connection on a bundle (changes) in nLab

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Idea

A connection on a bundle P→XP \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 XX.

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 11-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 11-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→XP \to X, for instance a GG-principal bundle or a vector bundle, a connection on PP is a prescription to associate with each path

γ:x→y \gamma : x \to y

in XX (which is a morphism in the path groupoid P 1(X)\mathbf{P}_1(X)) a morphism tra(γ)tra(\gamma) between the fibers of PP over these points

P x →tra(γ) P y x →γ y \array{ P_x &\stackrel{tra(\gamma)}{\to}& P_y \\ x &\stackrel{\gamma}{\to}& y }

such that

• this assignment respects the structure on the fibers P xP_x (for instance is GG-equivariant in the case that PP is a GG-bundle or that is linear in the case that PP is a vector bundle);

• this assignment is smooth in a suitable sense;

• this assignment is functorial in that for all pairs x→γyx \stackrel{\gamma}{\to} y, y→γ′zy \stackrel{\gamma'}{\to} z of composable paths in XX we have

  P x →tra(γ) P y →tra(γ′) P z x →γ y →γ′ z=P x →tra(γ′∘γ) P z x →γ′∘γ z \array{ P_x &\stackrel{tra(\gamma)}{\to}& P_y &\stackrel{tra(\gamma')}{\to}& P_z \\ x &\stackrel{\gamma}{\to}& y &\stackrel{\gamma'}{\to}& z } \;\;\; = \;\;\; \array{ P_x &\stackrel{tra(\gamma' \circ \gamma)}{\to}& P_z \\ x &\stackrel{\gamma'\circ \gamma}{\to}& z }

In other words, a connection on PP is a functor

tra:P 1(X)→At″(P) tra : \mathbf{P}_1(X) \to At''(P)

from the path groupoid of XX to the Atiyah Lie groupoid of PP that is smooth in a suitable sense and splits the Atiyah sequence in that P 1(X)→traAt″(X)→P 1(X)\mathbf{P}_1(X) \stackrel{tra}{\to} At''(X) \to \mathbf{P}_1(X) (see the notation at Atiyah Lie groupoid).

Terminology

The functor tratra is called the parallel transport of the connection. This terminology comes from looking at the orbits of all points in PP under tratra (i.e. from looking at the category of elements of tratra): these trace out paths in PP sitting over paths in XX and one thinks of the image of a point p∈P xp \in P_x under tra(γ)tra(\gamma) as the result of propagating pp parallel to these curves in PP.

Flat connections

It may happen that the assignment tra:γ↦tra(γ)tra : \gamma \mapsto tra(\gamma) only depends on the homotopy class of the path γ\gamma relative to its endpoints x,yx, y. In other words: that tratra factors through the functor P 1(X)→Π 1(X)P_1(X) \to \Pi_1(X) from the path groupoid to the fundamental groupoid of XX. In that case the connection is called a flat connection.

More concrete picture

By Lie differentiation the functor tratra, 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

∇:TX→at(P) \nabla : T X \to at(P)

of vector bundles. Locally on XX – meaning: when everything is pulled back to a cover Y→XY \to X of XX – this is a Lie(G)Lie(G)-valued 1-form A∈Ω 1(Y,Lie(G))A \in \Omega^1(Y, Lie(G)) with certain special properties.

In particular, since every GG-principal bundle canonically trivializes when pulled back to its own total space PP, a connection in this case gives rise to a 1-form A∈Ω 1(P)A \in \Omega^1(P) satisfying two conditions. This data is called an Ehresmann connection.

If instead P=EP = E is a vector bundle, then the two conditions satisfies by AA imply that it defines a linear map

∇:Γ(E)→Ω 1(X,E):=Γ(X,T X *⊗E) \nabla : \Gamma(E) \to \Omega^1(X,E):=\Gamma(X,T_X^*\otimes E)

from the space Γ(E)\Gamma(E) of section of EE 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 TXT X of XX on EE: it defines the action Lie algebroid which is the Lie version of the Lie groupoid classified by the parallel transport functor.

…

More abstract picture

Recall from the discussion at GG-principal bundle that the GG-bundle P→XP \to X is encoded in a a suitable morphism

X→BG X \to \mathbf{B}G

(namely a morphism in the right (infinity,1)-category which may be represented by an anafunctor).

It turns out that similarly suitable morphisms

P 1(X)→BG \mathbf{P}_1(X) \to \mathbf{B}G

encode in one step the GG-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… )

Definition

Let GG be a Lie group. We recall briefly the following discussion of GG-principal bundles. For an in-depth discussion see Smooth∞Grpd.

Write

BG:U↦(Hom Diff(U,G)→→*) \mathbf{B}G : U \mapsto ( Hom_{Diff}(U,G) \stackrel{\to}{\to} *)

for the functor that sends a Cartesian space UU to the delooping groupoid of the group of GG-valued smooth functions on UU: the groupoid with a single object and the group Hom Diff(U,G)Hom_{Diff}(U,G) of maps as its set of morphisms.

This is a groupoid-valued sheaf on the site CartSp smooth{}_{smooth} and in fact is a (2,1)-sheaf/stack.

For XX a paracompact smooth manifold, we may also regard it as a (2,1)-sheaf on CartSp in an evident way.

A detailed discussion of this is at Smooth∞Grpd in the section on Lie groups.

