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CREF895.tmp
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CREF895.tmp
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\documentclass[a4paper,oneside,11pt,bibliography=totoc]{scrartcl}
%Use bibliography=totoc or bibliography=totocnumbered to show the List of References in the table of contents, numbered if using the latter
\usepackage{CJKutf8}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
%\usepackage{exscale}
%\newcommand\hmmax{0}
%\newcommand\bmmax{1}
%\newcommand{\Plus}{\mathord{\tikz\draw[line width=0.3ex, x=1ex, y=1ex] (0.75,0) -- (0.75,1.5)(0,0.75) -- (1.5,0.75);}}
\def\RPlus{\ensuremath{\mathbin{\rule[.13em]{.66em}{.22em}\hspace{-.44em}\rule[-.08em]{.22em}{.66em}\,}}} %Fat plus symbol
\usepackage[english]{babel}
\usepackage{graphicx} %extended \includegraphics options
%\usepackage{CJKutf8} %Kanji signs
\usepackage{dsfont}%even more mathematical symbols
%\usepackage{bm} %use \bm to get bold math symbols
\usepackage{amssymb} %even more mathematical symbols
\usepackage{amsthm} %theorem style usually needed for AMS (american mathematical society)
\usepackage{amsmath} %important stuff like equation structures and so on
\usepackage{amsfonts} %new mathematical symbols like fractals
%\usepackage{amscd}
%\usepackage{amstext}
%\usepackage{mathabx}
\usepackage{booktabs} %enhanced table stuff
%\usepackage{amsbsy} %bold mathematical symbols, also loaded in amsmath
\usepackage{fancybox} %for theorem boxes
%\usepackage[hypcap=false]{caption} %new captions for floating options
%\captionsetup{font={small,sf},labelfont=bf,textfont=rm}
%\usepackage{float} %provides H for images-here
%\usepackage{subfigure} %citing subfigures in a figure
\usepackage{lmodern} %latin modern font
\usepackage{fixcmex} %Fixing the issue with scaling brackets etc. in lmodern
%\renewcommand\familydefault{lmr}% uses lm only for text
%\usepackage[numbib,nottoc]{tocbibind} %Options for including the bibliography to content list etc.; nottoc prevents listing of Contents in the list of contents, numbib also numbers the section of the biblio
%\usepackage[pdfstartview={FitH},linkbordercolor={0 1 0}]{hyperref}
\usepackage[pdfstartview={FitH},linkbordercolor={0 1 0},colorlinks]{hyperref}
\usepackage{esint} %für \oiint und andere Integralformen
%\usepackage{fancyhdr} %Different handling for footers and headers
\usepackage[headsepline=.5pt,footsepline=.5pt,automark]{scrlayer-scrpage}
\usepackage{pdfpages} %For \includepdf etc
%\usepackage{trfsigns} % Fourier and Laplacesymbols for engineering und für \e für \exp
\newcommand{\e}{\ensuremath{\mathrm{e\;\!}}}
%\usepackage{wasysym} %Smiley
%\usepackage{siunitx} %SI Einheiten
%\usepackage{eqnarray} %Gleichungen in Tabellenform
\usepackage{mathtools} %for mathclap and coloneqq
\usepackage{relsize} %Integralgröße
\usepackage{scrhack} %Warnung bei book bezüglich toc
\usepackage{mleftright} %Distanz zu \left \right weg
\usepackage{tikz} %Diagramme
\usepackage{stmaryrd} % fuer llbracket usw.
\usepackage[many]{tcolorbox} %Coloured boxes around theorems etc
\tcbuselibrary{theorems} % siehe oben
%\usepackage{float} %For H descriptor in figure, but also for allowing figure in minipages
%\usepackage{xfrac} %for \sfrac quotient of sets
\usetikzlibrary{cd}
%For small mathcal letters:
\makeatletter
\DeclareFontEncoding{LS1}{}{}
\DeclareFontSubstitution{LS1}{stix}{m}{n}
\DeclareMathAlphabet{\mathKel}{LS1}{stixscr}{m}{n}
\DeclareMathAlphabet{\mathcal}{LS1}{stixscr}{m}{n}
%Big tilde
\def\oversortoftilde#1{\mathop{\vbox{\m@th\ialign{##\crcr\noalign{\kern3\p@}%
\sortoftildefill\crcr\noalign{\kern3\p@\nointerlineskip}%
$\hfil\displaystyle{#1}\hfil$\crcr}}}\limits}
\def\sortoftildefill{$\m@th \setbox\z@\hbox{$\braceld$}%
\braceld\leaders\vrule \@height\ht\z@ \@depth\z@\hfill\braceru$}
\makeatother
\DeclareMathOperator{\sAut}{\mathKel{A\mkern-5.5mu u\mkern-4mu t\mkern-1.5mu}}
\DeclareMathOperator{\sAd}{\mathKel{A\mkern-5.5mu d}}
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\DeclareMathOperator{\sEnd}{\mathKel{E\mkern-4mu n\mkern-4.5mu d\mkern-1mu}}
%\tikzset{every node/.style={align=center}}
%\usepackage{pst-node} %Diagramme
%\usepackage{auto-pst-pdf}
%\usepackage{stackengine} %Kommawedge
%\usepackage{mathabx}
%\usepackage{here}
%\usepackage[style=authortitle-icomp]{biblatex}
%\usepackage[babel,german=guillemets]{csquotes}
\setcounter{tocdepth}{4}
\setcounter{tocdepth}{5}
\setcounter{secnumdepth}{4}
\setcounter{secnumdepth}{5}
\setlength{\jot}{12pt} % Zeilenabstand in der Align-Umgebung
\allowdisplaybreaks % mit \displaybreak einen Seitenumbruch in align-Umgebung erzeugen
%\pagestyle{headings}
\pagestyle{scrheadings}
%\pagestyle{fancy} %eigener Seitenstil
%\fancyhf{}
%\lhead{\leftmark} %\nouppercase, um auch Kleinbuchstaben zu bekommen
%\renewcommand{\sectionmark}[1]{\markright{#1}{}}
% \markboth{\thesection{} #1}}
%\fancyhead[L]{Titel} %Kopfzeile links
\clearpairofpagestyles
\ihead{\headmark} %Kopfzeile links, automark has to be loaded in the package scrlayer-scrpage
\ohead{Simon-Raphael Fischer} %Kopfzeile rechts
%\renewcommand{\headrulewidth}{0.4pt} %obere Trennlinie
%\fancyfoot[C]{\thepage} %Seitennummer
\cfoot{\pagemark}
%\renewcommand{\footrulewidth}{0.4pt} %untere Trennlinie
\renewcommand{\theequation}{\arabic{equation}}
%\renewcommand*{\chapterpagestyle}{scrheadings}
%Folgender Befehl dafür da, dass sections auf ungeraden Seiten anfangen
%\let\Section\section
%\renewcommand\section[2][]{%
%\cleardoublepage
%\def\myTEMP{#1}\ifx\myTEMP\empty\Section{#2}\else\Section[#1]{#2}\fi}
%\usepackage[automark]{scrlayer-scrpage}
%\ohead{\pagemark}
%\ihead{\leftmark}
\renewcommand{\listoffigures}{\begingroup
\tocsection
\tocfile{\listfigurename}{lof}
\endgroup}
\renewcommand{\listoftables}{\begingroup
\tocsection
\tocfile{\listtablename}{lot}
\endgroup}
\oddsidemargin -0.0 cm
\evensidemargin -0.5 cm
\topmargin -1.5 cm
\headheight 1.2 cm
\headsep 1.3 cm
\topskip 0.5 cm
\textheight 23.0 cm
\textwidth 16.0 cm
\parindent 0.0 cm
\renewcommand{\baselinestretch}{1.2}
%\renewcommand{\arraystretch}{1.6}
\setcounter{totalnumber}{2}
\def\tp{^{^{^{\leftrightarrow}}}\!\!\!\!\!\!\!}
%% Quick macros
\newcommand{\makecommand}[3]{%
\foreach \i in #3 {%
\expandafter\xdef\csname #1\i\endcsname{\noexpand#2{\unexpanded\expandafter{\i}}}%
}%
}
\newcommand{\latinalphabet}{A,a,B,b,C,c,d,D,E,e,F,f,G,g,H,h,I,i,J,j,K,k,L,l,M,m,N,n,O,o,P,p,Q,q,R,r,S,s,T,t,U,u,V,v,W,w,X,x,Y,y,Z,z}
\makecommand{fr}{\mathfrak}{\latinalphabet}
\makecommand{fr}{\mathfrak}{{der,gl,sl,so,osp,brst,su,sp,spin,string,Poisson,inn,at}}
\makecommand{sf}{\mathsf}{\latinalphabet}
\makecommand{sf}{\mathsf}{{id,String,Lie,Hom,SU,inv,SO,Map,Diff,HSymp,id,inn,Inn,CE,hom,Gpd,Bibun,pr}}
\makecommand{rm}{\mathrm}{\latinalphabet}
\makecommand{rm}{\mathrm}{{Ham,ad,id,dim,Ad,im,deg}}
\makecommand{ca}{\mathcal}{\latinalphabet}
\makecommand{I}{\mathbbm}{{N,Z,R,C}}
\makecommand{sc}{\mathscr}{\latinalphabet}
\makecommand{tt}{\mathtt}{\latinalphabet}
\newcommand{\ihom}{\underline{\sfhom}}
\newcommand{\iGpd}{\underline{\sfGpd}}
\DeclareMathOperator{\tr}{tr}
%%%%%%%%%%%%%%%%%%%%%%%%%%%% Definitionen %%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\renewcommand{\chapterheadstartvskip}{\vspace{0cm}} %für bookklasse
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\bs{\begin{subequations}}
\def\es{\end{subequations}}
\def\ba#1\ea{\begin{align}#1\end{align}}
\def\bes{\begin{equation*}}
\def\ees{\end{equation*}}
\def\bas#1\eas{\begin{align*}#1\end{align*}}
