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\section*{fiber bundles in physics}

\begin{quote}%
This entry is supposed to be a survey of and motivation for the role of [[fiber bundles]] in [[physics]].


\end{quote}

\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{bundles}{}\paragraph*{{Bundles}}\label{bundles}

[[!include bundles - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{BasicIdeaOfDefinition}{Basic idea of the definition of fiber bundles}\dotfill \pageref*{BasicIdeaOfDefinition} \linebreak
\noindent\hyperlink{examples_of_fiber_bundles_in_physics}{Examples of fiber bundles in physics}\dotfill \pageref*{examples_of_fiber_bundles_in_physics} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

All of [[physics]] has two aspects: a local or even [[infinitesimal object|infinitesimal]] aspect, and a global aspect. Much of traditional lore deals just with the local and infinitesimal aspects -- the \emph{[[perturbative field theory|perturbative]]} aspects -- and [[fiber bundles]] play little role there. But they are the all-important structure that govern the global -- the \emph{[[non-perturbative field theory|non-perturbative]]} -- aspects. [[bundles|Bundles]] are the \emph{global} structure of [[physical fields]] and they are irrelevant only for the crude local and perturbative description of reality.

\hypertarget{BasicIdeaOfDefinition}{}\subsection*{{Basic idea of the definition of fiber bundles}}\label{BasicIdeaOfDefinition}

One way to think of fiber bundles is that they are the data to \emph{globally twist functions} (on [[spacetime]], say) where ``global twist'' is much in the sense of ``global [[anomaly]]'' and the like, namely an effect visible on topologically nontrivial spaces when moving around non-contractible cycles. The concept of \emph{[[monodromy]]} -- which may be more familiar to physicists -- is closely related: monodromy is something exhibited by a [[connection on a bundle]] and specifically by a [[flat connection|flat bundle]]. For a [[discrete group|discrete]] structure group ([[gauge group]]) \emph{every} bundle is flat, and in this case non-trivial bundles and non-trivial [[monodromy]] come down to essentially the same thing (see also at \emph{[[local system]]}).

More explicitly, suppose $X$ denotes [[spacetime]] and $F$ denotes some space that one wants to map into. For instance $F$ might be the [[complex numbers]] and a [[free field|free]] [[scalar field]] would be a [[function]] $X\to F$. For the following it is useful to talk about functions a bit more indirectly: observe that the [[projection]] $pr_2 \colon F \times X \to X$ from the [[product]] of $F$ with $X$ down to $X$ is such that a [[section]] of this map, namely a $\Phi \colon X \to X\times F$ such that

\begin{displaymath}
pr_2 \circ \Phi = id_X
  \phantom{AAAAA}
  \itexarray{
     && F \times X
     \\
     &
     {}^{\mathllap{\Phi}}\nearrow
     & \downarrow\mathrlap{pr_2}
     \\
     X &=& X
  }
\end{displaymath}
is just the same as data a function $X \to F$. We think of the projection $X \times F \overset{pr_2}{\to} X$ as encoding the fact that there is one copy of $F$ associated with each point of $X$, and think of a function with values in $F$ as something that, of course, takes values in $F$ over each point of $X$. One says that this projection, suggestively shown vertically,

\begin{displaymath}
\itexarray{
    F \times X
    \\
    \downarrow\mathrlap{pr_2}
    \\
    X
  }
\end{displaymath}
is the \emph{[[trivial bundle|trivial]]} $F$-[[fiber bundle]] over $X$.

If $F$ is equipped with the [[structure]] of a [[vector space]] then one calls this a \emph{[[trivial vector bundle|trivial]] [[vector bundle]]}, etc.

The point being that more generally we may add a global ``twist'' to the $F$-valued functions by making the space $F$ vary to some degree as we move along $X$. For a [[fiber bundle]] one requires that it doesn't change \emph{much}: in fact the word ``fiber'' in ``[[fiber bundle]]'' refers to the fact that \emph{all} fibers (over all points of $X$) are \emph{[[equivalence|equivalent]]}. But the point is that any $F$ may be equivalent to \emph{itself} in more than one way (it may have ``[[automorphisms]]''), and this allows non-trivial global structure even though all fibers look alike.

