Obstructions to the integrability of VB-algebroids

VB-groupoids are vector bundle objects in the category of Lie groupoids: the total and the base spaces of the vector bundle are Lie groupoids and the vector bundle structure maps are required to define Lie groupoid morphisms. The infinitesimal version of VB-groupoids are VB-algebroids, namely, vector bundle objects in the category of Lie algebroids. Following recent developments in the area, we show that a VB-algebroid is integrable to a VB-groupoid if and only if its base algebroid is integrable and the spherical periods of certain underlying cohomology classes vanish identically. We illustrate our results in concrete examples. Finally, we obtain as a corollary computable obstructions for a 2-term representation up to homotopy of Lie algebroid to arise as the infinitesimal counterpart of a smooth such representation of a Lie groupoid.