Integrating representations of Lie algebroids

Question

If $A \to M$ is a Lie algebroid over a smooth manifold $M$ then a representation of $A$ is a vector bundle $E \to M$ with a flat $A$-connection $$ \nabla : \Gamma(E) \to \Gamma(E\otimes A^*). $$ If $G$ is a Lie groupoid over $M$ then a representation is a vector bundle $E\to M$ and a Lie groupoid homomorphism $G \to GL(E)$, where $GL(E)$ is the groupoid over $M$ whose arrows are linear isomorphisms of the fibers of $E$.

Now suppose $G$ is a Lie groupoid that integrates $A$.

Under what sort of conditions can every representation of $A$ be integrated to a representation of $G$?

Two extreme cases are well-known:

• If $M$ is a point then we're just talking about Lie groups and algebras and every representation of the algebra integrates to a representation of the group if the group is simply connected.

• If $A = TM$ is the tangent Lie algebroid and $G$ is the fundamental groupoid then the Riemann-Hilbert correspondence says that every representation of $TM$ (i.e. flat vector bundle over $M$) corresponds to a representation of $G$. On the other hand, the pair groupoid also integrates $A$ but it doesn't seem like it has the same representations (I think a representation of the pair groupoid is given by a vector bundle $E\to M$ and a section of $p_1^* E^* \otimes p_2^* E \to M\times M$, where $p_j : M\times M \to M$ is the projection on the $j$th factor).

I would be interested in seeing any references that talk about this.

Answer 1

The usual condition that one uses to ensure that all representations of $A$ integrate to representations of $G$ is that $G$ be "source 1-connected", meaning that all of the fibres of the source map $s : G \to M$ are connected and simply connected.

That the source 1-connected condition is sufficient is a consequence of a more general theorem which asserts that if $G$ and $H$ are Lie groupoids with $G$ source 1-connected, and if $\phi : Lie(G) \to Lie(H)$ is a morphism between their Lie algebroids, then $\phi$ integrates to a unique Lie groupoid morphism $G \to H$. You can find this result, for example, in the papers

• "Integrations of Lie bialgebroids" by Mackenzie and Xu (Topology 39 (2000), no. 3, 445–467)

• "On integrability of infinitesimal actions" by Moerdijk and Mrčun (Amer. J. Math. 124 (2002), no. 3, 567–593)

Your question deals with the specific case when $H = GL(E)$ is the gauge groupoid of a vector bundle $E \to M$, since a flat $A$-connection on $E$ is the same thing as a morphism $E \to \mathfrak{gl}(E)$ to the Atiyah algebroid.

Regarding the specific case $A = TM$: it's a good exercise to a) convince yourself that the fundamental groupoid $\Pi_1(M)$ is source 1-connected, and b) convince yourself that a flat connection will integrate to a representation of the pair groupoid if and only if it has trivial holonomy along every loop. One way to think about b) is that the natural map $\Pi_1(M) \to Pair(M)$ expresses $Pair(M)$ as a quotient of $\Pi_1(M)$ and you're looking for representations that descend to the quotient.

Finally, I should point out that, while source 1-connectedness is sufficient to conclude that every representation of $A$ integrates to a representation of $G$, it is, in general, not necessary. The example I know of is when the manifold $M$ is the Riemann sphere $\mathbb{C}P^1$ and the Lie algebroid $A$ is generated by the real and imaginary parts of the holomorphic vector fields that vanish at $0 \in \mathbb{C}P^1$. A flat connection for this algebroid is basically a flat connection in the usual sense, but it is allowed to have a singularity of logarithmic type at $0$. In this case the source 1-connected groupoid integrating $A$ is not Hausdorff, but it has a quotient that is Hausdorff, and every representation of $A$ integrates to a representation of this Hausdorff quotient.