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We study correlation functions in topologically twisted $\mathcal{N}=2, d=4$supersymmetric Yang-Mills theory for gauge groups of rank larger than one oncompact four-manifolds $X$. We find that the topological invariance of thegenerator of correlation functions of BRST invariant observables is not spoiledby noncompactness of field space. We show how to express the correlators onsimply connected manifolds of $b_{2,+}(X)>0$ in terms of Seiberg-Witteninvariants and the classical cohomology ring of $X$. For manifolds $X$ ofsimple type and gauge group SU(N) we give explicit expressions of thecorrelators as a sum over $\mathcal{N}=1$ vacua. We describe two applicationsof 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.Comment: 58 pages, harvmac b-mode. Important corrections in sections 9.3 and  11.

The Donaldson-Witten Function for Gauge Groups of Rank Larger Than One