Supersymmetric theories on squashed five-sphere

TL;DR

This work constructs five-dimensional ${\cal N}=1$ SUSY gauge theories on the ${\rm SU}(3)\times {\rm U}(1)$-symmetric squashed ${\bm S}^5$, using a dimensional reduction from six dimensions to derive Killing equations and a Noether-based action construction, with the Yang–Mills sector arising from a SUSY Chern–Simons action via a constant vector multiplet. It analyzes how squashing, implemented through twisted boundary conditions, reduces the isometry to ${\rm SU}(3)\times {\rm U}(1)$ and yields distinct SUSY fractions ${\cal N}=3/4$ or ${\cal N}=1/4$, and further shows how adding an adjoint hypermultiplet at critical masses enhances SUSY to ${\cal N}=3/2$ or ${\cal N}=1/2$, all while embedding the framework in a 6d interpretation. The paper provides explicit transformation laws and actions, including ${\cal L}_{\rm YM}$, ${\cal L}_{\rm FI}$, and hypermultiplet couplings adapted to the squashed geometry, and discusses perturbative partition functions and potential localization, highlighting cases where squashing dependence can be absorbed or may persist. Overall, it advances the toolbox for exact results in 5d SUSY on curved backgrounds and clarifies the role of geometry and mass deformations in shaping the SUSY spectrum.

Abstract

We construct supersymmetric theories on the SU(3)xU(1) symmetric squashed five-sphere with 2, 4, 6, and 12 supercharges. We first determine the Killing equation by dimensional reduction from 6d, and use Noether procedure to construct actions. The supersymmetric Yang-Mills action is straightforwardly obtained from the supersymmetric Chern-Simons action by using a supersymmetry preserving constant vector multiplet.