Now write 𝔤\mathfrak{g} for the Lie algebra of GG. Then consider the functor

BG conn:U↦[P 1(U),BG]={A→g(g −1Ag+g −1dg)|A∈Ω 1(U,𝔤),g∈C ∞(U,G)} \mathbf{B} G_{conn} : U \mapsto [\mathbf{P}_1(U),\mathbf{B}G] = \left\{ A \stackrel{g}{\to} (g^{-1} A g + g^{-1} d g) | A \in \Omega^1(U,\mathfrak{g})\,, g \in C^\infty(U,G) \right\}

that sends a Cartesian space UU to the groupoid of Lie-algebra valued 1-forms over UU.

There is an evident morphism of (2,1)-sheaves

BG conn→BG \mathbf{B}G_{conn} \to \mathbf{B}G

that forgets the 1-forms on each object UU.

Explicitly, a morphism g:X→BGg : X \to \mathbf{B}G is a nonabelian Cech cohomology cocycle on XX with values in GG:

1. a choice of good open cover {U i→X}\{U_i \to X\} of XX;

2. a collection of smooth functions (g ij∈C ∞(U i∩U j),G)(g_{i j} \in C^\infty(U_i \cap U_j), G)

such that on U i∩U j∩U kU_i \cap U_j \cap U_k the equation

• g ijg jk=g ikg_{i j} g_{j k} = g_{i k}

holds.

A lift ∇:X→BG conn\nabla : X \to \mathbf{B}G_{conn} of this is in addition

1. a choice of Lie-algebra valued 1-forms (A i∈Ω 1(U i,𝔤))(A_i \in \Omega^1(U_i, \mathfrak{g}))

such that on U i∩U jU_i \cap U_j the equation

• A j=g −1A ig+g −1dgA_j = g^{-1} A_i g + g^{-1} d g

holds, where on the right we have the pullback g *θg^* \theta of the Maurer-Cartan form on GG (see there).

Properties

Existence of connections

Special cases

Connections on the tangent bundle

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.

Connections in physics

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.

Generalizations

Superconnections

Generalizing the parallel transport definition from ordinary manifolds to supermanifolds yields the notion of superconnection.

Simons-Sullivan structured bundles

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=UG = U the unitary group, the corresponding Grothendieck group of such bundles is a model for differential K-theory.

Connections on a principal ∞\infty-bundle

See connection on a principal ∞-bundle.

• connection on a bundle, universal connection

  • principal connection

    • affine connection, Levi-Civita connection Cartan connection

    • Chen connection

    • covariant derivative

    • equivariant connection

    • logarithmic connection

  • Chern connection

  • moduli space of connections

  • fiber bundles in physics

• connection on a 2-bundle / connection on a gerbe / connection on a bundle gerbe

• connection on a 3-bundle / connection on a bundle 2-gerbe

• connection on an ∞-bundle

• parallel transport, higher parallel transport

• holonomy, higher holonomy

  • holonomy group

  • special holonomy

  • nonabelian Stokes theorem

• fiber bundles in physics

higher Atiyah groupoid

higher Atiyah groupoid:	standard higher Atiyah groupoid	higher Courant groupoid	groupoid version of quantomorphism n-group
coefficient for cohomology:	B𝔾\mathbf{B}\mathbb{G}	B(B𝔾 conn)\mathbf{B}(\mathbf{B}\mathbb{G}_{\mathrm{conn}})	B𝔾 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

physics	differential geometry	differential cohomology
gauge field	connection on a bundle	cocycle in differential cohomology
instanton/charge sector	principal bundle	cocycle in underlying cohomology
gauge potential	local connection differential form	local connection differential form
field strength	curvature	underlying cocycle in de Rham cohomology
gauge transformation	equivalence	coboundary
minimal coupling	covariant derivative	twisted cohomology
BRST complex	Lie algebroid of moduli stack	Lie algebroid of moduli stack
extended Lagrangian	universal Chern-Simons n-bundle	universal characteristic map

References

The notion of Ehresmann connections originates with:

• Charles Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable,

  Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège; Masson et Cie., Paris (1951) 29–55

  Séminaire Bourbaki, 1 24 (1952) 153–168 [[numdam:SB_1948-1951__1__153_0](http://www.numdam.org/item/?id=SB_1948-1951__1__153_0), pdf]

The understanding of gauge potentials in Yang-Mills theory as connections on fiber bundles originates with (see also at fiber bundles in physics):

• Tai Tsun Wu, Chen Ning Yang, Concept of nonintegrable phase factors and global formulation of gauge fields, Phys. Rev. D 12 (1975) 3845 [[doi:10.1103/PhysRevD.12.3845](https://doi.org/10.1103/PhysRevD.12.3845)]

Textbook account:

• Werner Greub, Stephen Halperin, Ray Vanstone, Connections, Curvature, and Cohomology Academic Press (1973)

With an eye towards application in mathematical physics:

• Tohru Eguchi, Peter Gilkey, Andrew Hanson, Section 5 of: Gravitation, gauge theories and differential geometry, Physics Reports 66 6 (1980) 213-393 []

• Mikio Nakahara, Chapter 9 of: Geometry, Topology and Physics, IOP 2003 (doi:10.1201/9781315275826, pdf)

• Gerd Rudolph, Matthias Schmidt, Chapter 1 of: Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields, Springer 2017 (doi:10.1007/978-94-024-0959-8)

The formulation of connections in terms of their smooth parallel transport functors is in

• Urs Schreiber, Konrad Waldorf, Parallel Transport and Functors, J. Homotopy Relat. Struct. 4, 187-244 (2009) (arXiv:0705.0452)

based on a series of classical observations. I farily comprehensive commented list of related references is here:

• differential cohomology in a cohesive topos – References – Connections – In terms of parallel transport

Basic facts about connections, such as the existence proof in terms of Cech cocycles, are collected in the brief lecture note

• Theodore Voronov, Differential Geometry, §3 Connection on a vector bundle (pdf)

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)