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principal bundle

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Idea

For GG a group (internal to some category, traditionally that of topological spaces) and XX some other object, a GG-principal bundle over XX – also called a GG-torsor over XX – is a bundle PXP \to X equipped with a GG-action ρ:P×GP\rho : P \times G \to P on PP over XX, such that

  • the action is principal meaning that

    • the canonical morphism (p 1,ρ):P×GP× XP(p_1, \rho) : P \times G \to P \times_X P is an isomorphism, which in turn means that the action is free and transitive over XX, hence that each fiber of PXP \to X looks like GG once we choose a base point;

    and / or / equivalently (depending on technical details, see below)

    • PXP \to X is isomorphic to the quotient map PP/GP \to P/G.

and usually it is required that

  • the bundle is locally trivial in that there is a cover ϕ:UX\phi : U \to X and an isomorphism of GG-actions between the pullback ϕ *P\phi^* P and the trivial GG-principal bundle U×GUU \times G \to U over UU.

A central property of GG-principal bundles over XX is that they are a geometric model of the degree-1 nonabelian cohomology of XX with coefficients in GG. More precisely (subject to some technical details discussed below) there is a natural isomorphism

H 1(X,G)GBund(X)/ H^1(X, G) \simeq G Bund(X)/{\sim}

between the degree-1 GG-cohomology of XX and the isomorphism classes of GG-principal bundles over XX.

The “naturality” of this relation is more pronounced when one refines it from cohomology to cocycle groupoids. This is discussed below in some section,

Definition

We discuss first the definition of principal bundles

This is historically and traditionally the default setup. But the theory exists in and is usefully regarded from a more abstract perspective, which, most naturally, is that of a (2,1)-topos. This we introduce and discuss in detail in

Finally in

we discuss how the traditional setup and many other contexts are recovered from and illuminated by that abstract perspective.

In the category of topological spaces

We discuss here principal bundles in the context Top of topological spaces. So the group GG here is a topological group.

This is the original and oldest branch of the theory. There is a modern established default of the definition, but many slight but crucial variants exists in the literature and are relevant in applications. We start with the modern default notion and then look into its variants.

\,

Let GG be a topological group.

Definition

The trivial GG-principal bundle on a topological space XX is the productspace G×XG \times X equipped with

  • the projection map p 1:X×GXp_1 : X \times G \to X ;

  • the action of GG on X×GX \times G by right multiplication of GG on itself.

Definition

A GG-principal bundle over a topological space XX is a topological space PP equipped with

  • a continuous function p:PXp : P \to X;

  • an action ρ:P×GP\rho : P \times G \to P of GG on PP over XX, hence fitting into a coequalizing diagram

    P×G p 1 ρ P p X \array{ P \times G \\ {}^{\mathllap{p_1}} \downarrow \downarrow^{\mathrlap{\rho}} \\ P \\ \downarrow^{\mathrlap{p}} \\ X }

such that this is locally trivial in the sense that

U×G G P| Y P p 1 p U X. \array{ U \times G& \stackrel{\simeq_G}{\leftarrow} & P|_Y &\to& P \\ &{}_{\mathllap{p_1}}\searrow&\downarrow && \downarrow^{\mathrlap{p}} \\ &&U &\to& X } \,.

In the references listed below, this appears for instance as (Mitchell, section 2, …)

A central property of the above definition of principal bundle is

Proposition

For PXP \to X a GG-principal bundle, it is naturally isomorphic to the quotient projection PP/GXP \to P/G \simeq X of the GG-action.

Historically this quotient property of a free continuous action was sometimes taken as the very definition of “principal bundle” without requiring local triviality, e. g. in (Cartan, 1949-1950), where this perspective is attributed to Henri Cartan. A standard modern textbook following this tradition is (Husemöller).

Therefore in order to avoid ambiguous terminology in the following, we will now follow (Palais) and refer to this alternative definition of principal bundle as that of Cartan principal bundle.

Definition

Let GG be a locally compact topological group and PP a completely regular topological space equipped with a continuous function action ρ:P×GP\rho : P \times G \to P. If GG acts freely on PP, (no element gGg \in G except the neutral element has any fixed points in PP under the action) then the map

PP/G=:X P \to P/G =: X

to the topological quotient is called a GG-principal bundle in the wide sense. If furthermore the division map

P× XPG P \times_X P \to G

is a continuous function, then this is called a Cartan principal bundle (Palais, around theorem 1.1.3), following (Cartan).

