\boldsymbol{X}_{|p}+\big{[}\gamma(p)^{-1}d\gamma_{|p}\big{(}\boldsymbol{X}_{|p}\big{)}\big{]}^{v}_{|\psi(p)}\right),$
$\displaystyle=R_{\gamma(p)}^{*}\omega_{|\psi(p)}\big{(}\boldsymbol{X}_{|p}\big{)}+\omega_{|\psi(p)}\left(\big{[}\gamma(p)^{-1}d\gamma_{|p}\big{(}\boldsymbol{X}_{|p}\big{)}\big{]}^{v}_{|\psi(p)}\right).$
$\displaystyle=\text{Ad}_{\gamma(p)^{-1}}\omega_{|p}\big{(}\boldsymbol{X}_{|p}\big{)}+\gamma(p)^{-1}d\gamma_{|p}\big{(}\boldsymbol{X}_{|p}\big{)}.$

The generalised vertical transformation of a connection is thus,

$\displaystyle\omega^{\gamma}=\text{Ad}_{\gamma^{-1}}\omega+\gamma^{-1}d\gamma.$
(34)

It is all but identical to a standard gauge transformation under
$\psi\in\operatorname{Aut}_{v}(P)\sim\gamma\in\mathcal{H}$. But, doing the
same using (33), one obtains the less familiar result for two consecutive
general gauge transformations:

$\displaystyle(\omega^{\eta})^{\gamma}\vcentcolon=\psi^{*}(\varphi^{*}\omega)=(\varphi\circ\psi)^{*}\omega$
$\displaystyle=\text{Ad}_{[\gamma\,(\eta\circ
R_{\gamma})]^{-1}}\omega+[\gamma\,(\eta\circ
R_{\gamma})]^{-1}d[\gamma\,(\eta\circ R_{\gamma})],$ (35)
$\displaystyle=\text{Ad}_{(\eta\circ
R_{\gamma})^{-1}}\omega^{\gamma}+(\eta\circ R_{\gamma})^{-1}d(\eta\circ
R_{\gamma}).$

For $\varphi,\psi\in\operatorname{Aut}_{v}(P)\sim\eta,\gamma\in\mathcal{H}$ we
have $\gamma\,(\eta\circ R_{\gamma})=\eta\gamma$, so that we get the standard
result $(\omega^{\eta})^{\gamma}=\omega^{\eta\gamma}$, expressing the well-
known fact that the gauge group $\mathcal{H}$ acts on the right on the space
of connections $\mathcal{C}$ – Hence the fact that the latter can be seen
(under proper restrictions) as a principal bundle $\Phi=\mathcal{C}$ with
structure group $\mathcal{H}$ (see e.g. [12; 13], [14; 15], also [1]).

By definition of a tensorial form $\alpha\in\Omega^{\bullet}_{tens}(P,\rho)$:
$R^{*}_{h}\alpha_{|ph}=\rho(h^{-1})\alpha_{|p}$ and
$\alpha_{|p}(X^{v}_{|p})=0$. By the same method, one finds the general
vertical transformations,

$\displaystyle\alpha^{\gamma}$
$\displaystyle=\rho\big{(}\gamma^{-1}\big{)}\alpha,\qquad(\alpha^{\eta})^{\gamma}=\rho\big{(}\gamma\,(\eta\circ
R_{\gamma})\big{)}^{-1}\alpha,=\rho\big{(}\eta\circ
R_{\gamma}\big{)}^{-1}\alpha^{\gamma}.$ (36)

As a special case, this gives the general vertical transformation of the
curvature $\Omega\in\Omega^{2}_{tens}(P,\text{Ad})$ of $\omega$. Naturally,
(36) generalises the well-known gauge transformations of $\alpha$ under
$\operatorname{Aut}_{v}(P)\simeq\mathcal{H}$,
$(\alpha^{\eta})^{\gamma}=\alpha^{\eta\gamma}$, showing that $\mathcal{H}$
acts on the right on $\Omega^{\bullet}_{tens}(P,\rho)$ as it does on
$\mathcal{C}$.

###  3.2 Discussion

At this point, it should be highlighted that a priori
$\omega^{\gamma}\notin\mathcal{C}$ and
$\alpha^{\gamma}\notin\Omega_{tens}(P,\rho)$. To show this, let us first
derive the following lemma: the special case of (3.1) for
$\boldsymbol{X}=X^{v}$ is,

$\displaystyle\psi_{*}X^{v}_{|p}$
$\displaystyle=R_{\gamma(p)*}X^{v}_{|p}+\big{[}\gamma(p)^{-1}d\gamma_{|p}\big{(}X^{v}_{|p}\big{)}\big{]}^{v}_{|\psi(p)},$
$\displaystyle=\big{(}\text{Ad}_{\gamma(p)^{-1}}X\big{)}^{v}_{|\psi(p)}+\big{[}\gamma(p)^{-1}[X^{v}(\gamma)](p)\big{)}\big{]}^{v}_{|\psi(p)},\quad\text{using
\eqref{Gen.VVF.Equiv},}$
$\displaystyle=\left(\text{Ad}_{\gamma(p)^{-1}}X+\gamma(p)^{-1}[X^{v}(\gamma)](p)\right)^{v}_{|\psi(p)}.$
(37)

The definition of $\operatorname{Diff}_{v}(P)\simeq C^{\infty}(P,H)$ leaves
the H-equivariance of $\psi\sim\gamma$ unspecified, so $X^{v}(\gamma)$ remains
a priori unknown. Yet, there are two interesting special cases worth
emphasazing:

$\displaystyle\psi_{*}X^{v}_{|p}=\left\\{\begin{matrix}X^{v}_{|\psi(p)}\ \ \
\text{ for }\psi\in\operatorname{Aut}_{v}(P)\sim\gamma\in\mathcal{H}\ \text{
(gauge transformation)},\\\\[4.2679pt] \ \ 0\ \ \ \ \ \ \ \ \text{ for
}\psi(p)=f(p)\vcentcolon=pu(p)\text{ with }\gamma=u:P\rightarrow H\text{ s.t.
}R^{*}_{h}u=h^{-1}u\ \text{ (dressing field)}.\end{matrix}\right.$ (38)

The second case involves the _dressing map_ $f:P\rightarrow P$, $p\mapsto
pu(p)$, associated to the _dressing field_ $u$, satisfying $f\circ R_{h}=f$
and thus also $f\circ\psi=f$ for $\psi\in\operatorname{Diff}_{v}(P)$. It is
key to the “dressing field method” of gauge symmetry reduction, a tool to
build gauge-invariants in gauge field theory, see e.g. [15; 14; 11]. The map
$f$ is used to “ _dress_ ” forms on $P$, acting via $f^{*}$, turning them into
_basic_ forms (a.k.a “dressed/composite fields”).

Indeed, for some $\beta\in\Omega^{\bullet}(P)$ (horizontality and equivariance
unspecified), we define its dressing by $\beta^{u}\vcentcolon=f^{*}\beta$,
satisfying:

$\displaystyle\beta^{u}(X^{v},\ldots)\vcentcolon=$
$\displaystyle\,f^{*}\beta\,(X^{v}\ldots)=\beta(f_{*}X^{v},\ldots)=\beta(0,\ldots)=0,$
$\displaystyle R^{*}_{h}\,\beta^{u}\vcentcolon=$
$\displaystyle\,R^{*}_{h}\,f^{*}\beta=(f\circ
R_{h})^{*}\beta=f^{*}\beta=\beta^{u}.$ (39)

That is $\beta^{u}\in\Omega^{\bullet}_{\text{basic}}(P)$. For example, a
dressed connection is
$\omega^{u}\vcentcolon=f^{*}\omega=\text{Ad}_{u^{-1}}\omega u+u^{-1}du$, while
a dressed tensorial form is $\alpha^{u}=\rho(u^{-1})\alpha$ – e.g. the dressed
curvature is $\Omega^{u}=u^{-1}\Omega
u=d\omega^{u}+\nicefrac{{1}}{{2}}[\omega^{u},\omega^{u}]$. Any such dressed
form is invariant under $\operatorname{Diff}_{v}(P)\simeq C^{\infty}(P,H)$,
since

$\displaystyle(\beta^{u})^{\gamma}\vcentcolon=\psi^{*}(\beta^{u})=\psi^{*}(f^{*}\beta)=(f\circ\psi)^{*}\beta=f^{*}\beta\vcentcolon=\beta^{u},$
(40)

as can also be found via (3.1) and (3.2). Thus, dressed objects are in
particular gauge invariant, i.e. invariant under
$\operatorname{Aut}_{v}(P)\simeq\mathcal{H}$: hence their physical interest,
as they may encode physical degrees of freedom (d.o.f.).

