Elliptically Squashed 3-Spheres

• Elliptically and biaxially squashed three-spheres are compact Riemannian manifolds formed by deforming the standard S³ metric to create anisotropic geometries.
• They exhibit controlled symmetry breaking from SO(4) to subgroups like SU(2)×U(1) or U(1)×U(1), which alters their curvature, spectral properties, and Killing spinor configurations.
• These manifolds underpin key models in supergravity, supersymmetric localization, and holographic dualities, providing a versatile framework for studying quantum field theories on curved spaces.

An elliptically or biaxially squashed three-sphere is a compact, homogeneous Riemannian manifold obtained by deforming the standard round metric on $S^3 \simeq SU(2)$ along two or three inequivalent principal axes. These deformations break the full $SO(4)$ isometry group to a subgroup, typically $SU(2) \times U(1)$, and interpolate between the round geometry and squashed or even degenerate topologies. Such squashed geometries play a critical role in supergravity, Kaluza–Klein compactifications, quantum field theory on curved spaces, integrable sigma models, supersymmetric localization, and gauge/gravity dualities. The geometry, symmetry breaking, curvature properties, and role in physical models of squashed $S^3$ are comprehensively characterized by various explicit metric families and analytic results.

1. Definition and Metric Structures

The squashed $S^3$ is typically defined as a deformation of the left-invariant metric on $SU(2)$. Let $\{\sigma_1, \sigma_2, \sigma_3\}$ denote these left-invariant one-forms, satisfying $$d\sigma_i + \frac12\epsilon_{ijk}\sigma_j \wedge \sigma_k = 0$$. The round metric is

$ds^2_0 = \sigma_1^2 + \sigma_2^2 + \sigma_3^2.$

A biaxially squashed metric introduces a single real parameter $\lambda$ or $C$, yielding

$$ds^2 = \sigma_1^2 + \sigma_2^2 + \lambda\,\sigma_3^2 \qquad (\lambda \neq 1)$$

or explicitly in terms of Euler angles, in the $SU(2)$-invariant form,

$$ds^2 = \frac{L^2}{4}\left[ d\theta^2 + \sin^2\theta\,d\phi^2 + (1 + C)(d\psi + \cos\theta\,d\phi)^2 \right].$$

The most general (triaxial, or elliptic) squashed $S^3$ is given by

$$ds^2 = \ell_1^2 \sigma_1^2 + \ell_2^2 \sigma_2^2 + \ell_3^2 \sigma_3^2,$$

where $\ell_{i}>0$ are principal “squashing axes” and up to an overall scale two independent ratios label deformations (Kawaguchi et al., 2011).

An alternative ellipsoidal parametrization employs coordinates $(\theta, \varphi, \chi)$ with

$$ds^2 = f(\theta)^2 d\theta^2 + \ell^2 \cos^2\theta\,d\varphi^2 + \tilde{\ell}^2 \sin^2\theta\,d\chi^2,\qquad f(\theta) = \sqrt{\ell^2 \sin^2\theta + \tilde{\ell}^2 \cos^2\theta}$$

(Hama et al., 2011), or, in fully symmetric form for squashing along both $\sigma_2$ and $\sigma_3$,

$$ds^2_{(\alpha, \beta)} = \frac14\left[ \sigma_1^2 + \frac{1}{1+\beta}\,\sigma_2^2 + \frac{1}{1+\alpha}\,\sigma_3^2 \right],\qquad \alpha, \beta > -1$$

(Bobev et al., 2016, Conti et al., 2017). These forms admit a direct geometric interpretation as deformations away from the round three-sphere ($\ell_1=\ell_2=\ell_3$ or $\alpha=\beta=0$).

2. Symmetries and Isometry Breaking

For the round $S^3$ ($\lambda=1$, $C=0$, $\ell_1=\ell_2=\ell_3$), the isometry group is $SO(4) \simeq SU(2)_L \times SU(2)_R$. A one-parameter (biaxial) squash ($\lambda\neq1$ or $C\neq0$) reduces the isometry to $SU(2)_L \times U(1)_R$. In the generic triaxial case ($\ell_1\neq\ell_2\neq\ell_3$), only a $U(1) \times U(1)$ subgroup remains unbroken.

Transformation properties can be summarized as:

Geometry	Squashing Parameters	Isometry
Round sphere	$\ell_1=\ell_2=\ell_3$	$SU(2)_L \times SU(2)_R$
Biaxial squash	$\ell_1=\ell_2 \neq \ell_3$	$SU(2)_L \times U(1)_R$
Triaxial (elliptic) squash	$\ell_1 \neq \ell_2 \neq \ell_3$	$U(1)\times U(1)$

The breaking of symmetry has direct consequences for residual Killing spinors, preserved supercharges on backgrounds, and possible Wess-Zumino/WZNW-type constructions (Kawaguchi et al., 2011, Hama et al., 2011).

3. Curvature, Volume, and Geometric Invariants

The Ricci scalar $R$ and the total volume $V$ for the biaxially/elliptically squashed $S^3$ are explicitly computable for the general metric; for the double-squashed metric (Bobev et al., 2016, Conti et al., 2017): $$R(\alpha,\beta) = \frac{6 + 8\alpha + 8\beta + 2\alpha\beta(6-\alpha\beta)}{(1+\alpha)(1+\beta)},$$

$$\displaystyle \mathrm{Vol}\bigl(S^3_{(\alpha,\beta)}\bigr) = 2\pi^2\,\big[(1+\alpha)(1+\beta)\big]^{-1/2}.$$

For the biaxially squashed metric $ds^2 = a^2(\sigma_1^2+\sigma_2^2) + b^2 \sigma_3^2$ (Shiraishi, 2014, Canfora et al., 2023), the Ricci scalar is

$R_{3}(a, b) = \frac{b^2-4a^2}{2a^4},$

and the volume is $16\pi^2\, a^2 b$.