In this sense, a general $F$-[[fiber bundle]] on some $X$ is defined to be a [[space]] $P$ equipped with a map

\begin{displaymath}
\itexarray{
     P
     \\
     \downarrow \mathrlap{fb}
     \\ 
     X
  }
\end{displaymath}
to the base space $X$ (e.g. to spacetime), such that \emph{locally} it looks like the trivial $F$-fiber bundle, up to equivalence.

To say this more technically: $P \overset{fb}{\to} X$ is called an $F$-[[fiber bundle]] if there exists a [[cover]] ([[open cover]]) of $X$ by patches (e.g. [[coordinate charts]]!) $U_i \to X$ for some index [[set]] $I$, such that for each patch $U_i$ (with $i \in I$) there exists a fiberwise [[equivalence]] between the restriction $P|_{U_i}$ of $P$ to $U_i$, and the trivial $F$-fiber bundle $F\times U_i \to U_i$ over the patch $U_i$.

To say this again in terms of sections: this means that a [[section]] $\Phi$ of $P \overset{fb}{\to} X$

\begin{displaymath}
fb \circ \Phi = id_X
  \phantom{AAAAA}
  \itexarray{
     && P
     \\
     &
     {}^{\mathllap{\Phi}}\nearrow
     & \downarrow\mathrlap{fb}
     \\
     X &=& X
  }
\end{displaymath}
is locally on each ([[coordinate chart|coordinate]]) patch $U_i$ simply an $F$-valued function, but when we change patches (change coordinates) then there may be a non-trivial identifications (notably: [[gauge transformations]]) that relates the values of the function on one patch to that on another patch, where they overlap.

Even if this may seem a bit roundabout on first sight, this is actually something at the very heart of modern physics, in that it embodies the two central principles of modern physics, namely

\begin{enumerate}%
\item the \emph{principle of locality};


\item the \emph{gauge principle}.



\end{enumerate}
The first roughly says that every global phenomenon in physics must come from local data. In the above discussion this means that any ``globally $F$-valued thing on spacetime $X$'' must come from just $F$-valued functions on local (coordinate) charts $U_i \hookrightarrow X$ of spacetime. BUT -- and this is key now --, second, the \emph{gauge principle} says that we may never strictly identify any two phenomena in physics (neither locally nor globally) but we must always ask instead for [[gauge transformations]] connecting two maybe seemingly different phenomena. Hence combining the gauge principle with the locality principle means that if an $F$-valued something on spacetime is locally given by plain $F$-valued functions, then it should be globally given by gluing these $F$-valued functions together not by \emph{identification} but by [[gauge equivalence]]. The result may be a structure that has global twists, and the nature of these global twists is precisely what an $F$-fiber bundle embodies.

\hypertarget{examples_of_fiber_bundles_in_physics}{}\subsection*{{Examples of fiber bundles in physics}}\label{examples_of_fiber_bundles_in_physics}

For instance the [[gauge fields]] in [[Yang-Mills theory]], hence in [[electromagnetism]], in [[QED]] and in [[QCD]], hence in the [[standard model of particle physics|standard model of the known universe]], are not really just the local [[differential 1-forms]] ``$A_\mu^a$'' known from so many textbooks, but are \emph{globally} really [[connections on principal bundles]] (or their [[associated bundles]]) and this is all-important once one passes to [[non-perturbative field theory|non-perturbative]] [[Yang-Mills theory]], hence to the full story, instead of its infinitesimal or local approximation.