Returning to the generic case, (3.2), for $\omega\in\mathcal{C}$ we have:

$\displaystyle\omega^{\gamma}(X^{v})$
$\displaystyle=\psi^{*}\omega\,(X^{v})=\omega(\psi_{*}X^{v})=\text{Ad}_{\gamma^{-1}}X+\gamma^{-1}X^{v}(\gamma),\qquad\text{and}$
(41) $\displaystyle R^{*}_{h}\omega^{\gamma}$
$\displaystyle=R^{*}_{h}\psi^{*}\omega=(\psi\circ R_{h})^{*}\omega.$ (42)

In general then, $\omega^{\gamma}=\psi^{*}\omega\notin\mathcal{C}$ for
$\psi\in\operatorname{Diff}_{v}(P)\sim\gamma\in C^{\infty}(P,H)$. But per
(38), we see the especially significant role played by the groups
$\operatorname{Aut}_{v}(P)\simeq\mathcal{H}$, since only for them do we have
$\psi\circ R_{h}=R_{h}\circ\psi$ and $X^{v}(\gamma)=[\gamma,X]$, so that
$\omega^{\gamma}(X^{v})=X$ and
$R^{*}_{h}\omega^{\gamma}=\text{Ad}_{h^{-1}}\omega^{\gamma}$. That is,
$\operatorname{Aut}_{v}(P)\simeq\mathcal{H}$ is the only subgroup of vertical
transformations $\operatorname{Diff}_{v}(P)\simeq C^{\infty}(P,H)$ preserving
the space of connections $\mathcal{C}$ or $P$.222 This is to be expected as
$\operatorname{Aut}(P)\subset\operatorname{Diff}(P)$ is the largest natural
transformation group of a bundle $P$, preserving its fibration structure, and
$\operatorname{Aut}(P)\cap\operatorname{Diff}_{v}(P)=\operatorname{Aut}_{v}(P)$.
From above, it is obvious that the second special case in (38) gives us
$\omega^{u}\notin\mathcal{C}$: i.e. a dressed connection is not a connection.

In the same way, for $\alpha\in\Omega^{\bullet}_{\text{tens}}(P,\rho)$ we
have:

$\displaystyle\alpha^{\gamma}(X^{v})$
$\displaystyle=\psi^{*}\alpha\,(X^{v})=\alpha(\psi_{*}X^{v})=0,\qquad\text{and}$
(43) $\displaystyle R^{*}_{h}\alpha^{\gamma}$
$\displaystyle=R^{*}_{h}\psi^{*}\alpha=(\psi\circ R_{h})^{*}\alpha.$ (44)

So, if horizontality is preserved by the action of
$\operatorname{Diff}_{v}(P)\simeq C^{\infty}(P,H)$, this is a priori not the
case of $H$-equivariance. Again, only for $\psi\in\operatorname{Aut}_{v}(P)$
do we get indeed $R^{*}_{h}\alpha^{\gamma}=\rho(h^{-1})\,\alpha^{\gamma}$:
i.e. $\operatorname{Aut}_{v}(P)\simeq\mathcal{H}$ is the only subgroup of
vertical transformations $\operatorname{Diff}_{v}(P)\simeq C^{\infty}(P,H)$
preserving the space of tensorial forms
$\Omega^{\bullet}_{\text{tens}}(P,\rho)$. Again, from above it is obvious that
the second special case in (38) gives
$\alpha^{u}\notin\Omega^{\bullet}_{\text{tens}}(P,\rho)$.

This shows the importance of being mindful of the equivariance of
$\psi\sim\gamma$, if one wants to keep track of the mathematical space in
which one evolves: in particular one must keep in mind the clear distinction
between gauge transformations and dressing operations, as stressed by (38).

Connections $\omega\in\mathcal{C}$ can be understood as necessary to obtain
covariant derivatives preserving tensorial forms, i.e.
$D=d\,+\rho_{*}(\omega):\Omega^{\bullet}_{\text{tens}}(P,\rho)\rightarrow\Omega^{\bullet+1}_{\text{tens}}(P,\rho)$
– which the de Rham derivative $d$ does not ($d\alpha$ fails to be
horizontal). This implies that $\alpha$ and $D\alpha$ have the same gauge
transformations: this is a way to phrase physics’ _Gauge Principle_. But it
also means that $\alpha$ and $D\alpha$ have the same generalised vertical
transformations under $\operatorname{Diff}_{v}(P)\simeq C^{\infty}(P,H)$:

$\displaystyle
D\alpha\in\Omega^{\bullet+1}_{\text{tens}}(P,\rho)\quad\Rightarrow\quad(D\alpha)^{\gamma}=\rho(\gamma^{-1})D\alpha.$
(45)

Yet, it is also the case that:

$\displaystyle(D\alpha)^{\gamma}=\psi^{*}D\alpha=\psi^{*}\big{(}d\alpha+\rho_{*}(\omega)\alpha\big{)}$
$\displaystyle=d\psi^{*}\alpha+\rho_{*}(\psi^{*}\omega)\psi^{*}\alpha,$ (46)
$\displaystyle=\vcentcolon
d\alpha^{\gamma}+\rho_{*}(\omega^{\gamma})\alpha^{\gamma}.$

This goes to show that the covariant derivative $D$ is compatible with the
action of $\operatorname{Diff}_{v}(P)\simeq C^{\infty}(P,H)$, even if under
its action it may be that $\omega^{\gamma}\notin\mathcal{C}$ and
$\alpha^{\gamma}\notin\Omega^{\bullet}_{tens}(P,\rho)$: In this case, if
$D^{\gamma}\vcentcolon=d\,+\rho_{*}(\omega^{\gamma})$ is not a covariant
derivative in the geometrical sense of preserving
$\Omega^{\bullet}_{tens}(P,\rho)$, it is in the algebraic sense – dear to
physicists – of preserving the form of the $\operatorname{Diff}_{v}(P)\simeq
C^{\infty}(P,H)$ transformation of $\alpha^{\gamma}$ – by (35)-(36). If this
transformation is instead a dressing operation, we have in particular that:
$(D\alpha)^{u}=d\alpha^{u}+\rho_{*}(\omega^{u})\alpha^{u}$.

