Holonomy Saddles and Supersymmetry

Abstract

In gauge theories on a spacetime equipped with a circle, the holonomy variables, living in the Cartan torus, play special roles. With their periodic nature properly taken into account, we find that a supersymmetric gauge theory in $d$ dimensions tends to reduce in the small radius limit to a disjoint sum of multiple $(d-1)$ dimensional theories at distinct holonomies, called $H$-saddles. The phenomenon occurs regardless of the spacetime dimensions, and here we explore such $H$-saddles for $d=4$ $\mathcal{N}=1$ theories on $T^2$ fibred over ${\Sigma }_g$, in the limits of elongated $T^2$. This naturally generates novel relationships between 4d and 3d partition functions, including ones between 4d and 3d Witten indices, and also leads us to re-examine recent studies of the Cardy exponents and the Casimir energies and of their purported connections to the 4d anomalies.

66 pages; comments and references added, typos corrected

References in corpus (4)

• $\mathcal{N}=1$ supersymmetric indices and the four-dimensional A-model
• Cardy Formulae for SUSY Theories in d=4 and d=6
• Witten Index and Wall Crossing
• From 3d duality to 2d duality

Cited by in corpus (13)

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• Decomposition of BPS Moduli Spaces and Asymptotics of Supersymmetric Partition Functions
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• 3d TQFTs from Argyres-Douglas theories
• On Phases of 3d $\mathcal{N}=2$ Chern-Simons-Matter Theories
• Aspects of 5d Seiberg-Witten Theories on ${\mathbb{S}}^1$
• Superconformal index of higher derivative $\mathcal{N}=1$ multiplets in four dimensions
• Bridging 4D QFTs and 2D VOAs via 3D high-temperature EFTs
• Coulomb branch surgery: Holonomy saddles, S-folds and discrete symmetry gaugings
• The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals Arising in Supersymmetric Gauge Theory
• Universal Cardy-Like Behavior of 3D Partition Functions from Supersymmetric Localization