Notably what is called a \emph{[[Yang-Mills instanton]]} in general and the \emph{[[QCD instanton]]} in particular is nothing but the underlying nontrivial class of the principal bundle underlying the Yang-Mills [[gauge field]]. Specifically, what physicists call the \emph{[[instanton number]]} for [[special unitary group|SU(2)]]-[[gauge field theory]] in 4-dimensions is precisely what mathematically is called the [[Chern class|second Chern-class]], a ``[[characteristic class]]'' of these gauge bundles.

\begin{quote}%
\emph{YM Instanton = class of principal bundle underlying the non-perturbative gauge field}


\end{quote}
To appreciate the utmost relevance of this, observe that the non-perturbative vacuum of the [[experiment|observable world]] is a ``sea of instantons'' with about one [[Yang-Mills instanton|YM instanton]] per [[femtometer]] to the 4th. See for instance the first sections of (\href{instanton+in+QCD#SchaeferShuryak98}{Schaefer-Shuryak 98}) for a review of this fact. So the very substance of the physical world, the very vacuum that we inhabit, is all controled by non-trivial fiber bundles and is inexplicable without these.

Also [[monopole]] solutions in physics are mathematically nontrivial [[principal bundles]]. For instance the [[Dirac monopole]] (that appears in [[Dirac charge quantization]]) or the [[Yang monopole]].

Similarly [[fiber bundles]] control all other [[topology|topologically]] non-trivial aspects of [[physics]]. For instance most [[quantum anomalies]] are the statement that what looks like an [[action functional|action]] \emph{[[function]]} to feed into the [[path integral]], is globally really the [[section]] of a non-trivial bundle -- notably a [[Pfaffian line bundle]] resulting from the [[fermionic path integrals]]. Moreover \emph{all} [[classical anomalies]] are statements of nontrivializability of certain fiber bundles.

Indeed, as the discussion there shows, [[quantization]] as such, if done [[non-perturbative field theory|non-perturbatively]], is all about lifting [[differential form]] data to [[line bundle]] data, this is called the \emph{[[prequantum line bundle]]} which exists over any globally quantizable [[phase spaces]] and controls all of its [[quantum theory]]. It reflects itself in many [[central extensions]] that govern [[quantum physics]], such as the [[Heisenberg group]] central extension of the Hamiltonian translation and generally and crucially the [[quantomorphism group]] central extension of the [[Hamiltonian diffeomorphisms]] of [[phase space]]. All these [[central extensions]] are non-trivial fiber bundles, and the ``quantum'' in ``quantization'' to a large extent a reference to the discrete (quantized) [[characteristic classes]] of these bundles. One can indeed understand quantization as such as the lift of infinitesimal classical differential form data to global bundle data. This is described in detail at \emph{\href{quantization#MotivationFromClassicalMechanicsAndLieTheory}{quantization -- Motivation from classical mechanics and Lie theory}}.

But actually the role of fiber bundles reaches a good bit deeper still. Quantization is just a certain [[extension]] step in the general story, but already [[classical field theory]] cannot be understood globally without a notion of bundle. Notably the very formalization of what a \emph{[[field (physics)|classical field]]} really is says: a [[section]] of a \emph{[[field bundle]]}. For instance the global nature of [[spinors]], hence \emph{[[spin structures]]} and their subtle effect on [[fermion]] physics are all encoded by the corresponding [[spinor bundles]].

Two aspects of bundles in physics come together in the theory of [[gauge fields]] and combine to produce [[fiber infinity-bundle|higher fiber bundles]]: namely we saw above that a [[gauge field]] is itself already a bundle (with a [[connection on a bundle|connection]]), and hence the bundle of which a gauge field is a section has to be a ``second-order bundle''. This is called \emph{[[gerbe]]} or \emph{[[principal 2-bundle|2-bundle]]}: the only way to realize the [[Yang-Mills field]] both locally and globally accurately is to consider it as a section of a bundle whose typical fiber is $\mathbf{B}G$, the universal [[moduli stack]] of $G$-[[principal bundles]]. For more on this see on the nLab at \emph{\href{field%20%28physics%29#IdeaOfFieldBundlesAndItsProblems}{The traditional idea of field bundles and its problems}}.