%um die Theoreme etc. in das Inhaltsverzeichnis zu packen
%\let\amsthmhead\thmhead
%\let\amsswappedhead\swappedhead
%\makeatletter
%\renewcommand*\thmhead[3]{\amsthmhead{#1}{#2}{#3}%
%%\@ifnotempty{#2}{\addcontentsline{toc}{subsection}{#2 ~ #3}}{}}
%\renewcommand*\swappedhead[3]{\amsswappedhead{#1}{#2}{#3}%
%%\@ifnotempty{#2}{\addcontentsline{toc}{subsection}{#2 ~ #3}}{}}
%\makeatother
\renewcommand{\qedsymbol}{$\blacksquare$}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
%\newtcbtheorem[number within=section]{theorem}{Theorem}
%{colback=green!5,colframe=green!35!black,fonttitle=\bfseries}{th}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtcbtheorem
[use counter*=theorem,number within=section]% init options
{lemmata}% name
{Lemma}% title
{%
fontupper=\itshape,
breakable,
enhanced,
colback=gray!25,
colframe=gray!0!black,
fonttitle=\bfseries,
}% options
{lem}% prefix
\newtheorem{proposition}[theorem]{Proposition}
\newtcbtheorem
[use counter*=theorem,number within=section]% init options
{propositions}% name
{Proposition}% title
{%
fontupper=\itshape,
breakable,
enhanced,
colback=gray!25,
colframe=gray!0!black,
fonttitle=\bfseries,
}% options
{prop}% prefix
\newtcbtheorem
[use counter*=theorem,number within=section]% init options
{theorems}% name
{Theorem}% title
{%
fontupper=\itshape,
breakable,
enhanced,
colback=gray!25,
colframe=gray!0!black,
fonttitle=\bfseries,
}% options
{thm}% prefix
\newtcbtheorem
[use counter*=theorem,number within=section]% init options
{corollaries}% name
{Corollary}% title
{%
fontupper=\itshape,
breakable,
enhanced,
colback=gray!25,
colframe=gray!0!black,
fonttitle=\bfseries,
}% options
{cor}% prefix
\newtcbtheorem
[use counter*=theorem,number within=section]% init options
{conjectures}% name
{Conjecture}% title
{%
fontupper=\itshape,
breakable,
enhanced,
colback=gray!25,
colframe=gray!0!black,
fonttitle=\bfseries,
}% options
{con}% prefix
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{remark}
\newtcbtheorem
[use counter*=theorem,number within=section]% init options
{remarks}% name
{Remark}% title
{%
breakable,
enhanced,
colback=gray!5,
colframe=gray!50!black,
fonttitle=\bfseries,
}% options
{rem}% prefix
\newtheorem*{note}{Note}
\newtheorem{remark}[theorem]{Remarks}
\newtheorem{motivation}[theorem]{Motivation}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtcbtheorem
[use counter*=theorem,number within=section]% init options
{definitions}% name
{Definition}% title
{%
breakable,
enhanced,
colback=gray!25,
colframe=gray!0!black,
fonttitle=\bfseries,
}% options
{def}% prefix
\newtcbtheorem
[use counter*=theorem,number within=section]% init options
{examples}% name
{Example}% title
{%colback=black!5,colframe=red!35!black
breakable,
enhanced,
colback=gray!5,
colframe=gray!50!black,
fonttitle=\bfseries,
}% options
{ex}% prefix
\newtcbtheorem
[use counter*=theorem,number within=section]% init options
{situations}% name
{Situation}% title
{%colback=black!5,colframe=red!35!black
breakable,
enhanced,
colback=gray!5,
colframe=gray!50!black,
fonttitle=\bfseries,
}% options
{sit}% prefix
\newtcbtheorem
[use counter*=theorem,number within=section]% init options
{fieldredefinitions}% name
{Theorem}% title
{%colback=black!5,colframe=red!35!black
breakable,
enhanced,
colback=gray!25,
colframe=gray!0!black,
fonttitle=\bfseries,
}% options
{fieldredef}% prefix
\newtheorem{example}[theorem]{Example}
% hier Namen etc. einsetzen
%\newcommand{\fullname}{Simon-Raphael Fischer}
%\newcommand{\email}{sfischer@ncts.tw}
%\newcommand{\titel}{Examples and No-Gos of curves Yang-Mill-Higgs gauge theories}
%\newcommand{\titel}{Titel der Arbeit}
%\newcommand{\jahr}{2020}
%\newcommand{\matnr}{11424951}
%\newcommand{\gutachterA}{Prof. Dr. Mark John David Hamilton}
%\newcommand{\betreuer}{Professor Dr. Anna Dall'Acqua}
% hier richtige Fakultät auswählen
%\newcommand{\fakultaet}{Fakultät noch ergänzen}
%\newcommand{\fakultaet}{Mathematik und\\Wirtschaftswissenschaften}
%\newcommand{\fakultaet}{Naturwissenschaften}
%\newcommand{\fakultaet}{Medizin}
% nun noch unten das Institut einsetzen
%\newcommand{\institut}{Institut noch ergänzen}
% Für \widecheck: (umgekehrter Zirkumflex)
\DeclareFontFamily{U}{mathx}{\hyphenchar\font45}
\DeclareFontShape{U}{mathx}{m}{n}{
<5> <6> <7> <8> <9> <10>
<10.95> <12> <14.4> <17.28> <20.74> <24.88>
mathx10
}{}
\DeclareSymbolFont{mathx}{U}{mathx}{m}{n}
\DeclareFontSubstitution{U}{mathx}{m}{n}
\DeclareMathAccent{\widecheck}{0}{mathx}{"71}
\DeclareMathAccent{\wideparen}{0}{mathx}{"75}
%\newtheorem{cor}{Corollary}
%\newtheorem{ad}{Theorem}
%\newtheorem{theorem}{Theorem}
%\newtheorem{theorem}{Theorem}
%\newtheorem{theorem}{Theorem}
%\includeonly{PhysicalBasics/f(R)gravity} %nur das kompilieren
%\let\endtitlepage\relax %No page break after titlepage
\begin{document}
\pagenumbering{Roman}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\begin{titlepage}
%\thispagestyle{empty}
\author{Simon-Raphael Fischer\footnote{Email: \href{mailto:sfischer@ncts.tw}{sfischer@ncts.tw}; ORCiD: \href{https://orcid.org/0000-0002-5859-2825}{0000-0002-5859-2825}} }
\title{Integrating curved Yang-Mills gauge theories}
\subtitle{Gauge theories related to principal bundles equipped with Lie group bundle actions}
\date{\today}
\maketitle
\thispagestyle{empty}
\begin{center}
National Center for Theoretical Sciences (\begin{CJK*}{UTF8}{bkai}國家理論科學研究中心\end{CJK*}),
Mathematics Division,
\\
National Taiwan University
\\
No.\ 1, Sec.\ 4, Roosevelt Rd., Taipei City 106, Room 407, Cosmology Building, Taiwan
\ \\
\begin{CJK*}{UTF8}{bkai}106319 臺北市羅斯福路四段1號 (國立臺灣大學次震宇宙館407室)\end{CJK*}
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\textbf{Abstract}\footnote[2]{Abbreviations used in this paper: \textbf{LGB(s)} for Lie group bundle(s), \textbf{LAB(s)} for Lie algebra bundle(s).}
%\footnote[2]{Abbreviations used in this paper: \textbf{(C)YMH GT} for (curved) Yang-Mills-Higgs gauge theory.}
\begin{abstract}
\small{
We construct a gauge theory based on principal bundles $\mathcal{P}$ equipped with a right $\mathcal{G}$-action, where $\mathcal{G}$ is a Lie group bundle instead of a Lie group. Due to the fact that a $\mathcal{G}$-action acts fibre by fibre, pushforwards of tangent vectors via a right-translation act now only on the vertical structure of $\mathcal{P}$. Thus, we generalize pushforwards using sections of $\mathcal{G}$, and in order to provide a definition independent of the choice of section we fix a connection on $\mathcal{G}$, which will modify the pushforward by subtracting the fundamental vector field generated by a generalized Darboux derivative of the chosen section. A horizontal distribution on $\mathcal{P}$ invariant under such a modified pushforward leads to a parallel transport on $\mathcal{P}$ which is a homomorphism w.r.t.\ the $\mathcal{G}$-action and the parallel transport on $\mathcal{G}$. For achieving gauge invariance we impose conditions on the connection 1-form $\mu$ on $\mathcal{G}$: $\mu$ has to be a multiplicative form, \textit{i.e.}\ closed w.r.t.\ a certain simplicial differential $\delta$ on $\mathcal{G}$, and the curvature $R_\mu$ of $\mu$ has to be $\delta$-exact with primitive $\zeta$; $\mu$ will be the generalization of the Maurer-Cartan form of the classical gauge theory, while the $\delta$-exactness of $R_\mu$ will generalize the role of the Maurer-Cartan equation. For allowing curved connections on $\mathcal{G}$ we will need to generalize the typical definition of the curvature/field strength $F$ on $\mathcal{P}$ by adding $\zeta$ to $F$.
