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 $\Omega^G$ is computed by the partition function ${\cal Z}^G$ in one less dimensions. We show that this type of identification fails generically due to integrations over Wilson lines. Tracing the problem, physically, to saddles with reduced effective theories, we relate $\Omega^G$ to a sum of distinct ${\cal Z}^H$'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 ${\cal Z}^{G}$ had failed to capture the recently constructed bulk index ${\cal I}^G_{\rm bulk}$. The purported agreement between 4d and 5d instanton partition functions, despite such subtleties also present in the ADHM data, is explained.