SUSY Gauge Theories on Squashed Three-Spheres (1102.4716v1)

Abstract: We study Euclidean 3D N=2 supersymmetric gauge theories on squashed three-spheres preserving isometries SU(2) x U(1) or U(1) x U(1). We show that, when a suitable background U(1) gauge field is turned on, these squashed spheres support charged Killing spinors and therefore N=2 supersymmetric gauge theories. We present the Lagrangian and supersymmetry rules for general gauge theories. The partition functions are computed using localization principle, and are expressed as integrals over Coulomb branch. For the squashed sphere with U(1) x U(1) isometry, its measure and integrand are identified with the building blocks of structure constants in Liouville or Toda conformal field theories with b \neq 1.

• The paper introduces two squashing geometries that support charged Killing spinors, enabling the formulation of Euclidean 3D N=2 supersymmetric gauge theories.
• The paper employs localization to compute partition functions, revealing that determinants correspond to Liouville/Toda CFT structure constants.
• The paper explores extended supersymmetry and hints at AGT correspondence extensions, offering insights into dualities on curved manifolds.

Analysis of "SUSY Gauge Theories on Squashed Three-Spheres"

The paper "SUSY Gauge Theories on Squashed Three-Spheres," authored by Naofumi Hama, Kazuo Hosomichi, and Sungjay Lee, presents an in-depth paper of Euclidean 3D $\mathcal{N}=2$ supersymmetric gauge theories formulated on squashed three-spheres preserving specific isometries. The central focus lies on how these squashed spheres support charged Killing spinors necessary for formulating supersymmetric theories, with a rigorous computation of the partition functions using the localization technique. The significance of this work can be contextualized through its contribution to understanding the interplay between three-dimensional geometry and gauge field theories, specifically through the lens of the AGT correspondence, which connects supersymmetric gauge theories to two-dimensional conformal field theories (CFTs).

Key Contributions and Findings

1. Squashed Three-Spheres Geometry: The authors introduce two forms of squashed three-spheres: one preserving $SU(2) \times U(1)$ symmetry and the other preserving $U(1) \times U(1)$ symmetry. The metric of the squashed sphere, along with a suitable background $U(1)$ gauge field, is shown to support the existence of charged Killing spinors. This elegantly sets the stage for the construction of $\mathcal{N}=2$ supersymmetric gauge theories.
2. Localization and Partition Functions: Localization is employed to compute the partition functions of these theories, revealing that the determinants involved are expressible as integrals over the Coulomb branch. For spheres with the $U(1)\times U(1)$ isometry, a notable finding is that the measure and integrand correspond to the structure constants in Liouville or Toda CFTs for general $b$.
3. Extended Supersymmetry: By analyzing the special case of $\mathcal{N}=4$ hypermultiplets, the paper elucidates the complexity introduced when extending the results from $b=1$ to general values. The straightforwardness observed for round spheres is lost, indicating richer structures in squashed geometries.
4. Implications for AGT Correspondence: The work hints at possible extensions to the AGT correspondence for non-unit values of the parameter $b$, providing the groundwork for future theoretical explorations in how 3D SUSY theories on squashed spheres can serve as building blocks for mappings akin to those established for $S^4$ with Liouville/Toda CFTs.

Implications and Future Directions

The implications of this research lie in the broader understanding of supersymmetric geometry and field theories. The novel computation of partition functions on squashed spheres aids in the potential realization of generalized AGT correspondences. This could lead to new insights and more comprehensive application of these gauge theories in string theory and related areas.

Theoretical advancements in constructing richer geometries on which these field theories can exist allow for possible explorations of dualities and quantization in higher-dimensional spaces. The paper suggests an intriguing conjecture about squashing of $S^4$ and its potential application in extending the AGT relation, signaling a fertile area of research.

Conclusion

Hama, Hosomichi, and Lee's work on SUSY Gauge Theories on Squashed Three-Spheres highlights a sophisticated intersection of differential geometry, supersymmetry, and quantum field theory. With precise mathematical rigour, it opens pathways for further exploration into how complex geometrical structures like squashed spheres interact with field-theoretic constructs—holding potential for profound implications in the field of theoretical physics and beyond. The results showcase how altering the symmetry properties of underlying spaces can lead to novel theoretical insights and expand the toolkit available to physicists investigating the fundamental properties of our universe.