This leads to a generalized gauge theory with many similar, but generalized, statements, including Bianchi identity, gauge transformations and Darboux derivatives. An example for a gauge theory with a curved Maurer-Cartan form $\mu$ will be provided by the inner group bundle of the Hopf fibration $\mathds{S}^7 \to \mathds{S}^4$. We conclude that this gauge theory is an integral of an infinitesimal gauge theory developed by Alexei Kotov and Thomas Strobl.
}
\end{abstract}
\end{center}
\textit{2010 MSC:} Primary 53D17; Secondary 81T13, 22E99.
\textit{Keywords:} \texttt{Mathematical Gauge Theory}, Differential Geometry, High Energy Physics - Theory, Mathematical Physics
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\section{Introduction and summary}
This paper's research concerns curved Yang-Mills-Higgs gauge theories, originally introduced by Alexei Kotov and Thomas Strobl in \cite{CurvedYMH}, where essentially the structural Lie algebra together with its action on the manifold $N$ of values of the Higgs fields is replaced by a general Lie algebroid $E \to N$:
\begin{definitions*}{Lie algebroid, \cite[reduced definition of \S 16.1, page 113]{DaSilva}}
%\leavevmode\newline
Let $E \to N$ be a real vector bundle. Then $E$ is a smooth \textbf{Lie algebroid} if there is a bundle map $\rho: E \to \mathrm{T}N$, called the \textbf{anchor}, and a Lie algebra structure on $\Gamma(E)$ with Lie bracket $\left[ \cdot, \cdot \right]_E$ satisfying\footnote{With $\Gamma(E)$ I denote the space of sections of a vector bundle E, and with $\mathrm{T}N$ the tangent bundle of $N$.}
\begin{align*}
\left[\mu, f \nu\right]_E = f \left[\mu, \nu\right]_E + \mathcal{L}_{\rho(\mu)}(f) ~ \nu
\end{align*}
for all $f \in C^\infty(N)$ and $\mu, \nu \in \Gamma(E)$, where $\mathcal{L}_{\rho(\mu)}(f)$ is the action of the vector field $\rho(\mu)$ on the function $f$ by derivation.
\end{definitions*}
The idea of replacing Lie algebras with Lie algebroids was proposed by Thomas Strobl in \cite{OriginofCYMH}, with further understanding of the involved gauge transformations in \cite{mayer2009lie}; eventually, this type of infinitesimal gauge theory got summarised and finalised in \cite{CurvedYMH}. My Ph.D.\ thesis was devoted to this type of infinitesimal gauge theory, attempting to find new (physical) examples and understanding the geometry of this infinitesimal gauge theory; see \cite{My1stpaper} and \cite{MyThesis}.
A short summary of this type of generalized infinitesimal gauge theory follows; we have the following ingredients:
\begin{itemize}
\item $M$ a spacetime;
\item $N$ a smooth manifold, serves as set for the values of the Higgs field $\Xi: M \to N$;
\item $E \to N$ a Lie algebroid with anchor $\rho$, replacing the structural Lie algebra $\mathfrak{g}$ and its action $\gamma: \mathfrak{g} \to \mathfrak{X}(N)$ of the classical formulation, where $\mathfrak{X}$ denotes the set of vector fields in this work;
\item a vector bundle connection $\nabla$ on $E$;
\item a fibre metric $\kappa$ on $E$, as a substitute of the ad-invariant scalar product on $\mathfrak{g}$;
\item a Riemannian metric $g$ on $N$, replacing the scalar product on the vector space in which the Higgs field usually has values in and which is invariant under the action of $\gamma$, used for the kinetic term of $\Xi$ which is minimally coupled to the field of gauge bosons $A \in \Omega^1(M; \Xi^*E)$ (1-form on $M$ with values in $\Xi^*E$);
\item a 2-form on $N$ with values in $E$, $\zeta \in \Omega^2(N;E)$, an additional contribution to the field strength of $A$.
\end{itemize}
Infinitesimal gauge invariance of the Yang-Mills type functional leads to two \textbf{infinitesimal compatibility conditions} and two metric compatibilities to be satisfied between these structures; we will present those compatibility conditions later in this introduction. If the connection on $E$ is flat, the compatibilities imply that the Lie algebroid is locally an action Lie algebroid:
\begin{definitions*}{Action Lie algebroids, \cite[\S 16.2, Example 5; page 114]{DaSilva}}
Let $\mleft(\mathfrak{g}, \mleft[\cdot, \cdot \mright]_{\mathfrak{g}}\mright)$ be a Lie algebra equipped with a Lie algebra action $\gamma: \mathfrak{g} \to \mathfrak{X}(N)$ on a smooth manifold $N$. A \textbf{transformation Lie algebroid} or \textbf{action Lie algebroid} is defined as the bundle $E \coloneqq N \times \mathfrak{g}$ over $N$ with anchor
\bas
\rho(p, v) &\coloneqq \gamma(v)|_p
\eas
for $(p, v) \in E$, and Lie bracket
\bas
\mleft.\mleft[\mu, \nu\mright]_E\mright|_p
&\coloneqq
\mleft[\mu_p, \nu_p\mright]_{\mathfrak{g}}
+ \mleft.\mleft(\mathcal{L}_{\gamma(\mu(p))}(\nu^a) - \mathcal{L}_{\gamma(\nu(p))}(\mu^a) \mright)\mright|_p ~ e_a
\eas
for all $p \in N$ and $\mu, \nu \in \Gamma(E)$, where one views a section $\mu \in \Gamma(E)$ as a map $\mu: N \to \mathfrak{g}$, and $\mleft( e_a \mright)_a$ is some arbitrary frame of constant sections.
\end{definitions*}
As it is shown in the mentioned references for this type of infinitesimal gauge theory, one gets back to the standard Yang-Mills-Higgs gauge theory if additionally $\zeta \equiv 0$. Thus, the theory represents a covariantized version of gauge theory equipped with an additional 2-form $\zeta$. If $\nabla$ is flat we say in general that we have a \textbf{pre-classical} gauge theory, and if additionally $\zeta \equiv 0$ we have a \textbf{classical} gauge theory.
The 2-form $\zeta$ is needed to allow non-flat $\nabla$, because otherwise only flat $\nabla$ could satisfy the compatibility conditions. But $\zeta$ is not just an auxiliary map, in \cite{MyThesis} I have shown that there is also a class of \textbf{field redefinitions} for the classical formulation of gauge theory. This field redefinition breaks the gauge invariance; so, in order to keep gauge invariance, one needs to add $\zeta$ to the field strength, and at the same time one achieves a richer framework for gauge theories. It is then natural to study whether there is an infinitesimal gauge theory where $\zeta$ is non-zero and cannot be transformed to zero by the mentioned field redefinition.
In \cite{MyThesis} I first focused on Lie algebroids where the anchor map is an isomorphism and, thus, the Lie algebroid is just the tangent bundle of $M$. Locally, such examples can be excluded: The curvature can be always transformed to zero by the field redefinitions. However, globally, this is not always true: I have shown that the tangent bundle of the seven-dimensional sphere $\mathds{S}^7$ is an example of a curved gauge theory which cannot be transformed to a flat pre-classical gauge theory, otherwise it would be a Lie group which is not possible.
However, I also studied the other edge case of having Lie algebra bundles (LABs) instead of Lie algebroids: \cite{MyThesis} (also in \cite{My1stpaper}) points out that locally, \textit{i.e.}~over a contractible base manifold $N$, there is always a field redefinition transforming the initial gauge theory to a pre-classical one. Hence, we arrive at a similar situation as for tangent bundles. But globally there are examples given by the adjoint bundle of the Hopf fibration $\mathds{S}^7 \to \mathds{S}^4$, where $\mathds{S}^n$ ($n \in \mathbb{N}$) denotes the $n$-dimensional sphere.
This is now the starting point of this paper. We want to understand why the adjoint bundle of $\mathds{S}^7 \to \mathds{S}^4$ works as a curved example, and where it differs from the classical formalism of gauge theory. To truly understand this, and also to possibly get new information about the general case regarding Lie algebroids, we will integrate curved Yang-Mills-Higgs theories in the case of LABs in this paper. Using LABs means that there is no coupling to a Higgs field in the usual sense and therefore we will now just speak instead of a curved Yang-Mills gauge theory. In the following we will start outlining the paper's main results; we will focus on an easy presentation in the introduction, especially we will not always restate our assumptions, and the notation will be simplified w.r.t.\ the actual formulation.
\subsection{Summary}
The gauge theory presented here will be based on principal bundles $\mathcal{P} \stackrel{\pi}{\to} M$, $M$ a smooth manifold, related to a right action of a Lie group bundle (LGB) $\mathcal{G} \stackrel{\pi_{\mathcal{G}}}{\to} M$. The most important distinction to classical gauge theory is that the action $\Phi$ is a map
\bas
\mathcal{P} * \mathcal{G} \coloneqq \pi^*\mathcal{G} &\to \mathcal{P},\\
(p, g) &\mapsto p \cdot g,
\eas
where $\pi^*\mathcal{G}$ is the pullback LGB of $\mathcal{G}$ along $\pi$. Observe that $\mathcal{G}_x$ only acts on $\mathcal{P}_x$ as a Lie group, thus an element $g \in \mathcal{G}_x$ cannot act on all of $\mathcal{P}$, where notations like $\mathcal{G}_x$ denote the fibre of $\mathcal{G}$ over an $x \in M$. This leads to certain difficulties for defining gauge theory which we will resolve in this paper.
