For semisimple Lie algebra targets
For discrete group targets
For discrete 2-group targets
For Lie 2-algebra targets
For targets extending the super Poincare Lie algebra
(such as the supergravity Lie 3-algebra, the supergravity Lie 6-algebra)
Chern-Simons-supergravity
for higher abelian targets
for symplectic Lie n-algebroid targets
for the $L_\infty$-structure on the BRST complex of the closed string:
higher dimensional Chern-Simons theory
topological AdS7/CFT6-sector
A connection on a bundle $\nabla$ for $\pi : P \to X$ a $G$-principal bundle encodes data that allows to assigns to each path $\gamma : [0,1] \to X$ homomorphisms
between the fibers 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 parallel transport of the connection.
The term “parallel” comes from one of the many equivalent definitions of the notion of connection on a bundle: the original formulation of Ehresmann connections.
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 spaces, such that
$V_p = \her \pi_*|_p$ 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 $p : P \to X$);
$H_p \subset T_p P$ is a choice of complement, such that this choice varies smoothly over $P$ in an evident sense and is compatible with the $G$-action on $P$.
The vectors in $V_p$ are called vertical , the vectors in $H_p$ are called 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 unique lift $\hat \gamma : [0,1] \to P$ of the entire path to the total space:
such that $\hat \gamma$ is everywhere parallel (to $X$) in that all its tangent vectors sit in the horizontal subspaces choses:
In other words this means that given a path $\gamma$ down in $X$, we may transport any point $p \in P_{\gamma(0)}$ above its start point 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.
The parallel transport-assignment of fiber-homomorphisms to paths
enjoys the following properties:
it is invariant under thin homotopy of paths;
it is compatible with composition of paths and sends constant paths to identity homomorphisms;
it sends smooth families of paths to compatible smooth families of homomorphisms.
This may be equivalently but more succinctly be formulated as follows:
We say diffeological groupoid for an internal groupoid in the category of diffeological spaces.
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 $G$-torsors between the fibers over these points.
Then the above properties of parallel transport are equivalent to saying that we have an internal functor
that is the identity on objects. Moreover, this functor uniquely characterizes the 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 diffeological stacks – stacks 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
in the 2-topos. These are now given by anafunctors of internal groupoids, and one finds that they encode a 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
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 flat in that its curvature form vanishes.
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 forces exerted by such gauge fields on charged particles propagating on $X$ (i.e. electrons, quarks 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
where the first term is the standard kinetic action. If $\nabla$ is a (nontrivial) connection on a trivial bundle, then, as described below it is encoded by a differential form $A \in \Omega^1(X)$ – called the vector potential in physics – and we have
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 charghed 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 probe these background fields and obtain quantum observables for them.
This converse assignment where we fix a path $\gamma$ and regard the parallel transport then as a functional over the space of all connections over $X$
is called the 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:
Of $P \to X$ is a trivial bundle in that $P = X \times G$, then a connection on this is equivalently encoded in a Lie-algebra valued 1-form
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
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
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 $f(0)$.
The parallel transport of $A \in \Omega^1(X,\mathfrak{g})$ along a path $\gamma : [0,1] \to X$ which we write
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 $G$).
The notation here is motivated from the special case where $G = \mathbb{R}$ is the group of real numbers. In that case the Lie algebra $\mathfrak{g} \simeq \mathbb{R}$ is abelian, the differential equation above is simply
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 $A$:
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
and an even a bit more better approximation by
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 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.
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 higher parallel transport over manifolds of dimension greater than 1.
See higher parallel transport for details.
connection on a bundle, connection on a 2-bundle, connection on an infinity-bundle,
parallel transport, higher parallel transport, super parallel transport
A collection of references on the equivalent reformulation of connections in terms of their parallel transport is here.
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