parallel transport


\infty-Chern-Weil theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory






A connection on a bundle \nabla for π:PX\pi : P \to X a GG-principal bundle encodes data that assigns to each path γ:[0,1]X\gamma : [0,1] \to X a homomorphism

tra (γ):P γ(0)P γ(1) tra_\nabla(\gamma) : P_{\gamma(0)} \to P_{\gamma(1)}

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 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 connections.

In that formulation, the connection is encoded at each point pPp \in P in the total space by a decomposition of the tangent space T pPT_p P as a direct sum T pPV pH pT_p P \simeq V_p \oplus H_p of vector spaces, such that

  • V p=kerπ *| pV_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:PXp : P \to X);

  • H pT pPH_p \subset T_p P is a choice of complement, such that this choice varies smoothly over PP in an evident sense and is compatible with the GG-action on PP.

The vectors in V pV_p are called vertical , the vectors in H pH_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 PP.

More precisely, given such a choice of horizontal subspaces, there is for every path γ:[0,1]X\gamma : [0,1] \to X and every choice of lift γ^(0)P\hat \gamma(0) \in P of the start point γ(0)\gamma(0) to the total space of the bundle, a unique lift γ^:[0,1]P\hat \gamma : [0,1] \to P of the entire path to the total space:

γ^(0) γ^ γ^(1) P π γ(0) γ γ(1) X \array{ \hat \gamma(0) &\stackrel{\hat \gamma}{\to}& \hat \gamma(1) && \in & P \\ && &&& \downarrow^{\mathrlap{\pi}} \\ \gamma(0) &\stackrel{\gamma}{\to}& \gamma(1) && \in & X } \,

such that γ^\hat \gamma is everywhere parallel (to XX) in that all its tangent vectors sit in the horizontal subspaces chosen:

( σγ^)(σ)H γ(σ)T γ(σ)P. (\partial_\sigma \hat \gamma)(\sigma) \in H_{\gamma(\sigma)} \subset T_{\gamma(\sigma)} P \,.

In other words, this means that given a path γ\gamma down in XX, we may transport any point pP γ(0)p \in P_{\gamma(0)} above its start point parallely (with respect to the notion of parallelism determined by \nabla) along γ\gamma, to find a uniquely determined point tra (γ)(p)P γ(1)tra_\nabla(\gamma)(p) \in P_{\gamma(1)} over the endpoint.

The category-theoretic perspective

The parallel transport-assignment of fiber-homomorphisms to paths

(xγy)(P xtra (γ)y) (x \stackrel{\gamma}{\to} y) \mapsto ( P_x \stackrel{tra_\nabla(\gamma)}{\to} y )

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 XX naturally form the diffeological groupoid called the path groupoid P 1(X)P_1(X). Objects are points in XX, morphsims are thin homotopy-classes of smooth paths which are constant in a neighbourhood of their boundary, composition is concatenation of paths.

For PXP \to X any GG-bundle, there is also naturally the diffeological groupoid At(P)At(P) – the Atiyah Lie groupoid of PP. Objects are points in XX, morphisms are homomorphisms of GG-torsors between the fibers over these points.

Then the above properties of parallel transport are equivalent to saying that we have an internal functor

tra:P 1(X)At(P) tra : P_1(X) \to At(P)

that is the identity on objects. Moreover, this functor uniquely characterizes the connection on PP that it comes from. This means that we may identify connections on PP with their parallel transport functors.

But even the bundle PP 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 stacksstacks over CartSp.

Then we can take simply the diffeological delooping groupoid BG\mathbf{B}G, which has a single object and GG as its hom-set and consider morphisms

tra:P 1(X)BG tra : P_1(X) \to \mathbf{B}G

in the 2-topos. These are now given by anafunctors of internal groupoids, and one finds that they encode a Cech cocycle for a GG-principal bundle PP 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 Π 1(X)\Pi_1(X) of XX. Its morphisms are full homotopy-classes of paths. There is a canonical projection P 1(X)Π 1(X)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)Gtra : P_1(X) \to G factors through Π 1(X)\Pi_1(X) precisely if the corresponding conneciton is flat in that its curvature form vanishes.

In physics

In physics, a connection on a bundle over XX models a gauge field such as the electromagnetic field or more generally a Yang-Mills field or the field of gravity on a spacetime XX.

The forces exerted by such gauge fields on charged particles propagating on XX (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 XX in the presence of an electromagnetic field \nabla is the functional on the space of paths in XX given by

γexp(iS kin(γ))tra (γ), \gamma \mapsto \exp(i S_{kin}(\gamma)) \cdot tra_\nabla(\gamma) \,,

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 AΩ 1(X)A \in \Omega^1(X) – called the vector potential in physics – and we have

tra_\nabla(\gamma) = \exp(i \int_[0,1]} \gamma^* A) \,.