However, this paper aims to be accessible for all people knowing the basics of "classical" gauge theory. Thus, we start with an extensive introduction to LGBs, their actions and their infinitesimal analogues, the LABs, in Section \ref{LGBSection} to \ref{LGBActionIISection}. These sections introduce and generalize all the notions known in the "classical" Lie group situations in the context of gauge theory, most of which is straight-forward to generalize. We will especially follow \cite{mackenzieGeneralTheory}, but since this and other references are usually introducing these basic notions on (Lie) groupoids, we decided to reintroduce all the needed notions and proofs so that it should not be required for the reader to follow and learn about groupoids because we just need the much simpler situation regarding LGBs. The experienced reader may start directly with Section \ref{ConnCurvOnPrincLGBBundle}, but there is one partially new result in these preliminary sections: For gauge theory we will need to understand the differential of the $\mathcal{G}$-action $\Phi$ which we will derive in Thm.\ \ref{thm:DiffOfLGBAction} and which will form \textit{the} fundament for many calculations. For this we will need notations like $\mathrm{T}M$ which will denote a tangent bundle with fibre $\mathrm{T}_xM$ at $x \in M$; the fibres of other bundles with similar notation are denoted in the same fashion.
\begin{theorems*}{Differential of LGB action $\Phi$, simplified situation}
We have
\bas
\mathrm{D}_{(p, g)}\Phi(X, Y)
&=
\mathrm{D}_pr_\sigma(X)
+ \mleft.{\oversortoftilde{\mleft( \mu_{\mathcal{G}}\mright)_g \bigl(Y - \mathrm{D}_{x}\sigma (\omega)\bigr)}}\mright|_{p \cdot g}
\eas
for all $(p, g) \in \mathcal{P} * \mathcal{G}$ and $(X, Y) \in \mathrm{T}_{(p, g)}(\mathcal{P}*\mathcal{G})$, where $x \coloneqq \pi(p)$, $\sigma$ is any (local) section of $\mathcal{G}$ with $\sigma_{x} = g$ and $r_\sigma$ denotes the right-translation of $\sigma$ in $\mathcal{P}$, $\mu_{\mathcal{G}}$ is the fibre-wise defined Maurer-Cartan form of the fibres, $\oversortoftilde{\mleft( \mu_{\mathcal{G}} \mright)_g(\dotsc)}$ denotes its generated fundamental vector field on $\mathcal{P}_x$, and $\omega$ is an element of $\mathrm{T}_xM$ given by
\bas
\omega
&\coloneqq
\mathrm{D}_p \pi(X)
=
\mathrm{D}_g\pi_{\mathcal{G}}(Y).
\eas
\end{theorems*}
In Section \ref{ConnCurvOnPrincLGBBundle} we will introduce connections and generalized curvatures on principal LGB-bundles: Subsection \ref{PrincBundlLGBBased} starts with introducing principal $\mathcal{G}$-bundles $\mathcal{P}$. Such principal bundles were already introduced in \cite[\S 5.7, page 144f.]{GroupoidBasedPrincipalBundles}, but also here we rephrased it in such a way that these bundles' definition has a more familiar shape for readers not so proficient with the study of groupoids. The main difference to Lie group based principal bundles is the previously-mentioned LGB action $\Phi$. Since these bundles were not studied a lot before we introduce basic notions like morphisms and easy examples, for example LGBs $\mathcal{G}$ themselves are principal $\mathcal{G}$-bundles, so that the gauge theory presented in this paper can also be understood as a gauge theory based not only on "classical" principal bundles but also on LGBs which were excluded in general in the classical formalism. Furthermore we generalize certain statements known about classical principal bundles, \textit{e.g.}\ we observe that a section of $\mathcal{P}$ does not trivialize $\mathcal{P}$ in general, however it introduces an isomorphism to $\mathcal{G}$, as will be pointed out in Lemma \ref{lem:SectionsNowInduceIsomToLGBsNotNecTriv}.
\begin{lemmata*}{Local sections of principal bundles induce isomorphisms to the structural LGB, simplified language}
Let $s: U \to \mathcal{P}$ be a smooth local section of $\mathcal{P}$ over an open subset $U$ of $M$. Then the orbit map through $s$,
\bas
\mathcal{G}|_U &\to \mathcal{P}|_U,\\
g &\mapsto s_{\pi_{\mathcal{G}}(g)} \cdot g,
\eas
is a base-preserving principal $\mathcal{G}$-bundle isomorphism.
\end{lemmata*}
In Subsection \ref{ConnectionSubsection} we finally turn to the notion of connections on $\mathcal{P}$, described as horizontal subbundle $\mathrm{H}\mathcal{P}$ of the tangent bundle $\mathrm{T}\mathcal{P}$, complementary to the vertical bundle $\mathrm{V}\mathcal{P}$, and equivalently we want to describe $\mathrm{H}\mathcal{P}$ as a connection 1-form $A$. As one can already see in the last theorem about the derivative of the action $\Phi$, due to the fact that right-translations $r_g$ of an element $g \in \mathcal{G}_x$ now only acts on $\mathcal{P}_x$, its derivative $\mathrm{D}r_g$ only acts on the tangent bundle of $\mathcal{P}_x$, the vertical tangent vectors of $\mathcal{P}$ (over $x$). Hence, in order to define the pushforward of non-vertical vectors we make use of auxiliary sections. However, this leads to the problem that there are many different sections with the same value over a fixed base point $x \in M$, all of whose pushforwards of non-vertical vectors are in general different. To get rid of the ambiguity in the choice of section we are going to modify and generalize the pushforward of tangent vectors by right-translations. To do so, we fix a horizontal distribution $\mathrm{H}\mathcal{G}$ on $\mathcal{G}$, without further assumptions than the bare-bones for horizontal distributions. The \textbf{(modified) right-pushforward} on $\rmT P$ is then
Then we start to construct the horizontal distribution $\mathrm{H}\mathcal{P}$ initially by its associated parallel transport, shortly denoted now by $\mathrm{PT}^{\mathcal{P}}$; similarly we denote the parallel transport associated to $\mathrm{H}\mathcal{G}$ by $\mathrm{PT}^{\mathcal{G}}$. We want a connection on $\mathcal{P}$ characterised by
\bas
\mathrm{PT}^{\mathcal{P}}(p \cdot g)
&=
\mathrm{PT}^{\mathcal{P}}(p) \cdot \mathrm{PT}^{\mathcal{G}}(g)
\eas
for all $(p, g) \in \mathcal{P}*\mathcal{G}$. However, in order to define the connection 1-form it is useful to find a definition of such $\mathrm{H}\mathcal{P}$ via a symmetry w.r.t.\ a certain map acting on $\mathrm{T}\mathcal{P}$ which we will motivate by . As already high-lighted, this will be a modification of the pushforward via right-translations. Differentiating the condition about parallel transports w.r.t.\ the curve parameter achieves this modification, summarised in Prop.\ \ref{prop:IsomorphismRightPushAndDarboux}; we explain the second summand of the following proposition afterwards.
\begin{propositions*}{The modified right-pushforward}
For $g \in \mathcal{G}_x$ ($x \in M$) define the map
\bas
\mleft.\mathrm{T}\mathcal{P}\mright|_{\mathcal{P}_x} &\to \mleft.\mathrm{T}\mathcal{P}\mright|_{\mathcal{P}_x},\\
X
&\mapsto
\mathcal{r}_{g*}(X)
\coloneqq
\mathrm{D}_pr_\sigma\mleft(
X
\mright)
- \mleft.{\oversortoftilde{
\mleft. \mleft( \pi^!\Delta\sigma \mright) \mright|_p(X)
}}\mright|_{p \cdot g},
\eas
where $p \in \mathcal{P}_x$, $X \in \mathrm{T}_p \mathcal{P}$, $\pi^!$ denotes the pull-back of forms with $\pi$, and $\sigma$ is any (local) section of $\mathcal{G}$ with $\sigma_x = g$. Then $\mathcal{r}_{g*}$ is independent of the choice of the local section $\sigma$, and it is a vector bundle automorphism over the right-translation $r_g$.
Furthermore, if we have a horizontal distribution $\mathrm{H}\mathcal{P}$ on $\mathcal{P}$, then $\mathrm{H}_p\mathcal{P}$ is isomorphic via $\mathcal{r}_{g*}$ to a complement of $\mathrm{V}_{p \cdot g} \mathcal{P}$ in $\mathrm{T}_{p \cdot g}\mathcal{P}$ (this complement is not necessarily $\mathrm{H}_{p \cdot g}\mathcal{P}$).
\end{propositions*}
This can be generalized to sections $\sigma$ of $\mathcal{G}$ in a straight-forward manner, giving rise to an automorphism $\mathcal{r}_{\sigma*}$ of $\mathrm{T}\mathcal{P}$ over the right-translation $r_\sigma$.
