A connection on a bundle $P \to X$ – a principal bundle or an associated bundle like a vector bundle – is a rule that identifies fibers of the bundle along paths in the base space $X$.
There are several different but equivalent formalizations of this idea:
as a parallel transport functor,
as a rule for a covariant derivative,
as a distribution (field) of horizontal subspaces – an Ehresmann connection – and via a connection $1$-form which annihilates the distribution of horizontal subspaces. The connection in that sense induces a smooth version of Hurewicz connection.
The usual textbook convention is to say just connection for the distribution of horizontal subspaces, and the objects of the other three approaches one calls more specifically covariant derivative, connection $1$-form and parallel transport.
In the remainder of this Idea-section we discuss a bit more how to understand connections in terms of parallel transport.
Given a smooth bundle $P \to X$, for instance a $G$-principal bundle or a vector bundle, a connection on $P$ is a prescription to associate with each path
in $X$ (which is a morphism in the path groupoid $\mathbf{P}_1(X)$) a morphism $tra(\gamma)$ between the fibers of $P$ over these points
such that
this assignment respects the structure on the fibers $P_x$ (for instance is $G$-equivariant in the case that $P$ is a $G$-bundle or that is linear in the case that $P$ is a vector bundle);
this assignment is smooth in a suitable sense;
this assignment is functorial in that for all pairs $x \stackrel{\gamma}{\to} y$, $y \stackrel{\gamma'}{\to} z$ of composable paths in $X$ we have
In other words, a connection on $P$ is a functor
from the path groupoid of $X$ to the Atiyah Lie groupoid of $P$ that is smooth in a suitable sense and splits the Atiyah sequence in that $\mathbf{P}_1(X) \stackrel{tra}{\to} At''(X) \to \mathbf{P}_1(X)$ (see the notation at Atiyah Lie groupoid).
Terminology
The functor $tra$ is called the parallel transport of the connection. This terminology comes from looking at the orbits of all points in $P$ under $tra$ (i.e. from looking at the category of elements of $tra$): these trace out paths in $P$ sitting over paths in $X$ and one thinks of the image of a point $p \in P_x$ under $tra(\gamma)$ as the result of propagating $p$ parallel to these curves in $P$.
Flat connections
It may happen that the assignment $tra : \gamma \mapsto tra(\gamma)$ only depends on the homotopy class of the path $\gamma$ relative to its endpoints $x, y$. In other words: that $tra$ factors through the functor $P_1(X) \to \Pi_1(X)$ from the path groupoid to the fundamental groupoid of $X$. In that case the connection is called a flat connection.
By Lie differentiation the functor $tra$, i.e. by looking at what it does to very short pieces of paths, one obtains from it a splitting of the Atiyah Lie algebroid sequence, which is a morphism
of vector bundles. Locally on $X$ – meaning: when everything is pulled back to a cover $Y \to X$ of $X$ – this is a $Lie(G)$-valued 1-form $A \in \Omega^1(Y, Lie(G))$ with certain special properties.
In particular, since every $G$-principal bundle canonically trivializes when pulled back to its own total space $P$, a connection in this case gives rise to a 1-form $A \in \Omega^1(P)$ satisfying two conditions. This data is called an Ehresmann connection.
If instead $P = E$ is a vector bundle, then the two conditions satisfies by $A$ imply that it defines a linear map
from the space $\Gamma(E)$ of section of $E$ that satisfies the properties of a covariant derivative.
If again the connection is flat, then this is manifestly the datum of a Lie infinity-algebroid representation of the tangent Lie algebroid $T X$ of $X$ on $E$: it defines the action Lie algebroid which is the Lie version of the Lie groupoid classified by the parallel transport functor.
…
Recall from the discussion at $G$-principal bundle that the $G$-bundle $P \to X$ is encoded in a a suitable morphism
(namely a morphism in the right (infinity,1)-category which may be represented by an anafunctor).
It turns out that similarly suitable morphisms
encode in one step the $G$-bundle together with its parallel transport functor.
This is described in great detail in the reference by Schreiber–Waldorf below.
(…am running out of time… )
Let $G$ be a Lie group. We recall briefly the following discussion of $G$-principal bundles. For an in-depth discussion see Smooth∞Grpd.
Write
for the functor that sends a Cartesian space $U$ to the delooping groupoid of the group of $G$-valued smooth functions on $U$: the groupoid with a single object and the group $Hom_{Diff}(U,G)$ of maps as its set of morphisms.