###  3.3 Infinitesimal vertical transformations

Naturally, the infinitesimal version of the general vertical transformation
$\beta^{\gamma}=\psi^{*}\beta$ for
$\psi\in\operatorname{Diff}_{v}(P)\sim\gamma\in C^{\infty}(P,H)$ is given by
the Lie derivative along the associated general vertical vector field
$X^{v}\in\mathfrak{diff}_{v}(P)\sim X\in C^{\infty}(P,\text{Lie}H)$,
(13)-(14):

$\displaystyle
L_{X^{v}}\,\beta\vcentcolon=\tfrac{d}{d\tau}\,\beta^{\gamma_{\tau}}\,\big{|}_{|\tau=0}=\tfrac{d}{d\tau}\,\psi_{\tau}^{*}\,\beta\,\big{|}_{|\tau=0}.$
(47)

The Lie derivative belongs to the Lie algebra of derivations of
$\Omega^{\bullet}(P)$, it is the case that it is defined by
$L_{X^{v}}=[\iota_{X^{v}},d]$ (Cartan’s formula). Since, $X^{v}$ can be seen
as an element of $\Omega^{0}(P,VP)$, $L_{X^{v}}$ is to be understood as a
special case of the Nijenhuis-Lie derivative along the vertical vector-valued
$0$-form $X^{v}$. Naturally then, by (30), the commutator of Nijenhuis-Lie
derivatives involves the Frölicher-Nijehnhuis brachet on $\Omega^{0}(P,VP)$,
i.e. the extended bracket $\\{\,,\,\\}$ on $C^{\infty}(P,\text{Lie}H)$
(16)-(23):

$\displaystyle[L_{X^{v}},L_{Y^{v}}]\,\beta=L_{[X^{v},Y^{v}]_{\text{\tiny{FN}}}}\,\beta=L_{\\{X,Y\\}^{v}}\,\beta.$
(48)

To illustrate, let us have the
$\mathfrak{diff}_{v}(P)\simeq\Omega^{0}(P,VP)\sim\mathcal{C}^{\infty}(P,\text{Lie}H)$
transformations of $\omega\in\mathcal{C}$ and
$\alpha\in\Omega^{\bullet}_{tens}(P,\rho)$. For a connection, using (34), we
get:

$\displaystyle
L_{X^{v}}\omega\vcentcolon=\tfrac{d}{d\tau}\,\psi_{\tau}^{*}\,\omega\,\big{|}_{|\tau=0}=\tfrac{d}{d\tau}\,\text{Ad}_{\gamma^{-1}_{\tau}}\omega+\gamma^{-1}_{\tau}d\gamma_{\tau}\,\big{|}_{|\tau=0}=-\text{ad}_{X}\omega+dX=DX.$
(49)

This is cross-checked via Cartan’s formula, using Cartan’s structure equation
for the curvature, and the latter’s tensoriality: For
$\boldsymbol{X}\in\Gamma(TP)$, we have in general

$\displaystyle
L_{\boldsymbol{X}}\omega=\iota_{\boldsymbol{X}}d\omega+d(\iota_{\boldsymbol{X}}\omega)=\iota_{\boldsymbol{X}}\big{(}\Omega-\tfrac{1}{2}[\omega,\omega]\big{)}+d\big{(}\omega(\boldsymbol{X})\big{)}=\iota_{\boldsymbol{X}}\Omega+d\big{(}\omega(\boldsymbol{X})\big{)}+[\omega,\omega(\boldsymbol{X})].$
(50)

Thus, for $\boldsymbol{X}=X^{v}$ we get $L_{X^{v}}\omega=dX+[\omega,X]=DX$. We
must remain mindful that if $X\notin\Omega^{0}_{\text{tens}}(P,\text{Ad})=$
Lie$\mathcal{H}$ then
$DX=dX+[\omega,X]\notin\Omega^{1}_{\text{tens}}(P,\text{Ad})$: only for
$X^{v}\in\mathfrak{aut}_{v}(P)\sim X\in$ Lie$\mathcal{H}$ does $DX$ answers
the geometric definition of the covariant derivative on tensorial forms. This
echoes the observation made earlier that
$\omega^{\gamma}=\psi^{*}\omega\notin\mathcal{C}$ for a generic
$\psi\in\operatorname{Diff}_{v}(P)\sim\gamma\in C^{\infty}(P,H)$.

As $\mathcal{C}$ is an affine space modeled on
$\Omega^{1}_{\text{tens}}(P,\text{Ad})$, for $\omega^{\prime},\omega\in C$ it
must be that $\omega^{\prime}-\omega\in\Omega^{1}_{\text{tens}}(P,\text{Ad})$.
So, in particular the linear action of a transformation group of $\mathcal{C}$
would result in an element of $\Omega^{1}_{\text{tens}}(P,\text{Ad})$. As a
matter of fact, $\operatorname{Aut}(P)$ is the maximal such group: Indeed, for
$\psi\in\operatorname{Aut}(P)$ it is the case that,

$\displaystyle\psi_{*}X(p)^{v}_{|p}\vcentcolon=\tfrac{d}{d\tau}\,\psi\big{(}\phi_{\tau}(p)\big{)}\,\big{|}_{\tau=0}=\tfrac{d}{d\tau}\,\psi\big{(}R_{\gamma_{\tau}(p)}\,p\big{)}\,\big{|}_{\tau=0}=\tfrac{d}{d\tau}\,R_{\gamma_{\tau}(p)}\psi(p)\,\big{|}_{\tau=0}=\vcentcolon
X(p)^{v}_{|\psi(p)}.$ (51)

So we have first
$\psi^{*}\omega_{|p}(X^{v}_{|p})=\omega_{|\psi(p)}(X^{v}_{|\psi(p)})=X$, and
second
$R^{*}_{h}\psi^{*}\omega=\psi^{*}R^{*}_{h}\omega=\psi^{*}\text{Ad}_{h^{-1}}\omega=\text{Ad}_{h^{-1}}\psi^{*}\omega$:
i.e. $\psi^{*}\omega\in\mathcal{C}$. It follows indeed that
$L_{\boldsymbol{X}}\omega\in\Omega^{1}_{\text{tens}}(P,\text{Ad})$ for
$\boldsymbol{X}\in\mathfrak{aut}(P)$, thus also for the special case
$\boldsymbol{X}=X^{v}\in\mathfrak{aut}_{v}(P)$ with explicit result
$L_{X^{v}}\omega=DX\in\Omega^{1}_{\text{tens}}(P,\text{Ad})$.

Applying a second Nijenhuis-Lie derivative we get, using the fact that
$[L_{\boldsymbol{X}},d]=0$:

$\displaystyle
L_{Y^{v}}L_{X^{v}}\omega=d(L_{Y^{v}}X)+[L_{Y^{v}}\omega,X]+[\omega,L_{Y^{v}}X]$
$\displaystyle=d\big{(}Y^{v}(X)\big{)}+[DY,X]+[\omega,Y^{v}(X)],$
$\displaystyle=D\big{(}Y^{v}(X)\big{)}+[dY,X]+\big{[}[\omega,Y],X\big{]}$

From this follows that,

$\displaystyle[L_{X^{v}},L_{Y^{v}}]\omega$
$\displaystyle=D\big{(}X^{v}(Y)\big{)}+[dX,Y]+\big{[}[\omega,X],Y\big{]}-D\big{(}Y^{v}(X)\big{)}-[dY,X]-\big{[}[\omega,Y],X\big{]},$
$\displaystyle=D\big{(}X^{v}(Y)\big{)}-D\big{(}Y^{v}(X)\big{)}+d\big{(}[X,Y]\big{)}+\big{[}[\omega,X],Y\big{]}-\big{[}[\omega,Y],X\big{]},$
$\displaystyle=D\big{(}X^{v}(Y)\big{)}-D\big{(}Y^{v}(X)\big{)}+d\big{(}[X,Y]\big{)}+\big{[}\omega,[X,Y]\big{]},$
$\displaystyle=D\big{(}\\{X,Y\\}\big{)}=L_{\\{X,Y\\}^{v}}\omega=L_{[X^{v},Y^{v}]_{\text{
\tiny{FN} }}}\omega,$ (52)

by Jacobi identity in Lie$H$ from the second to third line, and by definition
of the extended/FN bracket (16)-(2.3) from the third to the last. This result
is as expected from (48).

