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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{parallel transport} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{chernweil_theory}{}\paragraph*{{$\infty$-Chern-Weil theory}}\label{chernweil_theory} [[!include infinity-Chern-Simons theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{the_idea_of_parallelism}{The idea of parallelism}\dotfill \pageref*{the_idea_of_parallelism} \linebreak \noindent\hyperlink{the_categorytheoretic_perspective}{The category-theoretic perspective}\dotfill \pageref*{the_categorytheoretic_perspective} \linebreak \noindent\hyperlink{in_physics}{In physics}\dotfill \pageref*{in_physics} \linebreak \noindent\hyperlink{special_cases}{Special cases}\dotfill \pageref*{special_cases} \linebreak \noindent\hyperlink{Of1Form}{Trivial bundle: parallel transport of a 1-form}\dotfill \pageref*{Of1Form} \linebreak \noindent\hyperlink{higher_parallel_transport}{Higher parallel transport}\dotfill \pageref*{higher_parallel_transport} \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} A [[connection on a bundle]] $\nabla$ for $\pi : P \to X$ a $G$-[[principal bundle]] encodes data that assigns to each path $\gamma : [0,1] \to X$ a [[homomorphism]] \begin{displaymath} tra_\nabla(\gamma) : P_{\gamma(0)} \to P_{\gamma(1)} \end{displaymath} between the [[fiber]]s of the bundle, such that this assignment depends well (e.g. smoothly) on the choice of path and is compatible with composition of paths. This assignment is called the \emph{parallel transport} of the connection. \hypertarget{the_idea_of_parallelism}{}\subsubsection*{{The idea of parallelism}}\label{the_idea_of_parallelism} The term ``parallel'' comes from one of the many equivalent definitions of the notion of [[connection on a bundle]]: the original formulation of [[Ehresmann connection]]s. In that formulation, the connection is encoded at each point $p \in P$ in the total space by a decomposition of the [[tangent space]] $T_p P$ as a [[direct sum]] $T_p P \simeq V_p \oplus H_p$ of [[vector space]]s, such that \begin{itemize}% \item $V_p = \ker \pi_*|_p$ is the [[kernel]] of the projection map that sends vectors in the total space to vectors in base space (this part is fixed by the choice of $p : P \to X$); \item $H_p \subset T_p P$ is a \emph{choice} of complement, such that this choice varies smoothly over $P$ in an evident sense and is compatible with the $G$-[[action]] on $P$. \end{itemize} The vectors in $V_p$ are called \emph{vertical} , the vectors in $H_p$ are called \emph{horizontal} . One may think of this as defining locally in which way the base space sits horizontally in the total space, equivalently as identifying locally a ``smoothly varying local trivialization'' of $P$. More precisely, given such a choice of horizontal subspaces, there is for every path $\gamma : [0,1] \to X$ and every choice of lift $\hat \gamma(0) \in P$ of the start point $\gamma(0)$ to the total space of the bundle, a \emph{unique lift} $\hat \gamma : [0,1] \to P$ of the entire path to the total space: \begin{displaymath} \itexarray{ \hat \gamma(0) &\stackrel{\hat \gamma}{\to}& \hat \gamma(1) && \in & P \\ && &&& \downarrow^{\mathrlap{\pi}} \\ \gamma(0) &\stackrel{\gamma}{\to}& \gamma(1) && \in & X } \, \end{displaymath} such that $\hat \gamma$ is everywhere \emph{parallel} (to $X$) in that all its tangent vectors sit in the horizontal subspaces chosen: \begin{displaymath} (\partial_\sigma \hat \gamma)(\sigma) \in H_{\gamma(\sigma)} \subset T_{\gamma(\sigma)} P \,. \end{displaymath} In other words, this means that given a path $\gamma$ down in $X$, we may \emph{transport} any point $p \in P_{\gamma(0)}$ above its start point \emph{parallely} (with respect to the notion of parallelism determined by $\nabla$) along $\gamma$, to find a uniquely determined point $tra_\nabla(\gamma)(p) \in P_{\gamma(1)}$ over the endpoint. \hypertarget{the_categorytheoretic_perspective}{}\subsubsection*{{The category-theoretic perspective}}\label{the_categorytheoretic_perspective} The parallel transport-assignment of fiber-homomorphisms to paths \begin{displaymath} (x \stackrel{\gamma}{\to} y) \mapsto ( P_x \stackrel{tra_\nabla(\gamma)}{\to} y ) \end{displaymath} enjoys the following properties: \begin{itemize}% \item it is invariant under [[thin homotopy]] of paths; \item it is compatible with composition of paths and sends constant paths to identity homomorphisms; \item it sends smooth families of paths to compatible smooth families of homomorphisms. \end{itemize} This may be equivalently but more succinctly be formulated as follows: We say \emph{diffeological groupoid} for an [[internal groupoid]] in the category of [[diffeological space]]s. The