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Atiyah Lie groupoid

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\infty-Lie theory

∞-Lie theory

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∞-Lie algebroids

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Contents

Idea

The Atiyah Lie groupoid At(P)At(P) of a GG-principal bundle PXP \to X is the Lie groupoid whose objects are the fibers of the bundle, and whose morphisms are the GG-equivariant morphisms between the fibers. Schematically:

At(P)={P xαP y|x,yX}. At(P) = \left\{ P_x \stackrel{\alpha}{\to} P_y | x,y \in X \right\} \,.

Its Lie algebroid is the Atiyah Lie algebroid at(P)at(P) of PP.

Both the Atiyah Lie groupoid and its Lie algebroid are used to characterize and are characterized by connections on PP.

Definition

As generally for every Lie algebroid, there are different Lie groupoids integrating the Atiyah Lie algebroid. We describe two of them.

The Aityah Lie algebroid at(P)at(P) of the principal bundle PXP \to X comes canonically with a morphism at(X)TXat(X) \to T X to the tangent Lie algebroid.

The simplest Lie integration of the tangent Lie algebroid is the pair groupoid X×XX \times X of XX. On the other hand, the universal integration is the fundamental groupoid Π(X)\Pi(X) (both coincide precisey if XX is a simply connected space).

Accordingly, there is a version of the Atiyah Lie groupoid over X×XX \times X, and a richer version over Π(X)\Pi(X).

Over the pair groupoid

For GG a Lie group and p:PXp \colon P \to X a GG-principal bundle, the Atiyah groupoid At(P)At(P) – also called the gauge groupoid or transport groupoid – of PP is the Lie groupoid with

  • the smooth manifold of objects is Obj(At(P))XObj(At(P)) \coloneqq X;

  • the smooth manifold of morphisms Mor(At(P))=(P×P)/GMor(At(P)) = (P \times P)/G, where the quotient is taken with respect to the diagonal action of GG on P×PP \times P;

  • the source/target maps are those induced by the bundle projection pp;

    notice that a point f:*(P×P)/Gf \colon * \to (P \times P)/G over (x 1=s(f),x 2=t(f))(x_1 = s(f), x_2 = t(f)), being an equivalence class of a pair (s 1,s 2)P×P(s_1, s_2) \in P \times P is canonically identified with the unique GG-equivariant function f:P x 1P x 2f \colon P_{x_1} \to P_{x_2} which sends s 1s_1 to s 2s_2;

  • composition is the given by ordinary composition of these functions.

The integrated Atiyah sequence

The Atiyah groupoid sits in an sequence of groupoids

Ad(P)At(P)Pair(X) Ad(P) \to At(P) \to Pair(X)

where

  • Ad(P)=P× GGAd(P) = P \times_G G is the adjoint bundle of groups associated via the adjoint action of GG on itself; regarded as a smooth union xXBP x× GG\coprod_{x \in X} \mathbf{B} P_x \times_G G of one-object groupoids coming from groups;

  • Pair(X)=(X×XX)Pair(X) = (X \times X \rightrightarrows X) is the pair groupoid of XX

  • the functor Ad(P)At(P)Ad(P) \to At(P) is the identity on objects and on morphisms given by the canonical identification P x× GG(P x×P x)GP_x \times_G G \stackrel{\simeq}{\to} (P_x \times P_x) G, where again we use the diagonal action of GG on P x×P xP_x \times P_x.

  • the functor At(P)Pair(X)At(P) \to Pair(X) is the unique one that is the identity on objects.

Notice that a splitting (a section)

Pair(X)At(P) Pair(X) \to At(P)

of the Atiyah groupoid is a trivialization of PP. On the other hand, locally on contractible UXU \subset X we have Pair(U)Π 1(U)Pair(U) \simeq \Pi_1(U) with UU the fundamental groupoid of UU, and a splitting Pair(U)Π 1(U)At(P)| UPair(U) \simeq \Pi_1(U) \to At(P)|_U is still a trivialization over UU but indicates now that one may want to interpret it as giving rise to a flat connection.

Over a path groupoid

We have the sequence of surjective and full functors of path categories

P 1(X)Π 1(X)Pair(X) P_1(X) \to \Pi_1(X) \to Pair(X)

with Π 1(X)\Pi_1(X) the fundamental groupoid and P 1(X)P_1(X) the smooth path groupoid and may refine the Atiyah groupoid by pulling back along these.

Write therefore At(P):=At(P)× Pair(X)Π 1(X)At'(P) := At(P) \times_{Pair(X)} \Pi_1(X) for the pullback

At(P) Π 1(X) At(P) Pair(X). \array{ At'(P) &\to& \Pi_1(X) \\ \downarrow && \downarrow \\ At(P) &\to& Pair(X) } \,.

A splitting Π 1(X)At(P)\Pi_1(X) \to At'(P) of the top row is now precisely a flat connection on PP.

If we pull back further to AA''

At(P) P 1(X) At(P) Π 1(X) At(P) Pair(X). \array{ At''(P) &\to& P_1(X) \\ \downarrow && \downarrow \\ At'(P) &\to& \Pi_1(X) \\ \downarrow && \downarrow \\ At(P) &\to& Pair(X) } \,.

then splittings of P 1(X)At(X)P_1(X) \to At''(X) are precisely (not necessarily flat) connections on PP.

All this is more well known in terms of the Lie algebroid underlying the Atiyah Lie groupoid, i.e. the Atiyah Lie algebroid sequence

ad(P)at(P)TX, ad(P) \to at(P) \to T X \,,

where

  • ad(P)=P× gLie(G)ad(P) = P \times_g Lie(G) is the adjoint bundle of Lie algebras, associated via the adjoint action of GG on its Lie algebra;

  • at(P)=(TP)/Gat(P) = (T P)/G is the Atiyah Lie algebroid

  • TXT X is the tangent Lie algebroid.

Indeed, a splitting flat:TXat(P)\nabla_{flat} : T X \to at(P) of this sequence in the category of Lie algebroids is precisely again a flat connection on PP and integrates under Lie integration to the splitting of At(P)At'(P) discussed above.

To get non-flat connections in the literature one often sees discussed splittings of the Atiyah Lie algebroid sequence in the category just of vector bundles. In that case one finds the curvature of the connection precisely as the obstruction to having a splitting even in Lie algebroids.

One can describe non-flat connections without leaving the context of Lie algebroids by passing to higher Lie algebroids, namely L L_\infty-algebroids.

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

Revised on February 20, 2013 03:24:07 by Urs Schreiber (80.81.16.253)