Similarly for a $\alpha\in\Omega^{\bullet}_{tens}(P,\rho)$ we have,

$\displaystyle
L_{X^{v}}\alpha\vcentcolon=\tfrac{d}{d\tau}\,\psi_{\tau}^{*}\,\alpha\,\big{|}_{|\tau=0}=\tfrac{d}{d\tau}\,\rho(\gamma^{-1}_{\tau})\alpha\,\big{|}_{|\tau=0}=-\rho_{*}(X)\alpha.$
(53)

This can be cross-checked via Cartan’s formula: For
$\boldsymbol{X}\in\Gamma(TP)$, we have in general

$\displaystyle L_{\boldsymbol{X}}\alpha$
$\displaystyle=\iota_{\boldsymbol{X}}d\alpha+d(\iota_{\boldsymbol{X}}\alpha)=\iota_{\boldsymbol{X}}\big{(}D\alpha-\rho_{*}(\omega)\alpha+d(\iota_{\boldsymbol{X}}\alpha)=\iota_{\boldsymbol{X}}D\alpha-\rho_{*}(\iota_{\boldsymbol{X}}\omega)\alpha+\rho_{*}(\omega)\iota_{\boldsymbol{X}}\alpha+d(\iota_{\boldsymbol{X}}\alpha)$
$\displaystyle=\iota_{\boldsymbol{X}}D\alpha+D(\iota_{\boldsymbol{X}}\alpha)-\rho_{*}(\iota_{\boldsymbol{X}}\omega)\alpha.$
(54)

Thus, for $\boldsymbol{X}=X^{v}$ we get $L_{X^{v}}\alpha=-\rho_{*}(X)\alpha$.
Here again, we notice that since generically
$X\notin\Omega^{0}_{\text{tens}}(P,\text{Ad})$, generically
$L_{X^{v}}\alpha\notin\Omega^{\bullet}_{tens}(P,\rho)$. Which echoes the
earlier observation that
$\alpha^{\gamma}=\psi^{*}\alpha\notin\Omega^{\bullet}_{tens}(P,\rho)$ for a
generic $\psi\in\operatorname{Diff}_{v}(P)\sim\gamma\in C^{\infty}(P,H)$. Upon
applying the Nijenhuis-Lie derivative again we get,

$\displaystyle
L_{Y^{v}}L_{X^{v}}\alpha=-\rho_{*}(L_{Y^{v}}X)\alpha-\rho_{*}(X)L_{Y^{v}}\alpha=-\rho_{*}\big{(}Y^{v}(X)\big{)}\alpha+\rho_{*}(X)\rho_{*}(Y)\alpha.$

Then it follows that,

$\displaystyle[L_{X^{v}},L_{Y^{v}}]\alpha$
$\displaystyle=-\rho_{*}\big{(}X^{v}(Y)\big{)}\alpha+\rho_{*}(Y)\rho_{*}(X)\alpha+\rho_{*}\big{(}Y^{v}(X)\big{)}\alpha-\rho_{*}(X)\rho_{*}(Y)\alpha,$
$\displaystyle=-\rho_{*}\big{(}[X,Y]+X^{v}(Y)-Y^{v}(X)\big{)}\alpha,$
$\displaystyle=-\rho_{*}\big{(}\\{X,Y\\}\big{)}\alpha=L_{\\{X,Y\\}^{v}}\alpha=L_{[X^{v},Y^{v}]_{\text{
\tiny{FN} }}}\alpha.$ (55)

Which is as expected from (48).

In the next and final section, we show how the global structures on $P$
exposed up to now descend as local structures on the base space $M$. These are
relevant to physics: e.g. vertical transformations generalise the usual gauge
transformations familiar in gauge field theory.

##  4 Local structure

We first summarize the standard local structure, and extend it to general
vertical transformations in the next section.

###  4.1 Gluings and local active gauge transformations

Given a open subset $U\subset M$ and a local section $\sigma:U\rightarrow
P_{|U}$, a local representative of $\beta\in\Omega^{\bullet}(P)$ is
$b\vcentcolon=\sigma^{*}\beta\in\Omega^{\bullet}(U)$. Any other local section
is $s^{\prime}=sg$ with $g:U\rightarrow H$ a transition function of the bundle
(encoding its topology, from a local viewpoint), so another local
representative is $b^{\prime}\vcentcolon={\sigma^{\prime}}^{*}\beta=b^{g}$.
The notation $b^{\prime}=b^{g}$ is meant to indicate that $b^{\prime}$ can be
seen as obtained from $b$ by a right action transformation by a transition
function $g$. For yet another local section
$s^{\prime\prime}=s^{\prime}g^{\prime}=sgg^{\prime}$, we have that
$b^{\prime\prime}=(b^{\prime})^{g^{\prime}}=b^{gg^{\prime}}$. These constitute
the _gluing relations_ of local representatives333The terminology stems from
considering $s$, $s^{\prime}$ and $s^{\prime\prime}$ as local sections over
overlapping opens $U$, $U^{\prime}$ and $U^{\prime\prime}$ respectively,
related by the transition functions $g$ and $g^{\prime}$. Then the gluing
relations reflects the fact that the local representatives on $M$ , the $b$’s,
comes from the same global object $\beta$ on $P$ – which can be reconstructed
from its local representatives. and are known in gauge field theory as
_passive gauge transformations_.

To illustrate, for $\beta=\omega\in\mathcal{C}$, we define
$A\vcentcolon=\sigma^{*}\omega\,\in\mathcal{A}$ (the gauge potential) and we
have the well known result
$A^{\prime}=\text{Ad}_{g^{-1}}A+g^{-1}dg=\vcentcolon A^{g}$, and
$A^{\prime\prime}=\text{Ad}_{{g^{\prime}}^{-1}}A^{\prime}+{g^{\prime}}^{-1}dg^{\prime}=A^{gg^{\prime}}$.
In the case of $\beta=\alpha\in\Omega^{\bullet}_{\text{tens}}(P,\rho)$, one
has $a\vcentcolon=\sigma^{*}\alpha\,\in\Omega^{\bullet}_{\text{tens}}(U,\rho)$
and it is well-known that $a^{\prime}=\rho(g^{-1})\,a=\vcentcolon a^{g}$, and
$a^{\prime\prime}=\rho({g^{\prime}}^{-1})\,a^{\prime}=\rho\big{(}(gg^{\prime})^{-1}\big{)}\,a=a^{gg^{\prime}}$.
The local representative of the curvature
$\Omega\in\Omega^{2}_{\text{tens}}(P,\text{Ad})$ of $\omega$ is
$F\vcentcolon=\sigma^{*}\Omega$. As a special case of the above, we have then
$F^{\prime}=\text{Ad}_{g^{-1}}F=\vcentcolon F^{g}$ and
$F^{\prime\prime}=\text{Ad}_{{g^{\prime}}^{-1}}F^{\prime}=F^{gg^{\prime}}$. A
matter field would be some $\alpha=\phi\in\Omega^{0}_{\text{tens}}(P,\rho)$
with local representative $a=\upphi$. Its covariant derivative
$D^{\omega}\phi=d\phi+\rho_{*}(\omega)\phi$ has local representative
$D^{A}\upphi=d\upphi+\rho_{*}(A)\upphi$, which represents the minimal coupling
to the gauge potential. It is well known that
$(D^{A}\upphi)^{g}=D^{A^{g}}\upphi^{g}=\rho(g^{-1})D^{A}\upphi$, which
expresses the requirement of the (passive) _gauge principle_.

Of course, the group $\operatorname{Aut}_{v}(P)$ cannot act on local
representatives, but there is naturally such a thing as the local
representative of the gauge transformed
$\beta^{\gamma}\vcentcolon=\psi^{*}\beta$, for
$\gamma\in\mathcal{H}\simeq\psi\in\operatorname{Aut}_{v}(P)$.

Given the defining equivariance of elements of $\mathcal{H}$, making them
tensorial 0-forms for the conjugate action of $H$, the
$\operatorname{Aut}_{v}(P)\simeq\mathcal{H}$-transformation of
$\eta\in\mathcal{H}$ is – as a special case of the one discussed below (36) –
$\eta^{\gamma}\vcentcolon=\psi^{*}\eta=\gamma^{-1}\eta\gamma$. This relation
defines the action of gauge group $\mathcal{H}$ on itself. Defining
$\upgamma\vcentcolon=\sigma^{*}\gamma$ and $\upeta\vcentcolon=\sigma^{*}\eta$,
we have
$\upeta^{\upgamma}\vcentcolon=\sigma^{*}(\eta^{\gamma})=\upgamma^{-1}\upeta\upgamma$.
Therefore, the local representative of $\mathcal{H}$, the _local gauge group_
, is defined as:
$\mathcal{H}_{\text{\tiny{loc}}}\vcentcolon=\\{\upeta,\upgamma:U\rightarrow
H\,|\,\upeta^{\upgamma}=\upgamma^{-1}\upgamma\upgamma\\}$.

