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 ΩG is computed by the partition function ZG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal{Z}}^G $$\end{document} 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 ΩG to a sum of distinct ZG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal{Z}}^G $$\end{document}’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 ZG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal{Z}}^G $$\end{document} had failed to capture the recently constructed bulk index ℐbulkG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathrm{\mathcal{I}}}_{\mathrm{bulk}}^G $$\end{document}. The purported agreement between 4d and 5d instanton partition functions, despite such subtleties also present in the ADHM data, is explained.