Notes on SUSY Gauge Theories on Three-Sphere (1012.3512v3)

Abstract: We extend the formula for partition functions of N=2 superconformal gauge theories on S^3 obtained recently by Kapustin, Willett and Yaakov, to incorporate matter fields with arbitrary R-charge assignments. We use the result to check that the self-mirror property of N=4 SQED with two electron hypermultiplets is preserved under a certain mass deformation which breaks the supersymmetry to N=2.

• The paper extends partition function formulas for N=2 gauge theories on S³ and confirms these results using a mass-deformed T[SU(2)] model.
• Localization techniques are applied to compute one-loop determinants and partition functions by focusing on saddle points dictated by vectormultiplet scalars.
• The derived integral formula for general 3D N=2 theories offers broad applicability, shedding light on duality properties and the role of S-duality in supersymmetric systems.

Overview of "Notes on SUSY Gauge Theories on Three-Sphere"

This paper investigates the supersymmetric gauge theories, particularly focusing on N = 2 superconformal gauge theories on a three-dimensional sphere, denoted as $S^3$. The authors build on previous work by extending methodologies to calculate partition functions, integral formulas, and explore various applications in the context of supersymmetry (SUSY).

Key Contributions and Methodologies

1. Extension of Partition Function Formulas: The authors extend the formula for partition functions of N = 2 superconformal gauge theories originally developed by Kapustin, Willett, and Yaakov. This extension incorporates matter properties of N = 4 SQED, showing self-mirror properties under certain mass deformations that break the supersymmetry to N = 2.
2. Localization Technique: The paper applies the localization principle to compute partition functions and one-loop determinants for N = 2 and N = 4 gauge theories on $S^3$. This involves a detailed analysis using the path integral that localizes onto saddle points specified by the expectation value of vectormultiplet scalars.
3. Integral Formula for General Gauge Theories: The authors derive a generalized integral formula for calculating exact partition functions for a broad class of 3D N = 2 gauge theories. This formula integrates contributions from vectormultiplets, matter chiral multiplets, Chern-Simons, and FI terms.
4. Application to Self-Mirror Theory: The paper examines an N = 4 SQED model known as T[SU(2)], which is established to be self-mirror. The partition function of this theory in the presence of mass deformations corroborates its duality properties, providing insight into the S-duality domain wall of 4D N = 2 SYM theory.

Numerical Results and Claims

• The paper emphasizes the correctness of their partition function formula through the application to specific models like the mass-deformed T[SU(2)], illustrating expected self-mirror behavior under mirror symmetry transformations.

Implications and Future Directions

Practical Implications: The methodologies developed are applicable to a wide range of problems in theoretical physics, especially in understanding duality and symmetry properties of gauge theories on non-trivial manifolds. They enable precise calculations of observables such as partition functions and Wilson loops for superconformal theories, thus having potential impacts on both string theory and condensed matter physics.

Theoretical Implications: The results contribute to the ongoing exploration of the AdS/CFT correspondence and its implications in lower-dimensional gauge theories. The connections with Toda Conformal Field Theories (CFTs) and the AGT correspondence underscore the broader implication in the landscape of integrable systems and quantum gravity.

Speculations for Future Development: With the established techniques, future research could delve into exploring gauge theories on more complex manifold structures or extension to higher supersymmetry. The paper of quantum field theories with varying boundary conditions on such manifolds can also open new avenues in both quantum field theory and mathematical physics.

Conclusion

This paper presents a meticulous extension of partition function calculations within the context of supersymmetric gauge theories on a three-sphere, offering theoretical insights that bridge several facets of quantum field theory and string theory. The techniques delineated hold significant promise for unraveling the structure of gauge theories in curved spacetime, thereby propelling further developments in both theoretical and mathematical investigations of such systems.