We consider 5d N=1 SU (2) super Yang-Mills theory on X S¹, with X a closed smooth four-manifold. A partial topological twisting along X renders the theory formally independent of the metric on X. The theory depends on the spin structure and the circumference R of S¹. The coefficients of the R-expansion of the partition function are Witten indices, which are identified with L²-indices of Dirac operators on moduli spaces of instantons. The partition function encodes BPS indices for instanton particles on a spatial manifold X, and these indices are special cases of K-theoretic Donaldson invariants. When the 't Hooft flux of the gauge theory is nonzero and X is not spin, the 5d theory can be anomalous, but this anomaly can be canceled by coupling to a line bundle with connection for the global U (1) ``instanton number symmetry''. For b₂^+ (X) >0 we can derive the partition function from integration over the Coulomb branch of the effective 4d low-energy theory. When X is toric we can also use equivariant localization with respect to the C^* C^* symmetry. The two methods lead to the same results for the wall-crossing formula. We also determine path integrals for four-manifolds with b₂^+ (X) >1. Our results agree with those for algebraic surfaces by Göttsche, Kool, Nakajima, Yoshioka, and Williams, but apply to a larger class of manifolds. When the circumference of the circle is tuned to special values, the path integral is associated with the 5d superconformal E₁ theory. Topological invariants in this case involve generalizations of Seiberg-Witten invariants.
Kim et al. (Sat,) studied this question.