This is a groupoid-valued sheaf on the site CartSp${}_{smooth}$ and in fact is a (2,1)-sheaf/stack.
For $X$ a paracompact smooth manifold, we may also regard it as a (2,1)-sheaf on CartSp in an evident way.
The groupoid $G Bund(X)$ of $G$-principal bundles on $X$ is equivalent to the hom-groupoid
taken in the (2,1)-topos of (2,1)-sheaves on CartSp${}_{smooth}$.
A detailed discussion of this is at Smooth∞Grpd in the section on Lie groups.
Now write $\mathfrak{g}$ for the Lie algebra of $\mathfrak{g}$. Then consider the functor
that sends a Cartesian space $U$ to the groupoid of Lie-algebra valued 1-forms over $U$.
There is an evident morphism of (2,1)-sheaves
that forgets the 1-forms on each object $U$.
(connection)
A connection on a smooth $G$-principal bundle $g : X \to \mathbf{B}G$ is a lift $\nabla$ to $\mathbf{B}G_{conn}$
The groupoid of $G$-principal bundles with connection on $X$ is
Explicitly, a morphism $g : X \to \mathbf{B}G$ is a nonabelian Cech cohomology cocycle on $X$ with values in $G$:
a choice of good open cover $\{U_i \to X\}$ of $X$;
a collection of smooth functions $(g_{i j} \in C^\infty(U_i \cap U_j), G)$
such that on $U_i \cap U_j \cap U_k$ the equation
holds.
A lift $\nabla : X \to \mathbf{B}G_{conn}$ of this is in addition
such that on $U_i \cap U_j$ the equation
holds, where on the right we have the pullback $g^* \theta$ of the Maurer-Cartan form on $G$ (see there).
(existence of connections)
Every $G$-principal bundle admits a connection. In other words, the forgetful functor
Choose a partition of unity $(\rho_i \in C^\infty(X,\mathbb{R}))$ subordinate to the good open cover $\{U_i \to X\}$ with respect to which a given cocycle $g : X \to \mathbf{B}G$ is expressed:
$(x \;not\; in\; U_i) \Rightarrow \rho_i(x) = 0$;
$\sum_i \rho_i = 1$.
Then set
By slight abuse of notation we shall write this and similar expressions simply as
Using the that $(g_{i j})$ satisfies its cocycle condition, one checks that this satisfies the cocycle condition for the 1-forms:
Connections on tangent bundles are also called affine connections, or Levi-Civita connections.
They play a central role for instance on Riemannian manifolds and pseudo-Riemannian manifolds. From the end of the 19th century and the beginning of the 20th centure originates a language to talk about these in terms of Christoffel symbols.
In physics connections on bundles model gauge fields.
The electromagnetic field is a connection on a circle group-principal bundle;
A Yang-Mills field more generally is a connection on a unitary group-principal bundle.
The field of gravity is encoded in a connection on the orthogonal group-principal bundle to which the tangent bundle is associated.
For more on this see higher category theory and physics.
Generalizing the parallel transport definition from ordinary manifolds to supermanifolds yields the notion of superconnection.
When the notion of connection on a principal bundle is slightly coarsened, i.e. when more connections are regarded as being ismorphic than usual, one arrives at a structure called a Simons-Sullivan structured bundle. This has the special property that for $G = U$ the unitary group, the corresponding Grothendieck group of such bundles is a model for differential K-theory.
See connection on a principal ∞-bundle.
connection on a bundle, universal connection
connection on a 2-bundle / connection on a gerbe / connection on a bundle gerbe
higher Atiyah groupoid: | standard higher Atiyah groupoid | higher Courant groupoid | groupoid version of quantomorphism n-group |
---|---|---|---|
coefficient for cohomology: | $\mathbf{B}\mathbb{G}$ | $\mathbf{B}(\mathbf{B}\mathbb{G}_{\mathrm{conn}})$ | $\mathbf{B} \mathbb{G}_{conn}$ |
type of fiber ∞-bundle: | principal ∞-bundle | principal ∞-connection without top-degree connection form | principal ∞-connection |
gauge field: models and components
A classical textbook reference is
The formulation of connections in terms of their smooth parallel transport functors is in
based on a series of classical observations. I farily comprehensive commented list of related references is here:
Basic facts about connections, such as the existence proof in terms of Cech cocycles, are collected in the brief lecture note
Discussion of connection form (mostly on trivial bundles) in synthetic differential geometry includes the following
Anders Kock, Connections and path connections in groupoids, 2006 (web)
Anders Kock, Group valued differential forms revisited, Feb 2007 (web)