$\Delta\sigma$ is the generalised version of the Darboux derivative, often simply denoted as $\sigma^{-1} \mathrm{d}\sigma$ in the classical formalism, but in some works like \cite[\S 5.1, page 182ff.]{mackenzieGeneralTheory}\ also already written as $\Delta\sigma$ in the classical formalism. As expected, this derivative plays an important role in gauge theory, which is why we will also discuss and introduce the Darboux derivative; usually it only appears in the gauge transformations, but in our case it will already appear now. In Def.\ \ref{def:TotMCFormOnLGB} we will first introduce the \textbf{total Maurer-Cartan form $\mu_{\mathcal{G}}^{\mathrm{tot}}$} as the connection 1-form associated to $\mathcal{G}$ with values in the LAB $\mathcal{g}$ of $\mathcal{G}$; the labelling comes from that this form will play a similar role like the Maurer-Cartan form in the typical formalism for gauge theory, and in contrast to $\mu_{\mathcal{G}}$ it acts on the whole of $\mathrm{T}\mathcal{G}$ instead of just the vertical subbundle.
\begin{definitions*}{Total Maurer-Cartan form}
Let us denote with $\pi^{\mathrm{vert}}$ the projection onto the vertical bundle $\mathcal{G}$, corresponding to its horizontal bundle $\mathrm{H}\mathcal{G}$. Then we define the \textbf{total Maurer-Cartan form $\mu_{\mathcal{G}}^{\mathrm{tot}} \in \Omega^1\mleft(\mathcal{G}; \pi_{\mathcal{G}}^*\mathcal{g}\mright)$ of $\mathcal{G}$} as the connection 1-form corresponding to $\mathrm{H}\mathcal{G}$, \textit{i.e.}\
\bas
\mleft( \mu_{\mathcal{G}}^{\mathrm{tot}} \mright)_g(Y)
&\coloneqq
\mleft.\mleft(\mu_\mathcal{G} \circ \pi^{\mathrm{vert}}\mright)\mright|_g(Y)
=
\mleft( \mathrm{D}_g L_{g^{-1}} \mright)\mleft(\pi^{\mathrm{vert}}(Y)\mright)
\eas
for all $g \in \mathcal{G}$ and $Y \in \mathrm{T}_g\mathcal{G}$.
\end{definitions*}
As expected, the Darboux derivative is the form-pullback of the total Maurer-Cartan form, as such we will introduce this in Def.\ \ref{def:DarbouxDerivativeOnLGBs}.
\begin{definitions*}{Darboux derivative}
For (local) $\sigma \in \Gamma(\mathcal{G})$ we define the \textbf{Darboux derivative $\Delta \sigma \in \Omega^1(M; \mathcal{g})$}
\bas
\Delta \sigma
&=
\sigma^! \mu_{\mathcal{G}}^{\mathrm{tot}}.
\eas
\end{definitions*}
Using the Darboux derivative, we defined the modified pushforward via right-translations (also called modified right-pushforward) which eventually leads to the definition of an Ehresmann connection on the principal $\mathcal{G}$-bundle $\mathcal{P}$, as pinpointed in Def.\ \ref{def:FinallyTheConnection}.
\begin{definitions*}{Ehresmann connection on principal LGB-bundles}
We call $\mathrm{H}\mathcal{P}$ an \textbf{Ehresmann connection} or a \textbf{connection on $\mathcal{P}$} if it is \textbf{right-invariant (w.r.t.\ modified right-pushforward)}, \textit{i.e.}\
\bas
\mathcal{r}_{g*}\mleft( \mathrm{H}_p\mathcal{P} \mright)
&=
\mathrm{H}_{p\cdot g}\mathcal{P}
\eas
for all $p \in \mathcal{P}_x$ and $g \in \mathcal{G}_x$ ($x \coloneqq \pi(p)$).
\end{definitions*}
$\mathrm{H}\mathcal{P}$ will now always be an Ehresmann connection in the following.
The basic idea now is to replace the typical right-pushforward with the modified one in all the involved definitions, starting with the needed pull-back of forms, which will be simply given by Def.\ \ref{def:PullbackOfFormsViaModRight}.
\begin{definitions*}{The pullback of forms via modified right-pushforward, the section formulation}
For $\omega \in \Omega^k(\mathcal{P}; \pi^*\mathcal{g})$ ($k \in \mathbb{N}_0$) we define the \textbf{pullback via the modified right-pushforward $\mathcal{r}^!_{\sigma}\omega$ with a (local) section $\sigma \in \Gamma(\mathcal{G})$} as an element of $\Omega^k(\mathcal{P}; \pi^*\mathcal{g})$ by
\bas
\mleft.\mleft(\mathcal{r}_\sigma^!\omega\mright)\mright|_p \mleft( Y_1, \dotsc, Y_k\mright)
&\coloneqq
\omega_{p \cdot \sigma_x} \bigl( \mathcal{r}_{\sigma*}(Y_1), \dotsc, \mathcal{r}_{\sigma*}(Y_k)\bigr)
\eas
for all $p \in \mathcal{P}_x$ ($x \in M$) and $Y_1, \dotsc, Y_k \in \mathrm{T}_p\mathcal{P}$.
\end{definitions*}
It is now straight-forward to define the connection 1-form $A$ on $\mathcal{P}$, which we will do in Def.\ \ref{def:GaugeBosonsOnLGBPrincies}.
\begin{definitions*}{Connection 1-forms on principal LGB-bundles, simplified notation}
A \textbf{connection 1-form} or \textbf{gauge field} on $\mathcal{P}$ is a 1-form $A \in \Omega^1(\mathcal{P}; \pi^*\mathcal{g})$ satisfying:
\begin{itemize}
\item \textbf{($\mathcal{G}$-equivariance, but w.r.t.\ modified right-pushforward)}
\bas
\mathcal{r}_\sigma^! A
&=
\mathrm{Ad}_{\sigma^{-1}} \circ A
\eas
for all (local) $\sigma \in \Gamma(\mathcal{G})$, where $\mathrm{Ad}$ is the adjoint representation of $\mathcal{G}$ on $\mathcal{g}$.
\item \textbf{(Identity on $\mathrm{V}\mathcal{P}$)}
\bas
A\mleft(\widetilde{\nu}\mright)
&=
\pi^*\nu
\eas
for all (local) $\nu \in \Gamma(\mathcal{g})$, where notations like $\pi^*$ denote the pull-back of sections.
\end{itemize}
\end{definitions*}
Of course we will achieve a 1:1 correspondence to Ehresmann connections in Thm.\ \ref{thm:OurConnectionHasAUniqueoneForm}.
\begin{theorems*}{1:1 correspondence of Ehresmann connections and connection 1-forms}
There is a 1:1 correspondence between Ehresmann connections and connection 1-forms on $\mathcal{P}$:
\begin{itemize}
\item Let $\mathrm{H}\mathcal{P}$ be an Ehresmann connection on $\mathcal{P}$. Then $\mathrm{H}\mathcal{P}$ defines a connection 1-form $A \in \Omega^1(\mathcal{P}; \pi^*\mathcal{g})$ by
\bas
A_p\bigl( \widetilde{v}_p + X_p \bigr)
&=
(p, v)
\eas
for all $p \in \mathcal{P}_x$ ($x \in M$), $v \in \mathcal{g}_x$ and $X \in \mathrm{H}_p\mathcal{P}$.
\item Let $A \in \Omega^1(\mathcal{P}; \pi^*\mathcal{g})$ be a connection 1-form on $\mathcal{P}$. Then $A$ defines an Ehresmann connection $\mathrm{H}\mathcal{P}$ on $\mathcal{P}$ via its kernel $\mathrm{Ker}(A)$, that is,
\bas
\mathrm{H}_p\mathcal{P}
&=
\mathrm{Ker}(A_p)
\eas
for all $p \in \mathcal{P}$.
\end{itemize}
\end{theorems*}
Not only these results are straight-forward once one has the definition of the modified right-pushforward, also other results will be simple to guess. For example in Subsection \ref{GaugeTrafoForA} we turn to the gauge transformations which will have a familiar shape but with the generalized Darboux derivative appearing, see Thm.\ \ref{thm:GaugeTrafoOfGaugeBoson}.
\begin{theorems*}{Gauge transformations of connection 1-forms, simplified notation}
Let $H$ be a (base-preserving) automorphism of $\mathcal{P}$. We then have that $H^!A$ is a connection 1-form on $\mathcal{P}$ and
\bas
H^!A
&=
{\mathrm{Ad}_{\mleft(\sigma^H\mright)^{-1}}} \circ A
+ \mleft(\pi^*\Delta\mright)\sigma^H,
\eas
where $\sigma^H \in \Gamma(\pi^*\mathcal{G})$ is uniquely defined by
\bas
H(p)
=
p \cdot \sigma_p^H
\eas
for all $p \in \mathcal{P}$, and $\pi^*\Delta$ is the Darboux derivative on $\pi^*\mathcal{G}$ naturally inherited by the pullback of the connection on $\mathcal{G}$.
\end{theorems*}
The 1:1 correspondence between $H$ and $\sigma^H$ arising here leads to similar statements as in the classical gauge theory which we will also shortly point out. We will show in Thm.\ \ref{thm:LocalGaugeTrafoChangeGauge} that this theorem about gauge transformations implies a pullback version, that is, one looks at the pullback $A_{s_i} \coloneqq s_i^!A$ of $A$ w.r.t.\ a section (also called gauge) $s_i$ of $\mathcal{P}$ defined over an open subset $U_i \subset M$, and then the change of $A$ by changing the gauge $s_i$ to another gauge follows by the previous theorem.
\begin{theorems*}{Gauge transformations as a change of gauge in the local gauge field}
Let $U_i$ and $U_j$ be two open subsets of $M$ so that $U_i \cap U_j \neq \emptyset$, two gauges $s_i \in \Gamma\mleft(\mathcal{P}|_{U_i}\mright)$ and $s_j \in \Gamma\mleft(\mathcal{P}|_{U_j}\mright)$, and the unique $\sigma_{ji} \in \Gamma\mleft( \mleft.\mathcal{G}\mright|_{U_i \cap U_j} \mright)$ with $s_i = s_j \cdot \sigma_{ji}$ on $U_i \cap U_j$. Then we have over $U_i \cap U_j$ that
\bas
A_{s_i}
&=
\mathrm{Ad}_{\sigma_{ji}^{-1}}\circ A_{s_j}
+ \Delta\sigma_{ji}.