We have therefore the local representative of a gauge transformed form:
$b^{\upgamma}\vcentcolon=\sigma^{*}(\beta^{\gamma})$. Given that
$(\beta^{\eta})^{\gamma}=\beta^{\eta\gamma}$, it is the case that
$(b^{\upeta})^{\upgamma}=b^{\upeta\upgamma}$. This secures the consistency of
a heurisitic “field theoretic rule”: Considering $\upgamma$ as acting on
$b^{\upeta}$ as a concatenation of the fields $b$ and $\upeta$, one may write
$(b^{\upeta})^{\upgamma}=(b^{\upgamma})^{\upeta^{\upgamma}}$, using then their
respective $\mathcal{H}_{\text{\tiny{loc}}}$-transformations, one writes the
concatenation as
$(b^{\upgamma})^{\upeta^{\upgamma}}=(b^{\upgamma})^{\upgamma^{-1}\upeta\upgamma}=b^{\upgamma\,\upgamma^{-1}\upeta\upgamma}=b^{\upeta\upgamma}$.

The relation $(b^{\upeta})^{\upgamma}=b^{\upeta\upgamma}$ expresses the fact
that the action of the local gauge group $\mathcal{H}_{\text{\tiny{loc}}}$ on
local representatives – the fields of physics – is a right action. Hence, the
field space $\Phi=\\{b\\}$ can be seen (under proper restrictions) as an
infinite dimensional bundle with structure group
$\mathcal{H}_{\text{\tiny{loc}}}$. A fact often overlooked in the covariant
phase space literature – with rare exceptions, see e.g. [1; 16; 15; 14; 11].

To illustrate as above, we have that
$A^{\upgamma}\vcentcolon=\sigma^{*}(\omega^{\gamma})=\text{Ad}_{\upgamma^{-1}}A+\upgamma^{-1}d\upgamma$.
And $(A^{\upeta})^{\upgamma}=A^{\upeta\upgamma}$, as can be checked from using
the previous definition and
$(A^{\upeta})^{\upgamma}=(A^{\upgamma})^{\upeta^{\upgamma}}$. Similarly for a
tensorial form we have
$a^{\upgamma}\vcentcolon=\sigma^{*}(\alpha^{\gamma})=\rho(\upgamma^{-1})\,a$,
from which one checks that $(a^{\upeta})^{\upgamma}=a^{\upeta\upgamma}$ via
$(a^{\upeta})^{\upgamma}=a^{\upeta\upgamma}$. As a special cases,
$F^{\upgamma}\vcentcolon=\sigma^{*}(\Omega^{\gamma})=\text{Ad}_{\upgamma^{-1}}F$,
and
$\upphi^{\upgamma}\vcentcolon=\sigma^{*}(\phi^{\gamma})=\rho(\upgamma^{-1})\,\upphi$
while its covariant derivative is
$(D^{A}\upphi)^{\upgamma}\vcentcolon=\sigma^{*}\big{(}(D^{\omega}\phi)^{\gamma}\big{)}=\rho(\upgamma^{-1})\,D^{A}\upphi$.
Here, the local covariant derivative operator can be _defined_ as the
expression $D\vcentcolon=d\ +\rho_{*}(A)$. It is then easily checked that the
previous result is also recovered from the field theoretic rule:
$(D^{A}\upphi)^{\upgamma}=D^{A^{\upgamma}}\upphi^{\upgamma}=\rho(\upgamma^{-1})\,D^{A}\upphi$,
which is yet another expression of the gauge principle.

Notice that, from the standpoint of gauge field theory, there is no way to
distinguish between local gluings $b^{g}$ and the action $b^{\upgamma}$ of
$\mathcal{H}_{\text{\tiny{loc}}}$, i.e. between passive gauge transformations
and local active gauge transformations. The “gauge principle” therefore
actually encapsulates two gauge principles: a passive and an active one. Yet,
the conceptual meaning of each is quite different, as different as coordinate
changes (a.k.a. passive diffeomorphisms) and (active) diffeomorphisms
$\operatorname{Diff}(M)$ are in GR.

A theory being given by a Lagrangian $L$, invariance under gluings,
$L(b)=L(b^{g})$, mean that a gauge field theory is only sensitive to the
intrinsic geometry of the bundle $P$, and to global object $\beta$ living in
it. Invariance under $\mathcal{H}_{\text{\tiny{loc}}}$,
$L(b)=L(b^{\upgamma})$, means that a gauge field theory is sensitive only to
the geometry of the $\operatorname{Aut}_{v}(P)$-class of $P$, i.e. to the
$\mathcal{H}$-orbits of global objects. In the same way that the combination
of the hole argument and the point-coincidence argument [17] clarified that
$\operatorname{Diff}(M)$-invariance in GR encodes the _relational_ nature of
spacetime, and of general relativistic physics more generally, arguably a
combination of an “internal hole argument” and “internal point-coincidence
argument” suggests that $\mathcal{H}_{\text{\tiny{loc}}}$-invariance encodes
the _relational_ character of the enriched spacetime (represented by the
principal bundle) and of gauge field physics more generally. We will elaborate
on this in a forthcoming work.

###  4.2 Linear versions

All of the above naturally have linear versions. Consider a transition
function $g_{\tau}:U\rightarrow H$, so that $\sigma^{\prime}_{\tau}=\sigma
g_{\tau}$, s.t. $g_{\tau=0}=\text{id}_{\text{\tiny{H}}}$. The object
$\lambda=\tfrac{d}{d\tau}g_{\tau}\,\big{|}_{\tau=0}:U\rightarrow$ Lie$H$ is an
infinitesimal transition function of the bundle $P$. We define
$\delta_{\\!\lambda\,}b\vcentcolon=\tfrac{d}{d\tau}\,b^{g_{\tau}}\,\big{|}_{\tau=0}$:
It is the limit of the difference between the local representatives obtained
via $\sigma^{\prime}_{\tau}$ and $\sigma$ respectively. Such infinitesimal
gluing can be called an infinitesimal passive gauge transformation.

A commutator
$[\delta_{\\!\lambda},\delta_{\\!\lambda^{\prime}}]b=\delta_{[\lambda,\,\lambda^{\prime}]_{\text{\tiny{Lie$H$}}}}b$
arises from the definition
$[\delta_{\\!\lambda},\delta_{\\!\lambda^{\prime}}]b\vcentcolon=\tfrac{d}{ds}\tfrac{d}{d\tau}\,b^{g_{\tau}g^{\prime}_{s}g^{-1}_{\tau}{g^{\prime}}^{-1}_{s}}\,\big{|}_{\tau=0}\big{|}_{s=0}$
– i.e. the commutator in $H$. It is of course recovered when one has concrete
expressions for the result of $\delta_{\\!\lambda\,}b$, and considering
$\delta_{\\!\lambda}$ as an even graded derivation on $\Omega^{\bullet}(U)$
s.t. $[\delta_{\\!\lambda\,},d]=0$, and whose action on $b$’s is defined by
such expressions. The commutator is then of course
$[\delta_{\\!\lambda},\delta_{\\!\lambda^{\prime}}]=\delta_{\\!\lambda}\delta_{\\!\lambda^{\prime}}-\delta_{\\!\lambda}\delta_{\\!\lambda^{\prime}}$.
Let us illustrate.

