Twisted partition functions and H-saddles

While studying supersymmetric G-gauge theories, one often observes that a zero-radius limit of the twisted partition function Ω^G is computed by the partition function Z^G in one less dimensions. We show how this type of identification fails generically due to integrations over Wilson lines. Tracing the problem, physically, to saddles with reduced effective theories, we relate Ω^G to a sum of distinct Z^H's and classify the latter, dubbed H-saddles. This explains why, in the context of pure Yang-Mills quantum mechanics, earlier estimates of the matrix integrals Z^G had failed to capture the recently constructed bulk index I_bulk^G. The purported agreement between 4d and 5d instanton partition functions, despite such subtleties also present in the ADHM data, is explained.