  We prove a conformally invariant estimate for the index of Schr\"odinger
operators acting on vector bundles over four-manifolds, related to the
classical Cwikel-Lieb-Rozenblum estimate. Applied to Yang-Mills connections we
obtain a bound for the index in terms of its energy which is conformally
invariant, and captures the sharp growth rate. Furthermore we derive an index
estimate for Einstein metrics in terms of the topology and the Einstein-Hilbert
energy. Lastly we derive conformally invariant estimates for the Betti numbers
of an oriented four-manifold with positive scalar curvature.