From above, we obtain well-known relations, for $A\in\mathcal{A}$ and
$a\in\Omega^{\bullet}_{\text{tens}}(U,\rho)$: First,
$\delta_{\\!\lambda\,}A\vcentcolon=\tfrac{d}{d\tau}\,A^{g_{\tau}}\,\big{|}_{\tau=0}=d\lambda-\text{ad}_{\lambda}A=d\lambda+[A,\lambda]=D^{A}\\!\lambda$.
Then,
$\delta_{\\!\lambda\,}a\vcentcolon=\tfrac{d}{d\tau}\,a^{g_{\tau}}\,\big{|}_{\tau=0}=-\rho_{*}(\lambda)\,a$.
As special case we have,
$\delta_{\\!\lambda\,}F=-\text{ad}_{\lambda}F=[F,\lambda]$,
$\delta_{\\!\lambda\,}\upphi=-\rho_{*}(\lambda)\upphi$ and
$\delta_{\\!\lambda\,}D^{A}\upphi=-\rho_{*}(\lambda)D^{A}\upphi$. The latter
result is recovered, seeing $\delta_{\\!\lambda}$ as an even derivation, via
$\delta_{\\!\lambda\,}D^{A}\upphi=d\delta_{\\!\lambda\,}\upphi+[\delta_{\\!\lambda\,}A,\upphi]-[A,\delta_{\\!\lambda\,}\upphi]$.
In the same algebraic way, it it easily verified on $A$ and $a$ that
$[\delta_{\\!\lambda},\delta_{\\!\lambda^{\prime}}]A=\delta_{\\!\lambda}\delta_{\\!\lambda^{\prime}}A-\delta_{\\!\lambda}\delta_{\\!\lambda^{\prime}}A=D^{A}\big{(}[\lambda,\,\lambda^{\prime}]_{\text{\tiny{Lie$H$}}}\big{)}=\delta_{[\lambda,\,\lambda^{\prime}]_{\text{\tiny{Lie$H$}}}}A$,
and that
$[\delta_{\\!\lambda},\delta_{\\!\lambda^{\prime}}]a=\delta_{\\!\lambda}\delta_{\\!\lambda^{\prime}}a-\delta_{\\!\lambda}\delta_{\\!\lambda^{\prime}}a=-\rho_{*}\big{(}[\lambda,\lambda^{\prime}]_{\text{\tiny{Lie$H$}}}\big{)}=\delta_{[\lambda,\,\lambda^{\prime}]_{\text{\tiny{Lie$H$}}}}a$.
These computations are familiar in gauge field theory.

Given the infinitesimal equivariance of elements of Lie$\mathcal{H}$ (see
below (17)), making them tensorial 0-form for the Ad-representation of $H$,
the $\operatorname{Aut}_{v}(P)\simeq\mathcal{H}$-transformation of $Y\in$
Lie$H$ is
$Y^{\gamma}\vcentcolon=\psi^{*}Y=\text{Ad}_{\gamma^{-1}}X=\gamma^{-1}X\gamma$.
The corresponding $\mathfrak{aut}_{v}(P)\simeq$
Lie$\mathcal{H}$-transformation is
$L_{X^{v}}Y=X^{v}(Y)=-\text{ad}_{X}Y=[Y,X]_{\text{\tiny{Lie$H$}}}$. Thus,
defining $\xi\vcentcolon=\sigma^{*}X$ and $\zeta\vcentcolon=\sigma^{*}Y$, we
have locally: $\zeta^{\upgamma}=\text{Ad}_{\upgamma^{-1}}\zeta$ and the
corresponding linearisation
$\delta_{\xi}\zeta\vcentcolon=\sigma^{*}(L_{X^{v}}Y)=[\zeta,\xi]_{\text{\tiny{Lie$H$}}}$.
We define the Lie algebra of the local gauge group as
Lie$\mathcal{H}_{\text{\tiny{loc}}}\vcentcolon=\\{\xi,\zeta:U\rightarrow\text{Lie}H\
|\ \delta_{\xi}\zeta=[\zeta,\xi]_{\text{\tiny{Lie$H$}}}\\}$.

Local active infinitesimal gauge transformations of $b$ are defined as
$\delta_{\xi}b\vcentcolon=\sigma^{*}(L_{X^{v}}\beta)$. Here, $\delta_{\xi}$ is
immediately seen as an even derivation on $\Omega^{\bullet}(U)$ s.t.
$[\delta_{\xi},d]=0$ arising from the Lie derivative on $P$ along
$X^{v}\in\mathfrak{aut}_{v}(P)$. So, naturally, a Lie bracket arises by
$[\delta_{\xi},\delta_{\zeta}]b\vcentcolon=\sigma^{*}\big{(}[L_{X^{v}},L_{Y^{v}}]\,\beta\big{)}=\sigma^{*}\big{(}L_{-[X,Y]_{\text{\tiny{Lie$H$}}}}\,\beta\big{)}=-\delta_{[\xi,\,\zeta]_{\text{\tiny{Lie$H$}}}}b$.
See again below (17). This is of course cross-checked by writing the bracket
as a commutator
$[\delta_{\xi},\delta_{\zeta}]=\delta_{\xi}\delta_{\zeta}-\delta_{\zeta}\delta_{\xi}$,
from explicit expressions for $\delta_{\xi}b$ seen as defining the action of
$\delta_{\xi}$ on the $b$’s and by using the action
$\delta_{\xi}\zeta=[\zeta,\xi]_{\text{\tiny{Lie$H$}}}$ of Lie$\mathcal{H}$ on
its elements. Let us illustrate.

For $A\in\mathcal{A}$ and $a\in\Omega^{\bullet}_{\text{tens}}(U,\rho)$, we
obtain: $\delta_{\xi}A\vcentcolon=\sigma^{*}(L_{X^{v}}\omega)=D^{A}\xi$ and
$\delta_{\xi}a\vcentcolon=\sigma^{*}(L_{X^{v}}\alpha)=-\rho_{*}(\xi)\,a$. As
special case we have, $\delta_{\xi}F=-\text{ad}_{\xi}F=[F,\xi]$,
$\delta_{\xi}\upphi=-\rho_{*}(\xi)\,\upphi$ and
$\delta_{\xi}\,D^{A}\upphi=-\rho_{*}(\xi)\,D^{A}\upphi$. This last result is
recovered algebraically by
$\delta_{\xi}\,D^{A}\upphi=d\delta_{\xi}\upphi+[\delta_{\xi}A,\upphi]-[A,\delta_{\xi}\upphi]$.
In the same way, one easily shows:
$[\delta_{\xi},\delta_{\zeta}]A=\delta_{\xi}\delta_{\zeta}A-\delta_{\zeta}\delta_{\xi}A=-D^{A}\big{(}[\xi,\zeta]_{\text{\tiny{Lie$H$}}}\big{)}=-\delta_{[\xi,\,\zeta]_{\text{\tiny{Lie$H$}}}}A$,
and that
$[\delta_{\xi},\delta_{\zeta}]a=\delta_{\xi}\delta_{\zeta}a-\delta_{\zeta}\delta_{\xi}a=\rho_{*}\big{(}[\xi,\zeta]_{\text{\tiny{Lie$H$}}}\big{)}\,a=-\delta_{[\xi,\,\zeta]_{\text{\tiny{Lie$H$}}}}a$.

We observe again that, from a gauge field theoretic perspective,
Lie$\mathcal{H}_{\text{\tiny{loc}}}$-transformations are indistinguishable
from infinitesimal gluings (apart from a sign difference in their
commutators). Both can be encapsulated via the BRST framework [18]: There, one
extends $\Omega^{\bullet}(U)$ to the bigraded complex
$\Omega^{\bullet}(U,\rho)\otimes\wedge^{\bullet}c$, where $c$ is the odd
degree (Grassmann) Lie$H$-valued ghost field, place-holder for the parameters
$\lambda$ or $\xi\in$ Lie$\mathcal{H}_{\text{\tiny{Loc}}}$. The BRST
differential $s$, s.t. $s^{2}=0$ and $sd=-ds$, is introduced, playing the role
of $\delta_{\\!\lambda\,}$ or $\delta_{\xi}$. The action of $s$ on
$b=\\{A,a\\}$ and $c$ is by definition: $sA\vcentcolon=-Dc=-dc-Ac-cA$ (a
bigraded bracket is understood in $Dc$), $sa\vcentcolon=-\rho_{*}(c)\,\phi$,
and $sc\vcentcolon=\nicefrac{{1}}{{2}}[c,c]_{\text{\tiny{Lie$H$}}}$. The third
relation enforces $s^{2}=0$ on all fields, while the first two reproduces
infinitesimal gauge transformations. This algebraic treatment, efficient as it
is, erases the key conceptual difference between passive and active local
gauge transformations, i.e. between gluings and
Lie$\mathcal{H}_{\text{\tiny{Loc}}}$.