(…)

In a (2,1)(2,1)-topos

It is no surprise that there is a good theory of principal bundles internal to every topos. However, it turns out that the most “natural home” of the theory is the higher category theoretic context of a (2,1)-topos H\mathbf{H}. This we discuss now, and then relate it to the traditional notion and to various other generalizations.

Notably the existence of universal principal bundles finds its fundamental “explanation” here, where they are seen to be but a presentation of the construction of the homotopy fiber functor, which establishes the equivalence of groupoids

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

where on the left we have the groupoid of cocycles with coefficients in the internal delooping BG\mathbf{B}G of the group object GG: the moduli stack of GG-principal bundles.

In this context, all of the non-natural aspects of the traditonal theory of principal bundles disappear, for instance

  • every GG-principal bundle is locally trivial in a (2,1)(2,1)-topos H\mathbf{H};

  • accordingly there is no mismatch between the various definitions anymore as in the context of topological spaces: the condition of principality becomes equivalent to the quotient space condition.

Moreover, all these fact are fairly direct consequences simply of the Giraud axioms that characterize (2,1)-toposes in the first place.

Conversely, the traditional theory nicely naturally embeds into a (2,1)-topos – for instance that of (2,1)-sheaves over the site Top (or rather some small dense subsite thereof) – and the higher topos theory helps to study it there.

The failure of various definitions to match in the traditonal context becomes the fact that the colimits involved get “corrected” to homotopy colimits after embedding into a (2,1)-topos. For instance if an GG-action on some object PP is not suitably free, then the (2,1)(2,1)-topos theory still produces a healthy principal bundle by replacing the base space by a base groupoid/stack. In fact, this way every action becomes principal over its homotopy quotient. Notably the trivial GG-action on the terminal object ** becomes principal over the action groupoid *//G*//G and the resulting GG-principal bundle is nothing but the universal one.

As the notation suggests, thus formulating the theory in (2,1)-topos theory immediately generalizes it to (∞,1)-topos theory. This is discussed at principal ∞-bundle.

(…)

The following is old material collected from elsewhere that is going to be rearranged….

In terms of fiber sequences

This indicates the more fundamental way to define GG-principal bundles in the first place:

Recall (from fiber sequence) that for every group there is the the one-object groupoid BG\mathbf{B}G. Under the Yoneda embedding this represents a prestack. Write BG¯\bar{\mathbf{B}G} for the corresponding stack obtained by stackification. This is our GBund()G Bund(-)

GBund()=BG¯(). G Bund(-) = \bar{\mathbf{B} G}(-) \,.

This perspective in turn is by general abstract nonsense equivalent to the following useful description:

Let HH be the suitable (∞,1)-topos internal to which one looks at GG-principal bundles. For instance for topological bundles this would be Top. For smooth bundles it would be the (∞,1)-category of (∞,1)-sheaves on Diff, etc.

Then every element in GBund(X)BG¯(X)G Bund(X) \simeq \bar{\mathbf{B} G}(X) is given by a morphism in H(X,BG)H(X,\mathbf{B}G), which may be thought of as an anafunctor to BG\mathbf{B}G from the (categorially) discrete category XX; the GG-principal bundle from the beginning of the above definition is just the homotopy pullback of the point along this map, i.e. the homotopy fiber of XBGX \to \mathbf{B}G:

P * X BG. \array{ P &\to& {*} \\ \downarrow && \downarrow \\ X &\to& \mathbf{B} G } \,.

This diagram, incidentally, directly tells us about another important property of GG-principal bundles: they all canonically trivialize when pulled back to their own total space PP.

This is what the homotopy commutativity of the above homotopy pullback diagram says: the cocycle XBGX \to \mathbf{B}G pulled back to the bundle PXP \to X that it classifies becomes PXBGP \to X \to \mathbf{B}G, which is homotopic to the trivial cocycle (the one that factors through the point) on PP.

The homotopy pullback here is conveniently and traditionally computed as an ordinary pullback of a fibrant replacement of the pullback diagram. The canonical such fibrant replacement is obtained by replacing *BG{*} \to \mathbf{B}G by EGBG\mathbf{E}G \to \mathbf{B}G, with EG\mathbf{E}G an object weakly equivalent to the point, called the GG-universal principal bundle.

With that the above homotopy pullback is computed as the ordinary pullback

P EG X BG \array{ P &\to& \mathbf{E}G \\ \downarrow && \downarrow \\ X &\to& \mathbf{B} G }

So every GG-principal bundle PXP \to X is the pullback along a classifying map XBGX \to \mathbf{B}G (in the right (,1)(\infty,1)-categorical context, otherwise a span such as an anafunctor) of the GG-universal principal bundle.