\eas
\end{theorems*}
This finished the discussion about $A$, but of course we also needs its field strength $F$ and a certain shape of gauge transformation for $F$. Up until now $\mathrm{H}\mathcal{G}$ was an arbitrary horizontal distribution, but we will need to fix certain conditions on it to assure gauge invariance later. We will discuss this in Subsection \ref{CurvatureSubsection}. On one hand, viewing $\mathcal{G}$ as the principal $\mathcal{G}$-bundle $\mathcal{P}$ itself whose horizontal distribution $\mathrm{H}\mathcal{P}$ aligns with $\mathrm{H}\mathcal{G}$, it is natural to guess that $\mathrm{H}\mathcal{G}$ should be an Ehresmann connection itself. Ehresmann connections on LGBs were already discussed in works like \cite{LAURENTGENGOUXStienonXuMultiplicativeForms}; and there is also a rather recent preprint related to a similar subject, see \cite{FernandesMarcutMultiplicativeForms}. One of the results of these works is that an Ehresmann connection on LGBs is characterized by the fact that the total Maurer-Cartan form $\mu_{\mathcal{G}}^{\mathrm{tot}}$ has to be a multiplicative form:
\begin{definitions*}{Multiplicative forms, simplified situation, \newline \cite[\S 2.1, special situation of Def.\ 2.1]{crainic2015multiplicative}}
We call an $\omega \in \Omega^1\mleft( \mathcal{G}; \pi_{\mathcal{G}}^*\mathcal{g} \mright)$ a \textbf{multiplicative form} if
\bas
\omega_{gq}\mleft( \mathrm{D}_{(g, q)}\Phi(X, Y) \mright)
&=
\mathrm{Ad}_{q^{-1}}\bigl( \omega_{g}(X) \bigr)
+ \omega_{q}(Y)
\eas
for all $(g, q) \in \mathcal{G}*\mathcal{G}$ and $(X, Y) \in \mathrm{T}_{(g, q)}(\mathcal{G}*\mathcal{G})$, where $\Phi: \mathcal{G} * \mathcal{G} \to \mathcal{G}$ is the multiplication in $\mathcal{G}$.
\end{definitions*}
\begin{theorems*}{Connection 1-forms on LGBs are multiplicative, \newline \cite[\S 4.4, implication of Lemma 4.14]{LAURENTGENGOUXStienonXuMultiplicativeForms}}
$\mathrm{H}\mathcal{G}$ is an (Ehresmann) connection on $\mathcal{G}$ as principal bundle if and only if $\mu_{\mathcal{G}}^{\mathrm{tot}}$ is multiplicative.
\end{theorems*}
Assuming that $\mathrm{H}\mathcal{G}$ is an Ehresmann connection will also allow us to formulate certain technical identities for the Darboux derivative like a Leibniz rule. In order to calculate the gauge transformation for the curvature/field strength $F$ of $A$ we need to understand the curvature of the total Maurer-Cartan form. Classically, this is described by the Maurer-Cartan equation, however, we generalize this condition, allowing non-flat connections on $\mathcal{G}$. For this we will need a connection on the LAB $\mathcal{g}$ of $\mathcal{G}$, naturally induced by $\mathrm{H}\mathcal{G}$; we will construct this connection in Prop.\ \ref{prop:FinallyTheNablaInduction} and Def.\ \ref{def:ConnectionOnLAB}, and our construction aligns with \cite[\S 4.5, Prop.\ 4.22]{LAURENTGENGOUXStienonXuMultiplicativeForms} even though we argue the following differently since we have shown the following statement actually for all horizontal distributions $\mathrm{H}\mathcal{G}$ instead of just Ehresmann connections.
\begin{propositions*}{LGB connection induces LAB connection, simplified notation}
The map $\nabla^{\mathcal{G}}: \Gamma(\mathcal{g}) \to \Omega^1(M; \mathcal{g})$, $\nu \mapsto \nabla^{\mathcal{G}}\nu$ denoted as an element of $\Omega^1(M; \mathcal{g})$ by $X \mapsto \nabla^{\mathcal{G}}_X \nu$, defined by
\bas
\mleft.\nabla^{\mathcal{G}}_X \nu\mright|_x
&\coloneqq
\mleft.\frac{\mathrm{d}}{\mathrm{d}t}\mright|_{t=0} \Bigl( \mleft(\Delta \e^{t \nu}\mright)_x (X) \Bigr)
\eas
for all $x \in M$, $X \in \mathrm{T}_xM$ and $\nu \in \Gamma(\mathcal{g})$, is a vector bundle connection on $\mathcal{g}$, where $t \in \mathbb{R}$. We will call $\nabla^{\mathcal{G}}$ the \textbf{$\mathcal{G}$-connection (on its LAB $\mathcal{g}$)}.
\end{propositions*}
In fact, $\nabla^{\mathcal{G}}$ will play the role of $\nabla$ from the beginning of this introduction. Furthermore, this connection will allow us to define the generalized Maurer-Cartan equation in Thm.\ \ref{thm:GenMCEq} and Def.\ \ref{def:NowReallyYangMillsConnectio}.
\begin{definitions*}{Yang-Mills connection, simplified notation}
We say that $\mathrm{H}\mathcal{G}$, and $\mu_{\mathcal{G}}^{\mathrm{tot}}$, is a \textbf{Yang-Mills connection (w.r.t.\ a $\zeta \in \Omega^2(M; \mathcal{g})$)}, if it satisfies the \textbf{compatibility conditions}:
\begin{enumerate}
\item $\mu_{\mathcal{G}}^{\mathrm{tot}}$ is multiplicative (\textit{i.e.}\ $\mathrm{H}\mathcal{G}$ is an Ehresmann connection),
\item $\mu_{\mathcal{G}}^{\mathrm{tot}}$ satisfies the \textbf{generalized Maurer-Cartan equation}, that is,
\bas
\mleft.\mleft(\mathrm{d}^{\pi_{\mathcal{G}}^*\nabla^{\mathcal{G}}} \mu_{\mathcal{G}}^{\mathrm{tot}}
+ \frac{1}{2} \mleft[ \mu_{\mathcal{G}}^{\mathrm{tot}} \stackrel{\wedge}{,} \mu_{\mathcal{G}}^{\mathrm{tot}} \mright]_{\pi_{\mathcal{G}}^*\mathcal{g}} \mright)\mright|_g
&=
\mathrm{Ad}_{g^{-1}} \circ \mleft.\pi_{\mathcal{G}}^!\zeta\mright|_g
- \mleft.\pi_{\mathcal{G}}^!\zeta\mright|_g
\eas
for all $g \in \mathcal{G}_x$ ($x \in M$), where $\mleft[\cdot, \cdot\mright]_{\pi_{\mathcal{G}}^*\mathcal{g}}$ is the Lie bracket of the pullback-LAB $\pi_{\mathcal{G}}^*\mathcal{g}$, and the $\stackrel{\wedge}{,}$ denotes the typical graded extension of tensors, that is, the second summand on the left hand side is an element of $\Omega^2(\mathcal{G}; \pi_{\mathcal{G}}^*\mathcal{g})$ given here by
\bas
\mleft(\frac{1}{2} \mleft[ \mu_{\mathcal{G}}^{\mathrm{tot}} \stackrel{\wedge}{,} \mu_{\mathcal{G}}^{\mathrm{tot}} \mright]_{\pi_{\mathcal{G}}^*\mathcal{g}} \mright) (X, Y)
&=
\mleft[ \mu_{\mathcal{G}}^{\mathrm{tot}}(X), \mu_{\mathcal{G}}^{\mathrm{tot}}(Y) \mright]_{\pi_{\mathcal{G}}^*\mathcal{g}}
\eas
for all $X, Y \in \mathfrak{X}(\mathcal{G})$.
\end{enumerate}
\end{definitions*}
Making a pull-back of the generalized Maurer-Cartan equation w.r.t.\ a section $\sigma \in \Gamma(\mathcal{G})$ is straight-forward and will be provided in Cor.\ \ref{cor:PullbackOfMCSupperEquation}.
\begin{corollaries*}{Pullback of generalized Maurer-Cartan equation}
Let $\mathrm{H}\mathcal{G}$ be a Yang-Mills connection w.r.t.\ a $\zeta \in \Omega^2\mleft( M; \mathcal{g} \mright)$. Then
\bas
\mathrm{d}^{\nabla^{\mathcal{G}}} \Delta \sigma
+ \frac{1}{2} \mleft[ \Delta \sigma \stackrel{\wedge}{,} \Delta\sigma \mright]_{\mathcal{g}}
+ \zeta
&=
\mathrm{Ad}_{\sigma^{-1}} \circ \zeta
\eas
for all $\sigma \in \Gamma(\mathcal{G})$.
\end{corollaries*}
We will now argue that those compatibility conditions are the integrals of the infinitesimal compatibility conditions proposed by Alexei Kotov and Thomas Strobl, mentioned earlier (without details). In \cite{LAURENTGENGOUXStienonXuMultiplicativeForms} it was already observed that an Ehresmann connection on an LGB naturally implies that $\nabla^{\mathcal{G}}$ is a Lie bracket derivation.
\begin{lemmata*}{Ehresmann connections induce Lie bracket derivations, \newline \cite[\S 4.5, Prop.\ 4.21]{LAURENTGENGOUXStienonXuMultiplicativeForms}}
If $\mathrm{H}\mathcal{G}$ is an (Ehresmann) connection on $\mathcal{G}$ as principal bundle, then
\bas
\nabla^{\mathcal{G}}\mleft( \mleft[ \mu, \nu \mright]_{\mathcal{g}} \mright)
&=
\mleft[ \nabla^{\mathcal{G}} \mu, \nu \mright]_{\mathcal{g}}
+ \mleft[ \mu, \nabla^{\mathcal{G}} \nu \mright]_{\mathcal{g}}
\eas
for all $\mu, \nu \in \Gamma(\mathcal{g})$.