All of this becomes even more pronounced as one digs deeper into \emph{[[local quantum field theory]]}, with locality formalized as in the [[cobordism hypothesis|cobordism theorem]] that classifies local [[topological field theories]]. Then already the [[local Lagrangians]] and [[local action functionals]] themselves are [[principal infinity-connection|higher connections]] on higher bundles over the [[moduli infinity-stack|higher moduli stack]] of fields. For instance the fully local formulation of [[Chern-Simons theory]] exhibits the Chern-Simons [[action functional]] --- with all its global [[gauge invariance]] correctly realized -- as a [[universal Chern-Simons circle 3-bundle]]. This is such that by [[transgression]] to lower [[codimension]] it reproduces all the global gauge structure of this field theory, such as in codimension 2 the \emph{[[WZW gerbe]]} (itself a fiber 2-bundle: the [[background gauge field|background]] [[B-field]] of the [[WZW model]]!), in codimension 1 the [[prequantum line bundle]] on the [[moduli space of connections]] whose sections in turn yield the [[Hitchin connection|Hitchin bundle]] of [[conformal blocks]] on the [[moduli space of Riemann surfaces]].

And so on and so forth. In short: all global structure in [[field theory]] is controled by [[fiber bundles]], and all the more the more the field theory is [[quantum field theory|quantum]] and [[gauge field theory|gauge]]. The only reason why this can be ignored to some extent is because field theory is a complex subject and maybe the majority of discussions about it concerns really only a small little perturbative local aspect of it. But this is not the reality. The [[QCD]] [[vacuum]] that we inhabit is filled with a sea of non-trivial bundles and the whole quantum structure of the laws of nature are bundle-theoretic at its very heart.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[geometry of physics]]


\item [[twisted smooth cohomology in string theory]]


\item [[motivation for sheaves, cohomology and higher stacks]]


\item [[higher category theory and physics]]


\item [[string theory FAQ]]


\item [[motivation for higher differential geometry]]


\item [[applications of (higher) category theory]]


\item [[motivation for cohesion]]


\item [[Hilbert's sixth problem]]


\item [[motives in physics]]


\item [[model theory and physics]]


\item [[L-infinity algebras in physics]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item L. Mangiarotti, [[Gennadi Sardanashvily]], \emph{Connections in Classical and Quantum Field Theory}, World Scientific, 2000


\item Luciano Boi, \emph{Geometrical and topological foundations of theoretical physics: From gauge theories to string program}, 2003 (\href{http://www.emis.de/journals/HOA/IJMMS/2004/33-361777.pdf}{pdf})


\item [[A. P. Balachandran]], G. Marmo, B.-S. Skagerstam, A. Stern, \emph{Gauge Theories and Fibre Bundles - Applications to Particle Dynamics}, Springer ``Lecture Notes in Physics'', 188 (1983) (\href{https://arxiv.org/abs/1702.08910}{arXiv:1702.08910})


\item P. Balachandran, G. Marmo, B.-S. Skagerstam, A. Stern, \emph{Classical Topology and Quantum States} (World Scientific, Singapore, 1991).


\item Adam Marsh, \emph{Gauge Theories and Fiber Bundles: Definitions, Pictures, and Results} (\href{https://arxiv.org/abs/1607.03089}{arXiv:1607.03089})


\item [[Dale Husemoeller]], [[Michael Joachim]], [[Branislav Jurco]], [[Martin Schottenloher]], \emph{[[Basic Bundle Theory and K-Cohomology Invariants]]}, Lecture Notes in Physics, Springer 2008 (\href{http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726corr1.pdf}{pdf})


\item Gerd Rudolph, Matthias Schmidt, \emph{Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields}, Theoretical and Mathematical Physics series, Springer 2017 (\href{https://link.springer.com/book/10.1007/978-94-024-0959-8}{doi:10.1007/978-94-024-0959-8})



\end{itemize}
category: motivation

[[!redirects fiber bundles and physics]]



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