Twisted Partition Functions and $H$-Saddles

Abstract

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 ${\mathcal{Z}}^G$ 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 ${\mathcal{Z}}^H$'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 ${\mathcal{Z}}^G$ had failed to capture the recently constructed bulk index ${\mathcal{I}}_{\mathrm{bulk}}^G$. The purported agreement between 4d and 5d instanton partition functions, despite such subtleties also present in the ADHM data, is explained.

37 pages; a new appendix on examples with high rank classical groups

References in corpus (3)

• Counting Gauge Invariants: the Plethystic Program
• Witten Index and Wall Crossing
• M5-branes, orientifolds, and S-duality

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