\end{lemmata*}
That $\nabla^{\mathcal{G}}$ is a Lie bracket derivation is one of the infinitesimal compatibility conditions.
Furthermore, we will show in Thm.\ \ref{thm:GenMCEq} that the generalized Maurer-Cartan equation is actually equivalent to its infinitesimal version.
\begin{theorems*}{Generalized Maurer-Cartan equation, changed formulation}
Let $\mathrm{H}\mathcal{G}$ be an Ehresmann connection on $\mathcal{G}$, and $\zeta \in \Omega^2\mleft( M; \mathcal{g} \mright)$. Then $\mathrm{H}\mathcal{G}$ satisfies the generalized Maurer-Cartan equation w.r.t.\ $\zeta$ if and only if
\bas
R_{\nabla^{\mathcal{G}}}(X, Y)\mu
&=
\mleft[ \zeta(X, Y), \mu \mright]_{\mathcal{g}}
\eas
for all $X, Y \in \mathfrak{X}(M)$ and $\nu \in \Gamma(\mathcal{g})$, where $R_{\nabla^{\mathcal{G}}}$ is the curvature of $\nabla^{\mathcal{G}}$.
\end{theorems*}
The condition about the curvature is in fact the second infinitesimal compatibility condition proposed by Alexei Kotov and Thomas Strobl, so that we can conclude that Yang-Mills connections on LGBs serve as an integral of the infinitesimal compatibility conditions which we will point out in Def.\ \ref{def:YangMillsConnection} and Rem.\ \ref{rem:YangMillsEqualsInfYM}.
\begin{theorems*}{Yang-Mills connections satisfy the infinitesimal compatibility conditions}
Every Yang-Mills connection (w.r.t.\ $\zeta \in \Omega^2(M; \mathcal{g})$) $\mathrm{H}\mathcal{G}$ is an \textbf{infinitesimal Yang-Mills connection (on $\mathcal{G}$)}, that is, it satisfies the \textbf{infinitesimal compatibility conditions}
\bas
\nabla^{\mathcal{G}}\mleft( \mleft[ \mu, \nu \mright]_{\mathcal{g}} \mright)
&=
\mleft[ \nabla^{\mathcal{G}} \mu, \nu \mright]_{\mathcal{g}}
+ \mleft[ \mu, \nabla^{\mathcal{G}} \nu \mright]_{\mathcal{g}},
\\
R_{\nabla^{\mathcal{G}}}(X, Y)\mu
&=
\mleft[ \zeta(X, Y), \mu \mright]_{\mathcal{g}}
\eas
for all $\mu, \nu \in \Gamma(\mathcal{g})$ and $X, Y \in \mathfrak{X}(M)$.
\end{theorems*}
We will denote such connections $\nabla^{\mathcal{G}}$ by $\nabla^{\mathrm{YM}}$. Additionally we will mention in Rem.\ \ref{rem:SimplicialDifferentialStuff} that multiplicativity of the total Maurer-Cartan form is in fact a closedness condition w.r.t.\ a certain simplicial differential $\delta$, introduced in \cite[beginning of \S 1.2]{crainic2003differentiable}. The generalized Maurer-Cartan equation is then an exactness-condition of the "classical" curvature for the total Maurer-Cartan form, its $\delta$-primitive given by $\zeta$. This is in alignment with the infinitesimal compatibility conditions which are also equivalent to statements about closedness and exactness w.r.t.\ the Chevalley-Eilenberg complex.
This will finish the discussion about the connection on $\mathcal{G}$, and it will be important for defining the field strength which will be discussed afterwards, starting with Def.\ \ref{def:NewFieldStrength}. For this we will always assume now that $\mathrm{H}\mathcal{G}$ is a Yang-Mills connection w.r.t.\ a $\zeta \in \Omega^2(M; \mathcal{g})$ (while $\mathrm{H}\mathcal{P}$ is still an Ehresmann connection).
\begin{definitions*}{(Generalized) Field strength}
We define the \textbf{(generalized) curvature} or \textbf{(generalized) field strength $F$ (of $A$)} as an element of $\Omega^2\mleft( \mathcal{P}; \pi^*\mathcal{g} \mright)$ by
\bas
F
&\coloneqq
\mathrm{d}^{\pi^*\nabla^{\mathrm{YM}}} A \circ \mleft( \pi^{\mathrm{H}\mathcal{P}}, \pi^{\mathrm{H}\mathcal{P}} \mright)
+ \pi^!\zeta,
\eas
where $\mathrm{d}^{\pi^*\nabla^{\mathrm{YM}}}$ is the exterior covariant derivative related to the pullback connection $\pi^*\nabla^{\mathrm{YM}}$, and $\pi^{\mathrm{H}\mathcal{P}}: \mathrm{T}\mathcal{P} \to \mathrm{H}\mathcal{P}$ is the canonical projection onto the associated Ehresmann connection $\mathrm{H}\mathcal{P}$ on $\mathcal{P}$; that is,
\bas
F(X, Y)
&=
\mathrm{d}^{\pi^*\nabla^{\mathrm{YM}}} A \mleft( \pi^{\mathrm{H}\mathcal{P}}(X), \pi^{\mathrm{H}\mathcal{P}}(Y) \mright)
+ \mleft(\pi^*\zeta\mright) \bigl( \mathrm{D}\pi(X), \mathrm{D}\pi(Y) \bigr)
\eas
for all $X, Y \in \mathfrak{X}(\mathcal{P})$.
\end{definitions*}
With a lengthy calculation evolving around that $\mathrm{H}\mathcal{G}$ is a Yang-Mills connection, especially using the generalized Maurer-Cartan equation, we will show in Prop.\ \ref{prop:NewFieldStrengthWithCoolProps} that $F$ transforms in a suitable way when making a pull-back w.r.t.\ the modified right-pushforward, which is the last step needed for defining the physical theory and its associated Lagrangian.
\begin{propositions*}{Properties of the generalized field strength, simplified notation}
We have the following properties of the field strength:
\begin{itemize}
\item \textbf{(Form of type $\mathrm{Ad}$)}
\bas
\mathcal{r}_{\sigma}^!F
&=
\mathrm{Ad}_{\sigma^{-1}} \circ F
\eas
for all (local) $\sigma \in \Gamma(\mathcal{G})$.
\item \textbf{(Horizontal form)}
\newline
For $X, Y \in \mathrm{T}_p\mathcal{P}$ ($p \in \mathcal{P}$) we have
\bas
F(X, Y)
&=
0
\eas
if either of $X$ and $Y$ is vertical.
\end{itemize}
\end{propositions*}
We will also derive a structure equation in Thm.\ \ref{thm:StructureEq}.
\begin{theorems*}{Structure equation of the generalized field strength}
We have the \textbf{structure equation}
\bas
F
=
\mathrm{d}^{\pi^*\nabla^{\mathrm{YM}}} A
+ \frac{1}{2} \mleft[ A \stackrel{\wedge}{,} A \mright]_{\pi^*\mathcal{g}}
+ \pi^!\zeta.
\eas
\end{theorems*}
We will argue that $\zeta$ of course also affects the Bianchi identity, which we have in fact already derived in earlier works.
\begin{theorems*}{Generalized Bianchi identity, \newline \cite[\S 7, Thm.\ 7.3]{My1stpaper} \& \cite[\S 5, Thm.\ 5.1.42]{MyThesis}}
We have the \textbf{(generalized) Bianchi identity}
\bas
\mathrm{d}^{\pi^*\nabla^{\mathrm{YM}}}F
+ \mleft[ A \stackrel{\wedge}{,} F \mright]_{\pi^*\mathcal{g}}
&=
\pi^! \mathrm{d}^{\nabla^{\mathrm{YM}}} \zeta.
\eas
\end{theorems*}
Similarly to the discussion about connections, Subsection \ref{CurvatureSubsection} will be concluded with a discussion about the gauge transformation of $F$ in Thm.\ \ref{thm:GaugeTrafoOfCurv}; as for $A$ the gauge transformations of $F$ are given by form-pullbacks with a (base-preserving) principal bundle automorphism $H$ with associated unique $\sigma^H \in \Gamma(\pi^*\mathcal{G})$ given by $H(p) = p \cdot \sigma^H_p$ for all $p \in \mathcal{P}$.
\begin{theorems*}{Gauge transformation of the generalized field strength, simplified notation}
We have that $H^!F$ is the field strength related to $H^!A$ and
\bas
H^!F
&=
{\mathrm{Ad}_{\mleft(\sigma^H\mright)^{-1}}} \circ F.
\eas
\end{theorems*}
Again as for $A$, we are interested about what this formula for the gauge transformation implies for pull-backs $F_s \coloneqq s^!F$ w.r.t.\ a gauge $s$ of $\mathcal{P}$ when changing the choice of $s$. This is straight-forward to calculate and will be stated in Thm.\ \ref{thm:LocalGaugeTrafoChangeGaugeFieldStrength}.
\begin{theorems*}{Gauge transformations again as a change of gauge}
Let $U_i$ and $U_j$ be two open subsets of $M$ so that $U_i \cap U_j \neq \emptyset$, two gauges $s_i \in \Gamma\mleft(\mathcal{P}|_{U_i}\mright)$ and $s_j \in \Gamma\mleft(\mathcal{P}|_{U_j}\mright)$, and the unique $\sigma_{ji} \in \Gamma\mleft( \mleft.\mathcal{G}\mright|_{U_i \cap U_j} \mright)$ with $s_i = s_j \cdot \sigma_{ji}$ on $U_i \cap U_j$.
