N=2 supersymmetric theories on squashed three-sphere (1109.4734v3)

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.

• The paper demonstrates that N=2 supersymmetry is preserved on a squashed three-sphere by employing dimensional reduction from 4D theories to derive transformation laws and Lagrangians.
• The paper employs localization techniques to compute the partition function, revealing a distinct dependency on the squashing parameter unlike in round sphere setups.
• The paper extends the analysis to large N quiver gauge theories, confirming that the free energy scales as N^(3/2) with a universal 1/v² dependence.

Analysis of ${\cal N}=2$ Supersymmetric Theories on Squashed Three-Spheres

The paper presents a nuanced examination of ${\cal N}=2$ supersymmetric theories confined to a uniquely squashed three-sphere (${\bf S}^3$) with preserved $SU(2)_L \times U(1)_r$ isometry. This specific deformation of the three-sphere is innovative as it retains the pairing of supercharges into two doublets under $SU(2)_L$, diverging from traditional setups.

Core Findings

1. Supersymmetry in Squashed Geometries: The authors illustrate that ${\cal N}=2$ supersymmetry, typically encapsulating four supercharges, can persist under certain geometric deformations of ${\bf S}^3$. They achieve this by deriving transformation laws and constructing Lagrangians for these setups via dimensional reduction from a 4D theory, which intriguingly uses an ${\bf S}^1$ compactification as a foundational technique.
2. Partition Function Computation: By employing localization techniques, the computation of the partition function on this squashed sphere configuration reveals a nuanced dependency on the squashing parameter $v$—a relationship absent in the standard round sphere configurations. The insights into the partition function highlight the sensitivity of theory under symmetries and topology changes.
3. Large $N$ Limit and Free Energy: Extending their analysis to the large $N$ limit of quiver gauge theories, the authors successfully adapt the work on round ${\bf S}^3$ to squashed configurations. They confirm that the free energy for these theories scales proportionally to $N^{3/2}$, justifying prior expectations from AdS/CFT correspondence. Notably, they identify a straightforward $1/v^2$ scaling of the free energy—the independence of this factor from specific theory details bolsters its universality.

Theoretical and Practical Implications

The computational framework laid out serves as robust evidence for the claims about dualities within field theories and their gravitational counterparts. The insight that ${\cal N}=2$ theories on squashed spheres depend non-trivially on the squashing parameters could prompt further exploration into how field theories behave under more general geometric flows and deformations.

Additionally, these findings may pave the way for a deeper understanding of how various symmetry-breaking phenomena and parameter dependencies manifest in higher-dimensional setups, potentially extending to practical computations in quantum field theories and holography.

Avenues for Future Research

1. Holographic Duals: Investigation into the holographic dual descriptions corresponding to these gauge theories on a squashed sphere remains an instructive endeavor. Verifying the predictions from the gravitational side may provide further validation of the $1/v^2$ scaling law for free energies.
2. Other Supersymmetric Configurations: Extending the methodology to explore different supersymmetric configurations, including non-abelian and more complex gauge groups, might unravel additional phenomena or invariants within these theories.
3. Applications to Quantum Gravity: Insights from the supersymmetry transformations and their dimensional reductions could inspire novel approaches for formulating quantum gravity theories or for understanding the impact of compactified dimensions on physical observables.

In conclusion, this paper offers significant theoretical advancements in understanding supersymmetric theories on shaped manifolds, equipping researchers with new tools to analyze both the mathematical and physical complexities of symmetry and its breaking within quantum field landscapes.