smooth paths in a smooth manifold $X$ naturally form the diffeological groupoid called the [[path groupoid]] $P_1(X)$. Objects are points in $X$, morphsims are [[thin homotopy]]-classes of smooth paths which are constant in a neighbourhood of their boundary, composition is concatenation of paths. For $P \to X$ any $G$-bundle, there is also naturally the diffeological groupoid $At(P)$ -- the [[Atiyah Lie groupoid]] of $P$. Objects are points in $X$, morphisms are homomorphisms of $G$-[[torsor]]s between the [[fiber]]s over these points. Then the above properties of parallel transport are equivalent to saying that we have an [[internal functor]] \begin{displaymath} tra : P_1(X) \to At(P) \end{displaymath} that is the identity on objects. Moreover, this functor \emph{uniquely} characterizes the [[connection on a bundle|connection]] on $P$ that it comes from. This means that we may identify connections on $P$ with their parallel transport functors. But even the bundle $P$ itself is encoded in such functors. If instead of looking at the category of internal groupoids and internal functors, we look at the larger [[2-topos]] of \emph{diffeological stacks} -- [[stack]]s over [[CartSp]]. Then we can take simply the diffeological [[delooping]] groupoid $\mathbf{B}G$, which has a single object and $G$ as its [[hom-set]] and consider morphisms \begin{displaymath} tra : P_1(X) \to \mathbf{B}G \end{displaymath} in the 2-topos. These are now given by [[anafunctors]] of internal groupoids, and one finds that they encode a [[Cech cohomology|Cech cocycle]] for a $G$-principal bundle $P$ together with the parallel transport of a connection over it. This is discussed in more detail at \begin{itemize}% \item [[∞-Chern-Weil theory -- preparatory concepts]]. \end{itemize} There is also the diffeological groupoid incarnation of the [[fundamental groupoid]] $\Pi_1(X)$ of $X$. Its morphisms are full [[homotopy]]-classes of paths. There is a canonical projection $P_1(X) \to \Pi_1(X)$ that sends a thin-homotopy class of paths to the corresponding full-homotopy class. A parallel transport functor $tra : P_1(X) \to G$ factors through $\Pi_1(X)$ precisely if the corresponding conneciton is \emph{flat} in that its [[curvature form]] vanishes. \hypertarget{in_physics}{}\subsubsection*{{In physics}}\label{in_physics} In [[physics]], a [[connection on a bundle]] over $X$ models a [[gauge field]] such as the [[electromagnetic field]] or more generally a [[Yang-Mills field]] or the field of [[gravity]] on a [[spacetime]] $X$. The [[force]]s exerted by such gauge fields on charged particles propagating on $X$ (i.e. [[electron]]s, [[quark]]s and generally massive particles, respectively) are encoded precisely in the parallel transport assignment of the gauge field connection to their trajectories. More precisely, the exponentiated [[action functional]] for the electron propagating on $X$ in the presence of an electromagnetic field $\nabla$ is the functional on the space of paths in $X$ given by \begin{displaymath} \gamma \mapsto \exp(i S_{kin}(\gamma)) \cdot tra_\nabla(\gamma) \,, \end{displaymath} where the first term is the standard [[kinetic action]]. If $\nabla$ is a (nontrivial) connection on a trivial bundle, then, as described \hyperlink{Of1Form}{below} it is encoded by a [[differential form]] $A \in \Omega^1(X)$ -- called the \emph{vector potential} in physics -- and we have \begin{displaymath} tra_\nabla(\gamma) = \exp(i \int_[0,1]} \gamma^* A) \,. \end{displaymath} The [[Euler-Lagrange equations]] induced by this functional express precisely the [[Lorentz force]] encoded by $A$ acting on the particle. If instead of looking at the [[quantum mechanics]] of the quantum particle charged under a fixed background gauge field look at the [[quantum field theory]] of that gauge field itself, we can use the action functional of particles to \emph{probe} these background fields and obtain quantum observables for them. This converse assignment where we \emph{fix} a path $\gamma$ and regard the parallel transport then as a functional over the space of all connections over $X$ \begin{displaymath} tra_{(-)}(\gamma) : connections \to fiber-homomorphisms \end{displaymath} is called the \textbf{[[Wilson line]]-observable} of the theory. Or rather its expectation value in the [[path integral]] weighted by the action functional of the gauge theory is called such, schematically: \begin{displaymath} W_\gamma := \langle tra_{(\nabla)}(\gamma)\rangle = \int D \nabla \; \exp(i S_{gauge\;theory}(\nabla)) tra_\nabla(\gamma) \,. \end{displaymath} \hypertarget{special_cases}{}\subsection*{{Special cases}}\label{special_cases} \hypertarget{Of1Form}{}\subsubsection*{{Trivial bundle: parallel transport of a 