Then we have for the fields strength of $A$ over $U_i \cap U_j$ that
\bas
F_{s_i}
&=
\mathrm{Ad}_{\sigma_{ji}^{-1}}\circ F_{s_j}.
\eas
\end{theorems*}
Eventually in Section \ref{CYMSection} we will start to introduce the physical theory, that is, first of all defining the Lagrangian of a gauge theory based on principal LGB-bundles in Subsection \ref{CYMDefGaugeInv}. Due to that we will have pointed out that this theory integrates the infinitesimal curved Yang-Mills-Higgs gauge theory developed by Alexei Kotov and Thomas Strobl (in the case of LABs), we are going to label this theory \textbf{curved Yang-Mills gauge theory} in Def.\ \ref{def:CYMGTFinally}. To do so, $M$ is now a spacetime, so that we can define the Hodge star operator $*$, and we assume that we have an $\mathrm{Ad}$-invariant fibre metric $\kappa$ on $\mathcal{g}$.
\begin{definitions*}{Curved Yang-Mills gauge theory}
Let $\mleft( U_i \mright)_i$ be an open covering of $M$ so that there are subordinate gauges $s_i \in \Gamma\mleft(\mathcal{P}|_{U_i}\mright)$. Then the top-degree form $\mathfrak{L}_{\mathrm{CYM}}[A] \in \Omega^{\mathrm{dim}(M)}(M; \mathbb{R})$, defined locally by
\bas
\mleft.\bigl(\mathfrak{L}_{\mathrm{CYM}}[A]\bigr)\mright|_{U_i}
&\coloneqq
- \frac{1}{2} \kappa \mleft( F_{s_i} \stackrel{\wedge}{,} *F_{s_i} \mright),
\eas
is called the \textbf{curved Yang-Mills Lagrangian}.
\end{definitions*}
We will have argued in Cor.\ \ref{cor:ContractionOfLocalFIeldWellDefined} that this Lagrangian is well-defined; a trivial consequence of the previously-highlighted gauge transformations. Similarly, we will achieve gauge invariance in Thm.\ \ref{thm:GaugeInvarianceOfLagrangian}.
\begin{theorems*}{Gauge invariance of the curved Yang-Mills Lagrangian}
We have
\bas
\mathfrak{L}_{\mathrm{CYM}}\mleft[ H^!A \mright]
&=
\mathfrak{L}_{\mathrm{CYM}}[A]
\eas
for all principal bundle automorphisms $H$.
\end{theorems*}
As the final part of statements we will discuss new examples of gauge theory in Subsection \ref{CYMExamples}, after pointing out that this gauge theory generalizes the classical formalism. We will also discuss this matter in the context of my previously-mentioned Ph.D.\ thesis, \cite{MyThesis}, and the field redefinitions. Especially, the paper concludes with the reason why this project started: The inner group bundle of the Hopf fibration $\mathds{S}^7 \to \mathds{S}^4$ is an example of curved Yang-Mills gauge theory, so that $\nabla^{\mathrm{YM}}$ is not curved; we will present this in Ex.\ \ref{ex:HopfEx}. We will point out that this is indeed due to the fact that involved LGB is non-trivial, so that we also conjecture that trivial LGBs imply the existence of a field redefinition flattening $\nabla^{\mathrm{YM}}$.
\begin{examples*}{Hopf fibration $\mathds{S}^7 \to \mathds{S}^4$ giving rise to a curved Yang-Mills gauge theory}
Let $P$ be the Hopf bundle
\begin{center}
\begin{tikzcd}
\mathrm{SU}(2) \cong \mathds{S}^3 \arrow{r} & \mathds{S}^7 \arrow{d} \\
& \mathds{S}^4
\end{tikzcd}
\end{center}
We define the principal bundle $\mathcal{P}$ and LGB $\mathcal{G}$ as the inner group bundle of $P$
\bas
\mathcal{P}
\coloneqq
\mathcal{G}
&\coloneqq
c_{\mathrm{SU}(2)}(P)
\coloneqq
(P\times G) \Big/ G,
\eas
where $(P\times G) \Big/ G$ is the associated Lie group bundle, where the action of $G$ on $G$ is given by its conjugation, $c_g(q) = gqg^{-1}$ for all $g, q \in G$. This principal $c_{\mathrm{SU}(2)}(P)$-bundle admits the structure as curved Yang-Mills gauge theory, which is neither classical nor pre-classical. Furthermore, all possible (curved) Yang-Mills gauge theory structures lead to a curved $\nabla^{\mathrm{YM}}$ and non-zero $\zeta$, which also implies that this gauge theory cannot be described with the typical formalism of gauge theory.
\end{examples*}
We will finish this paper with a short discussion about future prospects in Section \ref{conclusions}, highlighting further constructions and conjecturing how to integrate the full formalism of the gauge theory developed by Alexei Kotov and Thomas Strobl.
\subsection{Basic notations and remarks}\label{BasicNotations}
\begin{itemize}
\item The appendix serves for providing extra information or background knowledge which did not fit in the flow of this paper's text. We will sometimes refer to the appendix, but the experienced reader may be able to ignore the appendix.
\item In the main text we usually repeat and reintroduce needed objects for the statements (like "\textbf{Let $M$ be a manifold [...]}" in every statement) (almost) allowing to just read the statements without having read the text introducing it, while the appendix is written as a continuous text which has to be read as a whole in order to understand the essential statements.
\item As usual, there will be definitions of certain objects depending on other elements, and for keeping notations simple we will not always explicitly denote all dependencies. It will be clear by context on which it is based on, that is for example, if we define an object $A$ using the notion of Lie algebra actions $\gamma$ and we write "Let $X$ be an object $A$", then it will be clear by context which Lie algebra action is going to be used, for example given in a previous sentence writing "Let $\gamma$ be a Lie algebra action".
\item Throughout this work we always use Einstein's sum convention if suitable.
\item If not mentioned explicitly, we always assume finite dimensions and morphisms denote base-preserving ones.
\item With $f^*F$ we denote the pullback/pull-back of the fibre bundles $F \to M$ under a smooth map $f: N \to M$. Similarly we denote the pullbacks of sections of a fibre bundle.
%We will also have sections $F$ as an element of $\Gamma\left( \left(\bigotimes_{m=1}^{l} E_m^*\right) \otimes E_{l+1} \right)$, where $E_1, \dots E_{l+1}$ ($l \in \mathbb{N}$) are real vector bundles of finite rank over $N$. Those pull-back as section, denoted by $\Phi^*F$, we will view as an element of $\Gamma\left( \mleft(\bigotimes_{m=1}^{l} \mleft(\Phi^*E_m\mright)^*\mright) \otimes \Phi^*E_{l+1} \right)$, and it is essentially given by
%\bas
%(\Phi^*F)(\Phi^*\nu_1, \dotsc , \Phi^*\nu_l)
%&=
%\Phi^*\mleft( F\mleft( \nu_1, \dotsc, \nu_l \mright) \mright)
%\eas
%for all $\nu_1 \in \Gamma(E_1), \dotsc, \nu_l \in \Gamma(E_l)$, using that pullbacks of sections generate the sections of a pullback bundle.
\item For $V \to M$ a vector bundle over $M$ do not confuse the pull-back of sections with the pull-back of forms $\omega \in \Omega^l(M; V)$ ($l \in \mathbb{N}_0$), here denoted by $f^!\omega$, which is an element of $\Gamma\left( \mleft(\bigwedge_{m=1}^{l} \mathrm{T}^*M \mright) \otimes f^*V \right) \cong \Omega^l(M; f^*V)$, and not of $\Gamma\left( \mleft(\bigotimes_{m=1}^{l} \mleft(f^*\mathrm{T}N\mright)^*\mright) \otimes f^*V \right)$ like $f^*\omega$.
%$f^!\omega$ is defined by
%\ba
%\mleft.\mleft(f^!\omega\mright)(Y_1, \dots, Y_l)\mright|_p
%&\coloneqq
%\omega_{f(p)}\mleft(\mathrm{D}_pf\mleft(\mleft.Y_1\mright|_p\mright), \dots, \mathrm{D}_pf\mleft(\mleft.Y_l\mright|_p\mright)\mright)
%\ea
%for all $p \in M$ and $Y_1, \dots, Y_l \in \mathfrak{X}(M)$.
\item Let $F \stackrel{\pi_F}{\to} M$ and $G \stackrel{\pi_G}{\to} N$ be two fibre bundles over smooth manifolds $M$ and $N$, respectively, and let $\phi: N \to M$ be a smooth map. Furthermore, let us assume we have a morphism $\Phi: G \to F$ of fibre bundles over $\phi$, that is, $\Phi$ is a smooth map such that the following diagram commutes
\begin{center}
\begin{tikzcd}
G \arrow{d}{\pi_G} \arrow{r}{\Phi} & F \arrow{d}{\pi_F}\\
N \arrow{r}{\phi} & M
\end{tikzcd}
\end{center}
especially, $\pi_F \circ \Phi = \phi \circ \pi_G$.
We make often use of that such morphisms have a 1:1 correspondence to \textbf{base-preserving} fibre bundle morphisms $\widetilde{\Phi}: G \to \phi^*F$, \textit{i.e.}\ $\widetilde{\Phi}$ is a smooth map with $\phi^*\pi_F \circ \widetilde{\Phi} = \pi_G$. For $p \in N$ the morphism $\widetilde{\Phi}$ has the form
\bas
\widetilde{\Phi}_p
&=