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June 20, 2007

Curvature, the Atiyah Sequence and Inner Automorphisms

Posted by Urs Schreiber

I am still trying to better understand nn-curvature and its relation to inner automorphism (n+1)(n+1)-groups.

A while ago David Roberts emphasized that the parallel transport functor tra:P 1(X)GTor \mathrm{tra} : P_1(X) \to G\mathrm{Tor} of a principal GG-bundle PXP \to X with connection can be thought of as inducing a morphism of short exact sequences of groupoids: from the sequence of path groupoids of XX to the integrated Atiyah sequence of PP.

Here I take a fresh look at the curvature 2-functor of a parallel transport 1-functor from this point of view, emphasizing the role played by inner automorphisms in this construction.

Curvature 2-Transport, the Atiyah Sequence and Inner Automorphisms

Let P 1(X)P_1(X) be the groupoid of thin homotopy classes of paths andΠ 1(X)\Pi_1(X) is the fundamental groupoid.

Then we have a short exact sequence of path groupoids Ω 1(X)P 1(X)Π 1(X). \Omega_1(X) \to P_1(X) \to \Pi_1(X) \,. Ω 1(X)\Omega_1(X) is the groupoid of closed paths (loops).

On the other hand, the integrated Atiyah sequence of the principal bundle PP is the short exact sequence AdPP× GPΠ 1(X). \mathrm{Ad} P \to P \times_G P \to \Pi_1(X) \,. (Here I am for simplicitly assuming that XX is simply connected, so that Π 1(X)\Pi_1(X) is the codiscrete grupoid over XX. If XX is not simply connected there is a relatively straightforward generalization of everything in sight.)

For any connection on PP, its parallel transport is a functor tra:P 1(X)P× GPGTor. \mathrm{tra} : P_1(X) \to P \times_G P \hookrightarrow G\mathrm{Tor} \,.

This extends to a morphism of exact sequences of groupoids

\;\; (A)

One crucial point is that the groupoids in the middle of the sequences act by conjugation – hence by “inner” automorphisms – on those on the left.

This allows us to turn the parallel transport 1-functor tra\mathrm{tra} into a functor Adtra:P 1(X)INN P× GP(AdP). \mathrm{Ad}\circ \mathrm{tra} : P_1(X) \to \mathrm{INN}_{P \times_G P}(\mathrm{Ad} P) \,. This functor tells us how the monodromy groups of the original functor get transformed as we conjugate loops by paths.

The 2-groupoid INN P× GP(AdP)\mathrm{INN}_{P \times_G P}(\mathrm{Ad} P) is codiscrete at top level (all its Hom-categories are codiscrete). This is the 2-groupoid that the curvature 2-functor of tra\mathrm{tra} takes values in. Codiscreteness of INN P× GP(AdP)\mathrm{INN}_{P \times_G P}(\mathrm{Ad} P) then induces the Bianchi identity on curvature.

To see this – and that’s the point of this discussion here – we pull back the sequences (A) along themselves to obtain

Here the fundamental 2-groupoid Π 2(X)\Pi_2(X) appears in its incarnation where 1-morphisms are taken to be thin homotopy classes of paths. (And now I am even assuming that XX is 2-connected, for simplicity. Otherwise there is a more or less obvious way to adapt the entire construction.)

Then the two nested pullbacks give us a unique canonical factorization morphism. That’s our curvature 2-functor

So by regarding the connection as a morphism from the path sequence of the base to the Atiyah sequence of the bundle, also the discussion of curvature becomes a little more compelling.

More details in the above mentioned notes.

Posted at June 20, 2007 4:23 PM UTC

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Re: Curvature, the Atiyah Sequence and Inner Automorphisms

Here is yet another way to see the same. This is closely analogous the the discussion of (twisted) vector bundles as morphisms into 2-vector transport 2-functors which I gave here.

Let GrpSpc GrpSpc be the 2-category whose

- objects are groups

- morphisms S:GHS : G \to H are left GG- right HH-spaces

- 2-morphisms are maps of GG-HH-spaces, commuting with both actions.

This plays the analogous role for principal 1- and 2-bundles as Bimodules \mathrm{Bimodules} does for vector 1- and 2-bundles.

Curvature of a principal 2-bundles with connection is a 2-functor curv:P 2(X)GrpSpc \mathrm{curv} : P_2(X) \to \mathrm{GrpSpc} which is such that it trivializes in the sense that there is a morphism TRA:Icurv \mathrm{TRA} : I \to \mathrm{curv} into it.

Here I:P 2(X)GrpSpcI : P_2(X) \to \mathrm{GrpSpc} is the 2-functor which sends everything to the identity on the trivial group.

The component map of TRA\mathrm{TRA} – a functor on P 1(X)P_1(X) – gives the parallel transport tra:P 1(X)GTor\mathrm{tra} : P_1(X) \to G\mathrm{Tor}.

To see this, notice that curv\mathrm{curv} will send any surface

γ x Σ y γ \array{ & \nearrow \searrow^{\gamma} \\ x &\Downarrow^\Sigma& y \\ & \searrow\nearrow_{\gamma^\prime} } to Aut(P x) Ad(tra(γ)) Aut(P x) tra(γ 1γ) Aut(P y) Aut(P x) Ad(tra(γ)). \array{ & \nearrow && \searrow^{\mathrm{Aut}(P_x)_{\mathrm{Ad}(\mathrm{tra}(\gamma))}} \\ \mathrm{Aut}(P_x) & & \Downarrow^{\mathrm{tra}(\gamma^{-1}\circ \gamma')}& & \mathrm{Aut}(P_y) \\ & \searrow && \nearrow_{{\mathrm{Aut}(P_x)_{\mathrm{Ad}(\mathrm{tra}(\gamma'))}}} } \,.

(Here a group GG with an automorphism α\alpha in the subscript, G αG_\alpha denotes GG as a GG-GG space with the obvious left and the α\alpha-twisted right action.)

Here P xP_x, P yP_y are the fibers of a principal GG-bundle over XX and Aut(P x)\mathrm{Aut}(P_x) is their automorphism group in the category of GG-spaces. This is canonically isomorphic to the group AdP x:=P x× GG \mathrm{Ad}P_x := P_x \times_G G which appeared a lot in the above entry.

Notice how the fact that the 2-functor curv\mathrm{curv} admits a “2-section” IcurvI \to \mathrm{curv} forces it to takes values in a sub-2-category of GrpSpc\mathrm{GrpSpc} which in the above entry I denoted INN P× GP(AdP). \mathrm{INN}_{P \times_G P}(\mathrm{Ad}P) \,.

This is equivalent to INN(G) \mathrm{INN}(G) and hence, in particular, trivializable.

While curv\mathrm{curv} is hence trivializable, it is the choice of the trivialization, TRA\mathrm{TRA}, which encodes the interesting information:

one and the same curvature 2-functor may be trivialized by several non-equivalent morphisms TRA\mathrm{TRA} and TRA\mathrm{TRA}'.

I am emphasizing this point since David Corfield was wondering about this here. The above is one example for the answer which I gave here.

Posted by: urs on June 22, 2007 2:22 PM | Permalink | Reply to this

Re: Curvature, the Atiyah Sequence and Inner Automorphisms

Here is a pdf version of the above sketch of how curvature is a principal 2-transport trivialized by its parallel 1-transport tra:I Gcurv. \mathrm{tra} : I_G \stackrel{\sim}{\to} \mathrm{curv} \,.

Posted by: urs on June 26, 2007 9:25 PM | Permalink | Reply to this
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