∞-Lie theory (higher geometry)
The Atiyah Lie groupoid $At(P)$ of a $G$-principal bundle $P \to X$ is the Lie groupoid whose objects are the fibers of the bundle, and whose morphisms are the $G$-equivariant morphisms between the fibers. Schematically:
Its Lie algebroid is the Atiyah Lie algebroid $at(P)$ of $P$.
Both the Atiyah Lie groupoid and its Lie algebroid are used to characterize and are characterized by connections on $P$.
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 $at(P)$ of the principal bundle $P \to X$ comes canonically with a morphism $at(X) \to T X$ to the tangent Lie algebroid.
The simplest Lie integration of the tangent Lie algebroid is the pair groupoid $X \times X$ of $X$. On the other hand, the universal integration is the fundamental groupoid $\Pi(X)$ (both coincide precisey if $X$ is a simply connected space).
Accordingly, there is a version of the Atiyah Lie groupoid over $X \times X$, and a richer version over $\Pi(X)$.
For $G$ a Lie group and $p \colon P \to X$ a $G$-principal bundle, the Atiyah groupoid $At(P)$ – also called the gauge groupoid or transport groupoid – of $P$ is the Lie groupoid with
the smooth manifold of objects is $Obj(At(P)) \coloneqq X$;
the smooth manifold of morphisms $Mor(At(P)) = (P \times P)/G$, where the quotient is taken with respect to the diagonal action of $G$ on $P \times P$;
the source/target maps are those induced by the bundle projection $p$;
notice that a point $f \colon * \to (P \times P)/G$ over $(x_1 = s(f), x_2 = t(f))$, being an equivalence class of a pair $(s_1, s_2) \in P \times P$ is canonically identified with the unique $G$-equivariant function $f \colon P_{x_1} \to P_{x_2}$ which sends $s_1$ to $s_2$;
composition is the given by ordinary composition of these functions.
The Atiyah groupoid sits in a sequence of groupoids
where
$Ad(P) = P \times_G G$ is the adjoint bundle of groups associated via the adjoint action of $G$ on itself; regarded as a smooth union $\coprod_{x \in X} \mathbf{B} P_x \times_G G$ of one-object groupoids coming from groups;
$Pair(X) = (X \times X \rightrightarrows X)$ is the pair groupoid of $X$
the functor $Ad(P) \to At(P)$ is the identity on objects and on morphisms given by the canonical identification $P_x \times_G G \stackrel{\simeq}{\to} (P_x \times P_x) G$, where again we use the diagonal action of $G$ on $P_x \times P_x$.
the functor $At(P) \to Pair(X)$ is the unique one that is the identity on objects.
Notice that a splitting (a section)
of the Atiyah groupoid is a trivialization of $P$. On the other hand, locally on contractible $U \subset X$ we have $Pair(U) \simeq \Pi_1(U)$ with $U$ the fundamental groupoid of $U$, and a splitting $Pair(U) \simeq \Pi_1(U) \to At(P)|_U$ is still a trivialization over $U$ but indicates now that one may want to interpret it as giving rise to a flat connection.
We have the sequence of surjective and full functors of path categories
with $\Pi_1(X)$ the fundamental groupoid and $P_1(X)$ the smooth path groupoid and may refine the Atiyah groupoid by pulling back along these.
Write therefore $At'(P) := At(P) \times_{Pair(X)} \Pi_1(X)$ for the pullback
A splitting $\Pi_1(X) \to At'(P)$ of the top row is now precisely a flat connection on $P$.
If we pull back further to $A''$
then splittings of $P_1(X) \to At''(X)$ are precisely (not necessarily flat) connections on $P$.
All this is more well known in terms of the Lie algebroid underlying the Atiyah Lie groupoid, i.e. the Atiyah Lie algebroid sequence
where
$ad(P) = P \times_g Lie(G)$ is the adjoint bundle of Lie algebras, associated via the adjoint action of $G$ on its Lie algebra;
$at(P) = (T P)/G$ is the Atiyah Lie algebroid
$T X$ is the tangent Lie algebroid.
Indeed, a splitting $\nabla_{flat} : T X \to at(P)$ of this sequence in the category of Lie algebroids is precisely again a flat connection on $P$ and integrates under Lie integration to the splitting of $At'(P)$ 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 $L_\infty$-algebroids.
higher Atiyah groupoid: | standard higher Atiyah groupoid | higher Courant groupoid | groupoid version of quantomorphism n-group |
---|---|---|---|
coefficient for cohomology: | $\mathbf{B}\mathbb{G}$ | $\mathbf{B}(\mathbf{B}\mathbb{G}_{\mathrm{conn}})$ | $\mathbf{B} \mathbb{G}_{conn}$ |
type of fiber ∞-bundle: | principal ∞-bundle | principal ∞-connection without top-degree connection form | principal ∞-connection |