The Donaldson-Witten function for gauge groups of rank larger than one

We study correlation functions in topologically twisted $\mathcal{N}=2, d=4$ supersymmetric Yang-Mills theory for gauge groups of rank larger than one on compact four-manifolds $X$. We find that the topological invariance of the generator of correlation functions of BRST invariant observables is not spoiled by noncompactness of field space. We show how to express the correlators on simply connected manifolds of $b_{2,+}(X)>0$ in terms of Seiberg-Witten invariants and the classical cohomology ring of $X$. For manifolds $X$ of simple type and gauge group SU(N) we give explicit expressions of the correlators as a sum over $\mathcal{N}=1$ vacua. We describe two applications of our expressions, one to superconformal field theory and one to large $N$ expansions of SU(N) $\mathcal{N}=2, d=4$ supersymmetric Yang-Mills theory.