As we show below, the local version of general vertical transformations
clearly distinguishes itself from local gluings. This opens the possibility
for the BRST formalism to be modified accordingly, which we elaborate on in
the conclusion.

###  4.3 Generalised local active gauge transformations

Like $\operatorname{Aut}_{v}(P)\simeq\mathcal{H}$, the group
$\operatorname{Diff}_{v}(P)\simeq C^{\infty}(P,H)$ cannot act on local
representatives $b$ of forms $\beta$ on $P$. Yet, one can define the local
representatives of $\beta^{\gamma}\vcentcolon=\psi^{*}\beta$ for $\gamma\in
C^{\infty}(P,H)\sim\psi\in\operatorname{Diff}_{v}(P)$.

As the equivariance of elements in $C^{\infty}(P,H)$ are left unspecified, so
are both their $\operatorname{Aut}_{v}(P)\simeq\mathcal{H}$ and
$\operatorname{Diff}_{v}(P)$ transformations. We thus simply use the generic
notation $\eta^{\gamma}\vcentcolon=\psi^{*}\eta=\eta\circ R_{\gamma}$, for
$\gamma,\eta\in C^{\infty}(P,H)$. By (7) the composition law in
$\operatorname{Diff}_{v}(P)$ is thus represented by the element
$\gamma\,\eta^{\gamma}\in C^{\infty}(P,H)$.

We define $\upgamma\vcentcolon=\sigma^{*}\gamma$, and
$C^{\infty}(U,H)\vcentcolon=\\{\upgamma:U\rightarrow H\\}$ is the local
version of the group of vertical transformations, so we may call it the group
of _generalised local active gauge transformations_. Its action on local
representatives is given by definition as
$b^{\upeta}\vcentcolon=\sigma^{*}(\beta^{\eta})$. By (36), the iteration law
is given by
$(b^{\upeta})^{\upgamma}\vcentcolon=\sigma^{*}\big{(}(\beta^{\eta})^{\gamma}\big{)}=\sigma^{*}\big{(}\beta^{\gamma\,\eta^{\gamma}}\big{)}=b^{\upgamma\,\upeta^{\upgamma}}\\!$.
This again secures the consistency of “field theoretic rule”: Considering
$\upgamma$ as acting on $b^{\upeta}$ as a concatenation of the fields $b$ and
$\upeta$, one may write
$(b^{\upeta})^{\upgamma}=(b^{\upgamma})^{\upeta^{\upgamma}}$, which results
indeed in the concatenation $b^{\upgamma\,\upeta^{\upgamma}}\\!$.

For example, by (34)-(35) and (36), the generalised gauge transformations of
the gauge potential $A\in\mathcal{A}$ and
$a\in\Omega^{\bullet}_{\text{tens}}(U,\rho)$ are:

$\displaystyle A^{\upeta}$
$\displaystyle=\text{Ad}_{\upeta^{-1}}A+\upeta^{-1}d\upeta\quad$
$\displaystyle\text{and}\quad(A^{\upeta})^{\upgamma}=\text{Ad}_{(\upgamma\,\upeta^{\upgamma})^{-1}}A+(\upgamma\,\upeta^{\upgamma})^{-1}d(\upgamma\,\upeta^{\upgamma}),$
(56) $\displaystyle a^{\upeta}$ $\displaystyle=\rho(\upeta)^{-1}a\quad$
$\displaystyle\text{and}\quad(a^{\upeta})^{\upgamma}=\rho(\upgamma\,\upeta^{\upgamma})^{-1}a.$

The second line gives in particular the generalised transformations of the
field strength $F$, matter field $\upphi$ and its covariant derivative
$D^{A}\upphi$. The latter is also found via the field-theoretic computation:
$(D^{A}\upphi)^{\eta}=D^{A^{\upeta}}\upphi^{\upeta}$, idem for iterated
transformations. This is noteworthy: It shows that the operator
$D\vcentcolon=d\ +\rho_{*}(A)$ preserves the covariance of $a$, or $\phi$, and
deserve the name “covariant derivative”, even under the action
$C^{\infty}(U,H)$, extending that of $\mathcal{H}_{\text{\tiny{loc}}}$. And
this, as discussed at the end of section 3.2, despite the fact that
$A^{\upeta}$ is not the local representative of a connection, since
$\omega^{\eta}\notin\mathcal{C}$, and that $a^{\upeta}$ is not the local
representative of a tensorial form, since
$\alpha^{\eta}\notin\Omega^{\bullet}_{\text{tens}}(P,\rho)$. This shows that a
Gauge Principle, or Gauge Argument, applies still to generalised gauge
transformations.

A Lagrangian invariant under gluings, $L(b^{g})=L(b)$, will then not only be
invariant under $\mathcal{H}_{\text{\tiny{loc}}}$ as previously mentioned, but
also under $C^{\infty}(U,H)$:
$L\big{(}b^{\upgamma\,\upeta^{\upgamma}}\big{)}=L(b^{\upeta})=L(b)$. Gauge
field theories thus naturally enjoys a larger group of gauge symmetries,
arising from $\operatorname{Diff}_{v}(P)$.

Let us consider the infinitesimal counterpart of the above. The Lie algebra of
generalised gauge transformations is the local version of
$C^{\infty}(P,\text{Lie}H)$ equipped with the extended bracket (16). Given
that the infinitesimal equivariance of elements of $C^{\infty}(P,\text{Lie}H)$
is left unspecified, so is their their
$\operatorname{Aut}_{v}(P)\simeq\mathcal{H}$ and $\operatorname{Diff}_{v}(P)$
transformations. We thus have, for the action of $\gamma\in C^{\infty}(P,H)$
on $Y\in C^{\infty}(P,\text{Lie}H)$, the generic notation
$Y^{\gamma}\vcentcolon=\psi^{*}Y$. Correspondingly, the action of $X\in
C^{\infty}(P,H)$ is noted $L_{X^{v}}Y=X^{v}(Y)$. Thus, defining
$\xi\vcentcolon=\sigma^{*}X$ and $\zeta\vcentcolon=\sigma^{*}Y$, we have
locally: $\zeta^{\upgamma}\vcentcolon=s^{*}(Y^{\gamma})$ and the corresponding
linearisation $\delta_{\xi\,}\zeta\vcentcolon=\sigma^{*}(L_{X^{v}}Y)$. The Lie
algebra of infinitesimal generalised gauge transformations is then
$C^{\infty}(U,\text{Lie}H)$ equipped with the local Frölicher-Nijenhuis
bracket (16)-(2.3):

$\displaystyle\big{\\{}\xi,\zeta\big{\\}}=[\xi,\zeta]_{\text{\tiny{Lie$H$}}}+\delta_{\xi\,}\zeta-\delta_{\zeta\,}\xi,$
(57)

We may call this a generalised gauge algebra.

Local infinitesimal generalised gauge transformations of $b$ are defined by
$\delta_{\xi}b\vcentcolon=\sigma^{*}(L_{X^{v}}\beta)$, given (47), where
$\delta_{\xi}$ is immediately seen as an even derivation on
$\Omega^{\bullet}(U)$, s.t. $[\delta_{\xi},d]=0$, arising from the Nijenhuis-
Lie derivative on $P$ along $X^{v}\in\mathfrak{diff}_{v}(P)$. Naturally, a Lie
bracket for $\delta_{\xi}$’s arises via
$[\delta_{\xi},\delta_{\zeta}]\,b\vcentcolon=\sigma^{*}\big{(}[L_{X^{v}},L_{Y^{v}}]\,\beta\big{)}$,
i.e. from the commutator of the Nijenhuis-Lie derivatives (17), which by (48)
gives:

$\displaystyle\,b$ $\displaystyle=\delta_{\\{\xi,\,\zeta\\}\,}b.$ (58)

Local fields are representations for the generalised gauge algebra. This is
obtained algebraically too, writing the commutator
$[\delta_{\xi},\delta_{\zeta}]=\delta_{\xi}\delta_{\zeta}-\delta_{\zeta}\delta_{\xi}$,
and using explicit expressions for $\delta_{\xi}b$ seen as defining the action
of $\delta_{\xi}$ on the $b$’s, as well as the action $\delta_{\xi}\zeta$ of
$C^{\infty}(U,\text{Lie}H)$ on its elements.

