nLab
connection on a bundle

Context

\infty-Chern-Weil theory

Differential cohomology

Contents

Idea

A connection on a bundle PXP \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:

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 PXP \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:xy \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 xγyx \stackrel{\gamma}{\to} y, yγzy \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(γγ) P z x γγ 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 pP 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:γtra(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

:TXat(P) \nabla : T X \to at(P)

of vector bundles. Locally on XX – meaning: when everything is pulled back to a cover YXY \to X of XX – this is a Lie(G)Lie(G)-valued 1-form AΩ 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 AΩ 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)Γ(E) \nabla : \Gamma(E) \to \Omega^1(X) \otimes \Gamma(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 PXP \to X is encoded in a a suitable morphism

XBG 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.

Observation

The groupoid GBund(X)G Bund(X) of GG-principal bundles on XX is equivalent to the hom-groupoid

H(X,BG)GBund(X) \mathbf{H}(X,\mathbf{B}G) \simeq G Bund(X)

taken in the (2,1)-topos of (2,1)-sheaves on CartSp smooth{}_{smooth}.

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 𝔤\mathfrak{g}. Then consider the functor

BG conn:U[P 1(U),BG]={Ag(g 1Ag+g 1dg)|AΩ 1(U,𝔤),gC (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 connBG \mathbf{B}G_{conn} \to \mathbf{B}G

that forgets the 1-forms on each object UU.

Definition

(connection)

A connection on a smooth GG-principal bundle g:XBGg : X \to \mathbf{B}G is a lift \nabla to BG conn\mathbf{B}G_{conn}

BG conn X g BG. \array{ && \mathbf{B}G_{conn} \\ & {}^{\mathllap{\nabla}}\nearrow & \downarrow \\ X &\stackrel{g}{\to}& \mathbf{B}G } \,.

The groupoid of GG-principal bundles with connection on XX is

GBund (X):=Hom(X,BG conn). G Bund_\nabla(X) := Hom(X,\mathbf{B}G_{conn}) \,.

Explicitly, a morphism g:XBGg : 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 iX}\{U_i \to X\} of XX;

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

such that on U iU jU 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 :XBG conn\nabla : X \to \mathbf{B}G_{conn} of this is in addition

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

such that on U iU 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

Definition

(existence of connections)

Every GG-principal bundle admits a connection. In other words, the forgetful functor

Hom(X,B¯G conn)Hom(X,BG) Hom(X, \bar \mathbf{B}G_{conn}) \to Hom(X,\mathbf{B}G)

is an essentially surjective functor.

Proof

Choose a partition of unity (ρ iC (X,))(\rho_i \in C^\infty(X,\mathbb{R})) subordinate to the good open cover {U iX}\{U_i \to X\} with respect to which a given cocycle g:XBGg : X \to \mathbf{B}G is expressed:

  • (xnotinU i)ρ i(x)=0(x \;not\; in\; U_i) \Rightarrow \rho_i(x) = 0;

  • iρ i=1\sum_i \rho_i = 1.

Then set

A i:= i 0ρ i 0| U i 0(g i 0i| U i 0 1)d dR(g i 0i| U i 0). A_i := \sum_{i_0} \rho_{i_0}|_{U_{i_0}} (g_{i_0 i}|^{-1}_{U_{i_0}}) d_{dR} (g_{i_0 i}|_{U_{i_0}}) \,.

By slight abuse of notation we shall write this and similar expressions simply as

A i:= i 0ρ i 0(g i 0i 1d dRg i 0i). A_i := \sum_{i_0} \rho_{i_0}(g_{i_0 i}^{-1} d_{dR} g_{i_0 i}) \,.

Using the that (g ij)(g_{i j}) satisfies its cocycle condition, one checks that this satisfies the cocycle condition for the 1-forms:

A jg ij 1A ig ij = i 0ρ i 0(g i 0j 1dg i 0j(g i 0ig ij) 1(dg i 0i)g ij) = i 0ρ i 0(g ij 1dg ij) =g ij 1dg ij. \begin{aligned} A_j - g_{i j}^{-1} A_i g_{i j} &= \sum_{i_0} \rho_{i_0} ( g_{i_0 j}^{-1} d g_{i_0 j} - ( g_{i_0 i} g_{i j}) ^{-1} (d g_{i_0 i}) g_{i j} ) \\ & = \sum_{i_0} \rho_{i_0} ( g_{i j}^{-1} d g_{i j} ) \\ & = g_{i j}^{-1} d g_{i j} \end{aligned} \,.

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.

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.

higher Atiyah groupoid

higher Atiyah groupoid:standard higher Atiyah groupoidhigher Courant groupoidgroupoid 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 ∞-bundleprincipal ∞-connection without top-degree connection formprincipal ∞-connection

gauge field: models and components

physicsdifferential geometrydifferential cohomology
gauge fieldconnection on a bundlecocycle in differential cohomology
instanton/charge sectorprincipal bundlecocycle in underlying cohomology
gauge potentiallocal connection differential formlocal connection differential form
field strengthcurvatureunderlying cocycle in de Rham cohomology
gauge transformationequivalencecoboundary
minimal couplingcovariant derivativetwisted cohomology
BRST complexLie algebroid of moduli stackLie algebroid of moduli stack
extended Lagrangianuniversal Chern-Simons n-bundleuniversal characteristic map

References

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

Revised on September 11, 2014 19:28:07 by Urs Schreiber (89.204.135.191)