Exploring Curved Superspace (1205.1115v2)

Abstract: We systematically analyze Riemannian manifolds M that admit rigid supersymmetry, focusing on four-dimensional N=1 theories with a U(1)_R symmetry. We find that M admits a single supercharge, if and only if it is a Hermitian manifold. The supercharge transforms as a scalar on M. We then consider the restrictions imposed by the presence of additional supercharges. Two supercharges of opposite R-charge exist on certain fibrations of a two-torus over a Riemann surface. Upon dimensional reduction, these give rise to an interesting class of supersymmetric geometries in three dimensions. We further show that compact manifolds admitting two supercharges of equal R-charge must be hyperhermitian. Finally, four supercharges imply that M is locally isometric to M_3 x R, where M_3 is a maximally symmetric space.

• The paper demonstrates that manifolds with a single supercharge are Hermitian, while those with multiple supercharges require stricter structures like hyperhermitian and maximally symmetric geometries.
• It employs advanced mathematical tools, including twistor and Chern connections, to link supersymmetry with explicit geometric features such as torus fibrations and canonical compact spaces.
• These findings pave the way for applying supersymmetric field theories on curved backgrounds, offering practical insights that could inspire further research in theoretical and high-energy physics.

Systematic Analysis of Riemannian Manifolds with Rigid Supersymmetry

The paper authored by Dumitrescu, Festuccia, and Seiberg provides an in-depth exploration of Riemannian manifolds that accommodate rigid supersymmetry with strong emphasis on four-dimensional $\mathcal{N}=1$ theories and their U(1) R-symmetries. One notable achievement of this work is the characterization of the manifolds based on the number and type of supercharges they admit. This analysis significantly advances the understanding of the geometric prerequisites and implications of placing field theories with supersymmetry on curved backgrounds, which is of particular interest in theoretical physics and mathematical geometry circles.

Key Findings and Contributions

1. Single Supercharge Requirement: It is demonstrated that a Riemannian manifold $M$ admits a single supercharge if and only if $M$ is a Hermitian manifold. This finding underlines the crucial role that Hermitian geometry plays in the establishment of supersymmetry in four-dimensional manifold contexts.
2. Two Supercharges: In scenarios where two supercharges with opposite R-charge exist, the authors identify these geometries in terms of certain fibrations of a torus over a Riemann surface. This structural insight not only extends known geometries but also enriches the field's understanding of lower-dimensional supersymmetric models through dimensional reduction.
3. Two Equal R-charge Supercharges: For manifolds admitting two supercharges of equal R-charge, the constraints are even tighter. The research concludes that such manifolds must be hyperhermitian, with compact representatives being restricted to specific well-known spaces: flat T^3, K3 surfaces with Ricci-flat Kähler metrics, and S^3 × S with standard metrics.
4. Four Supercharges: The presence of four supercharges implies that $M$ is locally isometric to $M^3 \times \mathbb{R}$ where $M^3$ is a maximally symmetric space. This categorization offers a pathway to exploring highly symmetric configurations in supersymmetric field theories.

Theoretical Implications and Directions for Future Research

This paper holds significant implications for the theoretical understanding of supersymmetry on curved manifolds. By employing mathematical constructs such as twistor and Chern connections, the paper enriches the toolkit available for formulating and analyzing similar field theories. The authors note the potential overlaps with existing conformal supergravity literature, suggesting a broader applicability of their findings.

Future research can build upon these results to explore non-trivial supersymmetric field theories in diverse geometric settings beyond the conventional arena. The inclusion of more complicated symmetry groups or coupling to additional fields might yield new classes of solutions that contribute to both the mathematical and physical discourse on supersymmetry and its manifestations in varied dimensions and environments.

Practical Implications

Practically, this work points towards novel geometric setups where supersymmetric theories can be applied, providing potential avenues for constructing new computational models or simulations in high-energy physics, especially in string theory and quantum field theory. Moreover, this analysis reinforces the interplay between geometry and physics, where understanding the geometric properties of a manifold can lead to deeper insights into the physical theories defined over them.

Conclusion

Dumitrescu et al. present a comprehensive framework that is instrumental in delineating the types of Riemannian manifolds which accommodate different supersymmetry constraints. Their work represents a pivotal step in understanding the coalescence of geometry and physics and has laid a cornerstone for forthcoming inquisition into the realms where these disciplines converge. This systematic exploration deconstructs complex relationships between supercharges and manifold geometry, progressively charting a course toward elucidating the mathematical structure underpinning supersymmetric quantum field theories.