1-form}}\label{Of1Form} Of $P \to X$ is a \emph{trivial bundle} in that $P = X \times G$, then a connection on this is equivalently encoded in a [[Lie-algebra valued 1-form]] \begin{displaymath} A \in \Omega^1(X, \mathcal{g}) \end{displaymath} on $X$. In terms of this, parallel transport is a solution to a [[differential equation]]. For $\gamma : [0,1] \to X$ we have the pull-back 1-form $\gamma^* A \in \Omega^1([0,1])$. For $f \in C^\infty([0,1], G)$ a smooth function with values in the [[Lie group]] $G$, consider the differential equation \begin{displaymath} d f + \rho(f)_*(\gamma^*A) = 0 \,, \end{displaymath} where $d f : T [0,1] \to T G$ is the differential of $f$ and where $\rho : G \times G \to G$ is the left [[action]] of $G$ on itself (i.e. just the multiplication on $G$) and $r(f)_* : T G \to T G$ its differential and using the defining identification $\mathfrak{g} \simeq T_e G$ we take $r(f)_*(A)$ to be the composite $T [0,1] \stackrel{\gamma^* A}{\to} \mathfrak{g} \hookrightarrow T G \stackrel{r(f)_*}{\to} T G$. If $G$ is a [[matrix Lie group]] such as the [[orthogonal group]] $O(n)$ or the [[unitary group]] $U(n)$, then also its Lie algebra identifies with matrices, and we may write this simply as \begin{displaymath} d f + \gamma^*(A) \cdot f = 0 \,, \end{displaymath} where the dot is matrix multiplication. By general results on [[differential equations]], this type of equation has a unique solution for each choice of value of $f(0)$. \begin{udefn} The parallel transport of $A \in \Omega^1(X,\mathfrak{g})$ along a path $\gamma : [0,1] \to X$ which we write \begin{displaymath} tra_A(\gamma) := P \exp(\int_{[0,1]} \gamma^* A) \in G \end{displaymath} is the value $f(1) \in G$ for the unique solution of the equation $d f + \rho(f)_*(A) = 0$ with initial value $f(0) = e$ (the neutral element in $G$). \end{udefn} The notation here is motivated from the special case where $G = \mathbb{R}$ is the group of [[real number]]s. In that case the [[Lie algebra]] $\mathfrak{g} \simeq \mathbb{R}$ is abelian, the differential equation above is simply \begin{displaymath} d f = \gamma^*(A) \wedge f \end{displaymath} for a real valued function $f \in C^\infty([0,1])$, and the unique solution to that with $f(0) = e = 0$ is literally the exponential of the integral of $A$: \begin{displaymath} tra_A(\gamma) = \exp(\int_{[0,1]} \gamma^* A) \,. \end{displaymath} In the case of general nonabelian $\mathfrak{g}$ this simple exponential formula gives the wrong result. One can see that a slightly better approximation to the correct result is given by \begin{displaymath} \exp(\int_{[0,1/2]} \gamma^* A) \cdot \exp(\int_{[1/2,1]} \gamma^* A) \end{displaymath} and an even a bit more better approximation by \begin{displaymath} \exp(\int_{[0,1/3]} \gamma^* A) \cdot \exp(\int_{[1/3,2/3]} \gamma^* A) \cdot \exp(\int_{[2/3,1]} \gamma^* A) \end{displaymath} and so on, with the correct result being the limit of this sequence -- if one defines it carefully -- as we integrate piecewise over ever smaller pieces of the path. This is called a \textbf{path-ordered integral}. The ``P'' in the above formula is short for ``path ordering''. Possibly this notation originates in [[physics]] where the above is known as the [[Dyson formula]]. \hypertarget{higher_parallel_transport}{}\subsection*{{Higher parallel transport}}\label{higher_parallel_transport} The notion of [[connection on a bundle]] generalizes to that of [[connection on a 2-bundle]]. [[connection on a 3-bundle]] and generally to that of [[connection on an ∞-bundle]]. The come with a notion of \emph{higher parallel transport} over manifolds of dimension greater than 1. See [[higher parallel transport]] for details. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[connection on a bundle]], [[connection on a 2-bundle]], [[connection on an infinity-bundle]], \item \textbf{parallel transport}, [[higher parallel transport]], [[super parallel transport]] \begin{itemize}% \item [[nonabelian Stokes theorem]] \end{itemize} \item [[holonomy]] \begin{itemize}% \item [[holonomy group]] \item [[special holonomy]] \end{itemize} \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} A collection of references on the equivalent reformulation of connections in terms of their parallel transport is . For more see the references at [[connection on a bundle]]. A discussion of parallel transport in the [[tangent bundle]] in terms of [[synthetic differential geometry]] (motivated by a discussion of [[gravity]]) is in \begin{itemize}% \item [[Gonzalo Reyes]], \emph{General Relativity: Affine connections, parallel transport and sprays} (\href{https://marieetgonzalo.files.wordpress.com/2009/01/affineconnections.pdf}{pdf}) \end{itemize} \end{document}