The GG-action from the homotopy pullback

Given the definition of the bundle PP in terms of a homotopy pullback of *BG{*} \to \mathbf{B}G we re-obtain the GG-action on PP as follows (with an eye towards its generalization to principal ∞-bundles).

Let

G×Gd 0d 1d 2Gd 0d 1*BG \cdots G \times G \stackrel{\stackrel{d_2}{\to}}{\stackrel{\stackrel{d_1}{\to}}{\stackrel{d_0}{\to}}} G \stackrel{\stackrel{d_1}{\to}}{\stackrel{d_0}{\to}} {*} \to \mathbf{B}G

be the effective groupoid object in an (∞,1)-category that exhibits the delooping BG\mathbf{B}G of GG.

Form the homotopy pullback of the classifying morphism XBGX \to \mathbf{B}G along the d 0d_0-face maps of this diagram. This yields a diagram

P×G×G P×G P X G×G G * BG \array{ \cdots & P \times G \times G & \stackrel{\to}{\stackrel{\to}{\to}} & P \times G & \stackrel{\to}{\to} & P & \stackrel{}{\to} & X \\ & \downarrow && \downarrow && \downarrow && \downarrow \\ \cdots & G \times G & \stackrel{\to}{\stackrel{\to}{\to}} & G & \stackrel{\to}{\to} & {*} & \stackrel{}{\to} & \mathbf{B}G }

where all squares formed by the lowest horizontal morphisms are homotopy pullback squares, by construction, and where the remaining horizontal morphisms in the top row are induced by the universal property of the homotopy pullback and the morphisms downstairs.

The claim is that

  • the top row encodes the action of GG on PP in that the action is the morphism indicated ρ\rho in

    P×Gp 1ρPX \cdots \stackrel{\to}{\stackrel{\to}{\to}} P \times G \stackrel{\stackrel{\rho}{\to}}{\stackrel{p_1}{\to}} P \to X
  • and it exhibits P×G ×(n1)P \times G^{\times (n-1)} as the groupoid object in an (∞,1)-category being the Čech nerve of PXP \to X:

    P× XP× XP P× XP P X P×G×G P×G ρ P X G×G G * BG \array{ \cdots & P \times_X P \times_X P & \stackrel{\to}{\stackrel{\to}{\to}} & P \times_X P & \stackrel{\to}{\to} & P & \stackrel{}{\to} & X \\ & \uparrow^{\simeq} && \uparrow^{\simeq} && \uparrow^{\simeq} && \uparrow^{\simeq} \\ \cdots & P \times G \times G & \stackrel{\to}{\stackrel{\to}{\to}} & P \times G & \stackrel{\stackrel{\rho}{\to}}{\to} & P & \stackrel{}{\to} & X \\ & \downarrow && \downarrow && \downarrow && \downarrow \\ \cdots & G \times G & \stackrel{\to}{\stackrel{\to}{\to}} & G & \stackrel{\to}{\to} & {*} & \stackrel{}{\to} & \mathbf{B}G }

Here the second statement in particular encodes the familiar way to formulate principality of the action ρ\rho, in that it says that

P×Gp 1×ρP× XP P \times G \stackrel{p_1 \times \rho}{\to} P \times_X P

is an isomorphism.

We now unwrap the first statement in gory detail to make clear that this abstract nonsense does reproduce the familiar definition of the action of GG on PP.

Unwinding the abstract description

We now rederive the action ρ\rho of GG on PP given just the classifying map XBGX \to \mathbf{B} G by spelling out the details implied by the above abstract description.

Whatever the precise context is (topological, smooth, etc.) we may assume that we are at least in a category of fibrant objects. Then the classifying morphism XBGX \to \mathbf{B}G is represented by an anafunctor, namely a cocycle

C(U) g BG WF X \array{ C(U) &\stackrel{g}{\to} & \mathbf{B}G \\ \downarrow^{\mathrlap{\in W \cap F}} \\ X }

in Čech cohomology coming from some cover {U iX}\{U_i \to X\} of XX.

The Čech nerve C(U)C(U) has

  • objects = {(x,i)|xU i}\{(x,i) | x \in U_i\}

  • morphisms = {(x,i)(x,j)|xU ij}\{ (x,i) \stackrel{}{\to} (x,j) | x \in U_{i j}\} .

The functor g:C(U)BGg : C(U) \to \mathbf{B}G sends

g:((x,i)(x,j))(g ij(x)) g : ((x,i) \to (x,j)) \mapsto (\bullet \stackrel{g_{i j}(x)}{\to}\bullet )

for g ijFunctions(U ij,G)g_{i j} \in Functions(U_{i j}, G) as described in detail at Čech cohomology.

