N=2 supersymmetric theories on squashed three-sphere

TL;DR

The paper develops ${ m N}=2$ supersymmetric theories on a squashed ${f S}^3$ that preserves ${ m SU}(2)_L imes{ m U}(1)_r$ and four supercharges, deriving transformation laws and Lagrangians through a 4d ${ m N}=1$ origin and a twisted ${ m S}^1$ reduction.Using localization, it computes the exact partition function, expressing 1-loop determinants in terms of the double-sine function $s_b$, with explicit vector and chiral multiplet contributions on the squashed background.In the large-$N$ regime for a class of quiver theories, the free energy scales as $N^{3/2}$ and factorizes with the squashing as $F_{ m squashed} = rac{1}{v^2}F_{ m round}$, independent of detailed quiver structure under standard non-chiral and balance conditions.The results imply nontrivial squashing dependence of the 3d partition function and a universal gravity-side scaling, suggesting a consistent holographic interpretation and motivating future dual constructions.

Abstract

We investigate a squashing deformation of 3d N=2 supersymmetric theories on three-sphere, which have four supercharges. The deformation preserves SU(2)_L x U(1)_r isometry and all four supersymmetries. We compute the partition function and find non-trivial dependence on the squashing parameter. We also consider the large N limit of a certain class of quiver gauge theories which have free energy of order N^{3/2}, and show that the free energy on the squashed sphere differs from that on round sphere by a certain factor depending only on the squashing parameter.

Figures (2)

• Figure 1: Twisted compactification of ${\bf S}^3\times{\bf R}$ is shown. Points A and B are identified.
• Figure 2: Integration contours $C_\pm$ and $C_0$ on the $t$-plane are shown. The crosses are poles of function $F(z,t)$.