The illustration with $A\in\mathcal{A}$ and
$a\in\Omega^{\bullet}_{\text{tens}}(U,\rho)$ is formally as before: we obtain
from (49)-(53),

$\displaystyle\delta_{\xi}A\vcentcolon=\sigma^{*}(L_{X^{v}}\omega)=D^{A}\xi,\quad\text{and}\quad\delta_{\xi}a\vcentcolon=\sigma^{*}(L_{X^{v}}\alpha)=-\rho_{*}(\xi)\,a.$
(59)

As special case of the second equation we have,
$\delta_{\xi}F=-\text{ad}_{\xi}F=[F,\xi]$,
$\delta_{\xi}\upphi=-\rho_{*}(\xi)\,\upphi$ and
$\delta_{\xi}\,D^{A}\upphi=-\rho_{*}(\xi)\,D^{A}\upphi$. The last result being
recovered algebraically by
$\delta_{\xi}\,D^{A}\upphi=d\delta_{\xi}\upphi+[\delta_{\xi}A,\upphi]-[A,\delta_{\xi}\upphi]$.
In the same algebraic way, one checks the commutator on $a$:

$\displaystyle\,a$
$\displaystyle=-\delta_{\xi}\,\rho_{*}(\zeta)\,a+\delta_{\zeta}\,\rho_{*}(\xi)\,a,$
(60)
$\displaystyle=-\rho_{*}(\delta_{\xi\,}\zeta)\,a+\rho_{*}(\zeta)\rho_{*}(\xi)\,a\
+\ \rho_{*}(\delta_{\zeta\,}\xi)\,a-\rho_{*}(\xi)\rho_{*}(\zeta)\,a,$
$\displaystyle=-\rho_{*}\left([\xi,\zeta]_{\text{\tiny{Lie$H$}}}+\delta_{\xi\,}\zeta-\delta_{\zeta\,}\xi]\right)\,a,$
$\displaystyle=-\rho_{*}\big{(}\big{\\{}\xi,\zeta\big{\\}}\big{)}\,a,$
$\displaystyle=\delta_{\\{\xi,\,\zeta\\}\,}a.$

The local version of (3.3). Similarly, one finds:

$\displaystyle[\delta_{\xi},\delta_{\zeta}]\,A=\delta_{\xi}\delta_{\zeta}A-\delta_{\zeta}\delta_{\xi}A=D^{A}\big{(}\\{\xi,\zeta\\}\big{)}=\delta_{\\{\xi,\,\zeta\\}\,}A,$
(61)

as the local counterpart of (3.3)

The bracket (58) is found in the physics literature, notably in the covariant
phase space literature, as pointed out already. It is often derived
heuristically: Terms like $\delta_{\xi\,}\zeta$ are assumed to arise because
of a field-dependence of the gauge parameters, i.e.
$\zeta=\zeta(b)=\zeta(A,\upphi)$ – itself arising either because of gauge
fixing or because boundary conditions must be preserved. As we have
demonstrated in this paper, such is not necessary the case.

Yet, there is indeed a geometric arena where field-dependent parameters
$\zeta=\zeta(b)$ arise naturally: When one considers the field space
$\Phi=\\{b\\}$ of a gauge theory as a principal bundle with structure group
$\mathcal{H}_{\text{\tiny{loc}}}$. The gauge group
$\boldsymbol{\operatorname{Aut}}_{v}(\Phi)$ of $\Phi$, or its group of
vertical automorphisms $\boldsymbol{\operatorname{Diff}}_{v}(\Phi)$, are the
geometric underpinning of the notion of field-dependent gauge transformations.
Then, objects like $\zeta=\zeta(b)$ belong to
$C^{\infty}\big{(}\Phi,\text{Lie}\mathcal{H}_{\text{\tiny{loc}}}\big{)}\simeq\boldsymbol{\mathfrak{diff}}_{v}(\Phi)$,
and (58)-(57) arise from the Nijenhuis-Lie derivative and Frölicher-Nijenhuis
bracket on $\Phi$. See for a [11] for a detailed treatment of the case where
$\operatorname{Diff}(M)$ is the structure group of $\Phi$.

##  5 Conclusion

In this paper, we have detailed the global and local geometry arising from the
group of vertical diffeomorphisms $\operatorname{Diff}_{v}(P)\simeq
C^{\infty}(U,H)$ of a principal bundle $P$. Notably, we have shown that its
Lie algebra is realised as by the Nijenhuis-Lie derivative: This shows that
the extended bracket often encountered in the gauge field theory literature,
mainly on gravity [7; 8] and its asymptotic symmetries (BMS and extensions)
[6; 1; 2; 3; 5; 4] is but an instance of the Frölicher-Nijenhuis bracket.

The local counterpart $C^{\infty}(U,H)$ of the general vertical
transformations of $P$, that we called generalised local active gauge
transformations, is manifestly distinct from the local gluings, contrary to
standard local active gauge transformations $\mathcal{H}_{\text{\tiny{loc}}}$
arising from vertical automorphisms $\operatorname{Aut}_{v}(P)$. In
consequence, since the usual BRST framework [18] indistinguishably encodes
both infinitesimal gluings and Lie$\mathcal{H}_{\text{\tiny{loc}}}$, one may
inquire as to the possibility that it must be adjusted to encode
$C^{\infty}(U,LieH)$. It is known that BRST cohomology is just the Chevaley-
Eilenberg (CE) cohomology of Lie$\mathcal{H}_{\text{\tiny{loc}}}$, or
$\mathcal{H}\simeq\operatorname{Aut}_{v}(\mathcal{P})$, with coefficient in
$\Omega^{\bullet}(U)$ (or local polynomials in the fields
$\\{b\\}=\\{A,\upphi,\cdots\\}$) [19; 20]. Perhaps the extended BRST framework
is just CE-cohomology of $C^{\infty}(U,H)\simeq\operatorname{Diff}_{v}(P)$. We
will investigate this further elsewhere.

As we observed, most available treatments of bundle geometry do not mention
$\operatorname{Diff}_{v}(P)$. A good reason for this may be that, since its
elements generically do no commute with the right action of the structure
group $H$, stricto sensu these are not bundle automorphisms: i.e. they are not
morphisms in the category of principal bundles. They are not “relevant
structures” from a categorical perspective. Indeed, the action of
$\operatorname{Diff}_{v}(P)$ on objects usually well defined for that
category, such as the spaces of connections, equivariant and tensorial forms,
is “problematic” (it doesn’t preserve those spaces) even if definable and
possible to work out. Still, these are maps from a bundle to itself,
preserving the fibration. One may thus enquire as to what category these are
natural arrows of.

At any rate, this is not an issue for local gauge field theory, which can
accommodate this extended notion of gauge transformation: a gauge argument
still carries through, and the covariant derivative still does its job of
preserving covariance under the action of $C^{\infty}(U,H)$. In a companion
paper [11], we show how the the finite dimensional geometry exposed here may
extend to the infinite dimensional bundle geometry of the field space
$\Phi=\\{b\\}$ of a gauge theory: there, it further clarifies the literature
mentioned above.

## Acknowledgment

This work was funded by the OP J.A.C MSCA grant, number
CZ.02.01.01/00/22_010/0003229, co-funded by the Czech government Ministry of
Education, Youth & Sports and the EU. This research was also funded in part by
the Austrian Science Fund (FWF), [P 36542]. Support from the service _Physics
of the Universe, Fields and Gravitation_ at UMONS (BE) is also acknowledged.

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