With EG={ghgh|g,hG}\mathbf{E}G = \{g \stackrel{h}{\to} g h | g,h \in G \} the fibrant replacement of the point, which we shall find it helpful to think of as given by

  • objects = { g }\left\{ \array{ && \bullet \\ & {}^g\swarrow \\ \bullet } \right\}

  • morphisms = { g g=gh h }\left\{ \array{ && \bullet \\ & {}^g\swarrow && \searrow^{g' = g h} \\ \bullet &&\stackrel{h}{\to}&& \bullet } \right\}

we compute the homotopy pullback as the homotopy fiber product given by the ordinary pullback (see category of fibrant objects for details)

P EG C(U) g BG. \array{ P &\to& \mathbf{E}G \\ \downarrow && \downarrow \\ C(U) &\stackrel{g}{\to}& \mathbf{B}G } \,.

So we read off that PP is the groupoid with

  • objects =

    { g (x,i)} \left\{ \array{ && \bullet \\ & {}^{g}\swarrow \\ \bullet \\ (x,i) } \right\}
  • morphisms =

    { g g g ij(x) (x,i) (x,j)} \left\{ \array{ && \bullet \\ & {}^{g}\swarrow && \searrow^{g'} \\ \bullet &&\stackrel{g_{i j }(x)}{\to}&& \bullet \\ (x,i) &&\stackrel{}{\to}&& (x,j) } \right\}

With PP determined as an ordinary pullback of a replacement it is convenient for the following to realize it in turn as the pullback-up-to-2-cell in

P * η P C(U) BG. \array{ P &\to& {*} \\ \downarrow &\Downarrow^{\eta_P}& \downarrow \\ C(U) &\to& \mathbf{B}G } \,.

A moment reflection shows that the component of the natural transformation η P\eta_P here is

η P:( g (x,i))(g) \eta_P : \left( \array{ && \bullet \\ & {}^{g}\swarrow \\ \bullet \\ (x,i) } \right) \;\;\; \mapsto \;\;\; (\bullet \stackrel{g}{\to}\bullet)

At the same time recall from the discussion at delooping that the component of the transformation η G\eta_G in

G * η G * BG \array{ G &\to& {*} \\ \downarrow &\Downarrow^{\eta_G}& \downarrow \\ {*} &\to& \mathbf{B}G }

is

η G:g(g). \eta_G : g \mapsto (\bullet \stackrel{g}{\to} \bullet) \,.

Taken together this shows that the universal morphism P×GPP \times G \to P induced from the commutativity of

P×G P G C(U) η P * η G * BG \array{ &&&& P \times G \\ && & \swarrow && \searrow \\ && P &&&& G \\ & \swarrow && \searrow && \swarrow && \searrow \\ C(U) &&\Downarrow^{\eta_P}&& {*} &&\Downarrow^{\eta_G}&& {*} \\ &&\searrow&& \downarrow && \swarrow \\ &&&& \mathbf{B}G }

and from the homotopy pullback property of

P * η P C(U) BG \array{ P &\to& {*} \\ \downarrow &\Downarrow^{\eta_P}& \downarrow \\ C(U) &\to& \mathbf{B}G }

is simply given by the composition of these two component maps

P×GP:(( g (x,i)), g )( g g (x,i)). P \times G \to P : \left( \left( \array{ && \bullet \\ & {}^{g}\swarrow \\ \bullet \\ (x,i) } \right) \,, \array{ \bullet \\ \downarrow^{g'} \\ \bullet } \right) \; \;\; \; \mapsto \; \;\; \; \left( \array{ && \bullet \\ && \downarrow^{g'} \\ && \bullet \\ & {}^{g}\swarrow \\ \bullet \\ (x,i) } \right) \,.

But this is manifestly the right (being both: from the right and correct :-) action ρ:P×GG\rho : P \times G \to G of GG on PP.

Other internalizations

We discuss here aspects of formulating a theory of principal bundles in contexts different from those already discussed above.

(…)

Higher generalizations

In higher category theory the notion of principal bundle has various vertical categorifications. See

Properties

Gauge groupoid

For PXP \to X a GG-principal bundle, its Atiyah Lie groupoid is

((P×P)/Gpp 2pp 1X) ((P \times P)/G \stackrel{\stackrel{p\circ p_1}{\to}}{\underset{p \circ p_2}{\to}} X )

with the evident composition operation.

The principal bundle PXP \to X is recovered from its Atiyah Lie groupoid, up to isomorphism, as the source fiber over any point.