The Euler-Lagrange equations induced by this functional express precisely the Lorentz force encoded by A tra_\nabla(\gamma) = \exp(i \int_[0,1]}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 probe these background fields and obtain quantum observables for them.

This converse assignment where we fix a path γ\gamma and regard the parallel transport then as a functional over the space of all connections over XX

tra ()(γ):connectionsfiberhomomorphisms tra_{(-)}(\gamma) : connections \to fiber-homomorphisms

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:

W γ:=tra ()(γ)=Dexp(iS gaugetheory())tra (γ). W_\gamma := \langle tra_{(\nabla)}(\gamma)\rangle = \int D \nabla \; \exp(i S_{gauge\;theory}(\nabla)) tra_\nabla(\gamma) \,.

Special cases

Trivial bundle: parallel transport of a 1-form

Of PXP \to X is a trivial bundle in that P=X×GP = X \times G, then a connection on this is equivalently encoded in a Lie-algebra valued 1-form

AΩ 1(X,) A \in \Omega^1(X, \mathcal{g})

on XX.

In terms of this, parallel transport is a solution to a differential equation.

For γ:[0,1]X\gamma : [0,1] \to X we have the pull-back 1-form γ *AΩ 1([0,1])\gamma^* A \in \Omega^1([0,1]). For fC ([0,1],G)f \in C^\infty([0,1], G) a smooth function with values in the Lie group GG, consider the differential equation

df+ρ(f) *(γ *A)=0, d f + \rho(f)_*(\gamma^*A) = 0 \,,

where df:T[0,1]TGd f : T [0,1] \to T G is the differential of ff and where ρ:G×GG\rho : G \times G \to G is the left action of GG on itself (i.e. just the multiplication on GG) and r(f) *:TGTGr(f)_* : T G \to T G its differential and using the defining identification 𝔤T eG\mathfrak{g} \simeq T_e G we take r(f) *(A)r(f)_*(A) to be the composite T[0,1]γ *A𝔤TGr(f) *TGT [0,1] \stackrel{\gamma^* A}{\to} \mathfrak{g} \hookrightarrow T G \stackrel{r(f)_*}{\to} T G.

If GG is a matrix Lie group such as the orthogonal group O(n)O(n) or the unitary group U(n)U(n), then also its Lie algebra identifies with matrices, and we may write this simply as

df+γ *(A)f=0, d f + \gamma^*(A) \cdot f = 0 \,,

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)f(0).


The parallel transport of AΩ 1(X,𝔤)A \in \Omega^1(X,\mathfrak{g}) along a path γ:[0,1]X\gamma : [0,1] \to X which we write

tra A(γ):=Pexp( [0,1]γ *A)G tra_A(\gamma) := P \exp(\int_{[0,1]} \gamma^* A) \in G

is the value f(1)Gf(1) \in G for the unique solution of the equation df+ρ(f) *(A)=0d f + \rho(f)_*(A) = 0 with initial value f(0)=ef(0) = e (the neutral element in GG).

The notation here is motivated from the special case where G=G = \mathbb{R} is the group of real numbers. In that case the Lie algebra 𝔤\mathfrak{g} \simeq \mathbb{R} is abelian, the differential equation above is simply

df=γ *(A)f d f = \gamma^*(A) \wedge f

for a real valued function fC ([0,1])f \in C^\infty([0,1]), and the unique solution to that with f(0)=e=0f(0) = e = 0 is literally the exponential of the integral of AA:

tra A(γ)=exp( [0,1]γ *A). tra_A(\gamma) = \exp(\int_{[0,1]} \gamma^* A) \,.

In the case of general nonabelian 𝔤\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

exp( [0,1/2]γ *A)exp( [1/2,1]γ *A) \exp(\int_{[0,1/2]} \gamma^* A) \cdot \exp(\int_{[1/2,1]} \gamma^* A)

and an even a bit more better approximation by

exp( [0,1/3]γ *A)exp( [1/3,2/3]γ *A)exp( [2/3,1]γ *A) \exp(\int_{[0,1/3]} \gamma^* A) \cdot \exp(\int_{[1/3,2/3]} \gamma^* A) \cdot \exp(\int_{[2/3,1]} \gamma^* A)

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.

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 higher parallel transport over manifolds of dimension greater than 1.

See higher parallel transport for details.


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

  • Gonzalo Reyes, General Relativity: Affine connections, parallel transport and sprays (pdf)

Last revised on August 8, 2018 at 10:03:56. See the history of this page for a list of all contributions to it.