Scalar curvature $R$ and spectrum become negative or degenerate at limiting values of the squashing parameters, e.g., in the limit $b\to0$ the manifold degenerates to $S^2\times S^1$ (Canfora et al., 2023).

4. Role in Quantum Field Theory, Localization, and Holography

Squashed $S^3$ backgrounds underpin the paper of supersymmetric quantum field theories via localization, AdS/CFT, dS/CFT, and integrable systems.

Supersymmetric Theories and Localization: For $\mathcal N=2$ 3D gauge theories localizable on a biaxially squashed $S^3$ with $SU(2)_L \times U(1)_R$ isometry, the partition function is given by matrix integrals with measure and integrand dependence on a squashing/biaxial parameter $v$ (Imamura et al., 2011, Martelli et al., 2011). The free energy in the large-$N$ limit for a class of quiver Chern–Simons–matter theories on a squashed $S^3$ is $F_{\rm squashed}(b) = F_{\rm round}/v^2$ with $v^2=1+u^2$ and $b=(1+iu)/v$.

Killing Spinors: Existence of charged Killing spinors and associated background gauge fields is essential for preserving supersymmetries under squashing. In the elliptic $b\neq1$ regime, determinant formulas for chiral multiplet contributions reduce to double-sine functions $s_b(z)$ that interpolate between Liouville/Toda CFT structure constants (Hama et al., 2011, Kawano et al., 2015). For hyper-ellipsoidal squashing, the background gauge field and Killing spinor equations also admit precise characterization (Park, 19 Nov 2025).

Gauge/Gravity Duality: The boundary metric in $\mathrm{AdS}_4$ or Taub–NUT–AdS compactifications can be a (biaxially) squashed $S^3$. Holographic free energies computed via gravitational action match precisely the localization results in the dual field theories: $F(v) = \frac{F_{\rm round}}{v^{2}}$ for the Taub–NUT–AdS/field theory match (Martelli et al., 2011). Double squashed boundaries ($\alpha, \beta$) are directly related to multiparameter minisuperspace wavefunctions in dS/CFT, with partition functions numerically and analytically tracking the phase structure and suppressing negative-curvature geometries (Bobev et al., 2016, Conti et al., 2017).

5. Spectral and Stability Properties

Squashing splits the Laplacian and Dirac spectra: for the Laplacian (Shiraishi, 2014, Conti et al., 2017): $$M_{L, m}^2(a,b) = \frac{1}{a^2}[L(L+1)-m^2] + \frac{1}{b^2} m^2 - \xi R_3(a,b).$$ Degeneracies are reduced as eigenvalues depend nontrivially on both $(L,m)$, and, for fermions in non-abelian backgrounds, explicit non-Abelian shifts appear (Canfora et al., 2023).

In Kaluza–Klein reductions, quantum stability analysis reveals that the round $S^3$ is only meta-stable against squashing for a narrow window of curvature coupling $\xi$ for nonminimally coupled scalars, $0.007\lesssim\xi\lesssim0.198$ (Shiraishi, 2014).

6. Integrable Sigma Models and T-Duality

The bi-axially squashed $S^3$ admits classical integrable structure for the sigma-model via two distinct Lax pair formulations (Orlando et al., 2010, Kawaguchi et al., 2011):

• Rational Lax pair: based on $SU(2)_L$ symmetry and Yangian extension.
• Trigonometric Lax pair: reflects $q$-deformation and quantum affine extension of $SU(2)_R$.

The integrability survives only for single-parameter squashing; the algebraic structure is strictly tractable in this regime, while for fully elliptic (two-parameter) squashing, coset structure and integrability machinery become more intricate or break down.

T-duality relates models on squashed $S^3$ to integrable models on $S^2 \times S^1$ and to warped $\mathrm{AdS}_3$ and Schrödinger spacetimes (Orlando et al., 2010).

7. Applications and Physical Contexts

Squashed $S^3$ manifolds are ubiquitous in:

• Compactifications in supergravity, especially Freund–Rubin mechanisms (e.g., AdS$_4 \times$ squashed $S^7$ and squashed $S^3$ toy models) (Avila et al., 2013).
• Gravity duals for supersymmetric field theories with deformed boundaries (Martelli et al., 2011).
• The geometry and dynamics of lower-dimensional quantum field theories, e.g., in the paper of integrable structures, partition functions, gauge theory index computations in Cardy-like limits, and Chern–Simons–matter theories (Park, 19 Nov 2025).
• Constructing explicit self-gravitating solitonic solutions (e.g., anisotropic merons in the Einstein–Yang–Mills–Chern–Simons theory) and exploring associated Dirac operator spectra (Canfora et al., 2023).
• Minisuperspace models for de Sitter cosmology and no-boundary quantum cosmological wavefunctions, where the anisotropic moduli of squashing play the role of dynamical fields (Conti et al., 2017, Bobev et al., 2016).

Squashed $S^3$ spaces, through their controllable symmetry breaking and spectral properties, thus serve as versatile models in the paper of symmetry, stability, integrability, and quantum geometry across high-energy theory and mathematical physics.