This is a classical statement due to Ehresmann … . See for instance (Androulidakis).

Examples

Hopf fibration

A standard example of a nontrivial circle group-principal bundle – a circle bundle – is the Hopf fibration S 3S 2S^3 \to S^2, which has the structure of an S 1S^1-principal bundle in topological spaces.

Pullbacks of universal bundles

Generally, if we accept that we have a large supply of continuous maps between topological spaces, we obtain a GG-principal bundle f *GXf^* \mathcal{E}G \to X on a space XX for each continuous map f:XGf : X \to \mathcal{B}G to the classifying space of GG, by pullback of the universal bundle GG\mathcal{E}G \to \mathcal{B}G along ff.

Quotients by Lie group actions

We consider topological group actions GG on PP where GG is equipped with the structure of a Lie group.

Theorem

Let PP be a completely regular topological space and let GG be a Lie group equipped with a free action on PP. Then the quotient map PP/GP \to P/G is a GG-principal bundle – in that it is locally trivial – precisely if the division map

P× P/GPG P \times_{P/G} P \to G

is a continuous function.

This is (Palais, theorem 4.1).

Coset spaces

We discuss principal bundles of the form PP/GP \to P/G for GPG \hookrightarrow P a subgroup of a topological group.

Proposition

If PP is a topological group and GPG \hookrightarrow P a closed Lie subgroup, then the quotient map PP/GP \to P/G is a locally trivial GG-principal bundle.

This is a corollary of theorem 1 (Palais).

Examples where PP/GP \to P/G is not locally trivial re in Karube. See also (Mostert).

Example

Let PP be the product of infinitly many circles, and let GG is the product of their order 2 subgroups. This cannot have local section because PP is locally connected and GG is not. Therefore PP is not even locally homeomorphic to (P/G)×G(P / G) \times G.

Gauge theory

In physics, principal bundles with connection and their higher categorical analogs model gauge fields. See at fiber bundles in physics.

In fact, the history of the development of the theory of principal bundles and gauge theory is closely related. In the early 1930s Dirac and Hopf independently introduced U(1)U(1)-principal bundles: Dirac, somewhat implicitly, in his study of the electromagnetic field as a background for quantum mechanics, Hopf in terms of the fibration named after him. However, from there it took apparently many years for the first publication to appear that explicitly states that these two considerations are aspects of the same phenomenon.

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

General

An original reference on the notion of a principal bundle as a quotient map by a free continuous action of a topological group is

  • Séminaire H. Cartan 1949-1950 (numdam)

some of which is recollected in (Palais).

A standard textbook following this original perspective is

For principal bundles in the smooth context see most textbooks on differential geometry, for instance

also around section 3.1 of

Lecture notes on the topic include

  • Stephan Mitchell, Notes on principal bundles and classifying spaces (2001) (pdf)

In relation to Lie groupoids

Discussion of Atiyah Lie groupoids associated to principal bundles and the reconstruction of principal bundles from their Atiyah Lie groupoids is due to

  • Ehresmann, …

Further discussion along these lines is for instance in

Examples

Detailed discussion of topological quotients of groups GG/HG \to G/H as principal HH-bundles is in

  • Richard Palais, On the existence of slices for actions of non-compact Lie groups, Annals of mathematics vol 73 no 2 (1961) (pdf)

Explicit examples and counter examples of coset principal bundles are discussed in

  • Takashi Karube, On the local cross-sections in locally compact groups, J. Math. Soc. Japan 10 (1958) 343–347 (Euclid)
  • Paul Mostert, Local cross sections in locally compact groups (pdf)

Relations between classes of continuous and of smooth principal bundles are discussed in

  • Christoph Müller, Christoph Wockel, Equivalences of smooth and continuous principal bundles with infinite-dimensional structure groups, Advances in Geometry. Volume 9, Issue 4, Pages 605–626 (2009)

Extensions

Extensions of principal bundles are discussed for instance in

  • Kirill Mackenzie, On extensions of principal bundles, Annals of Global Analysis and Geometry Volume 6, Number 2 (1988),

  • I. Androulidakis, Classification of extensions of principal bundles and transitive Lie groupoids with prescribed kernel and cokernel, J. Math. Phys. 45, 3995 (2004); (pdf)

Automorphism groups

The automorphism groups of principal bundles are discussed for instance in

  • M.C. Abbati, R. Cirelli, A. Mania, P. Michor The Lie group of automorphisms of a principal bundle 1989 JGP 6 215 (pdf, doi)

Revised on May 26, 2014 04:36:36 by Igor Khavkine (193.205.210.43)