Rigid Supersymmetric Theories in Curved Superspace (1105.0689v2)

Abstract: We present a uniform treatment of rigid supersymmetric field theories in a curved spacetime $\mathcal{M}$, focusing on four-dimensional theories with four supercharges. Our discussion is significantly simpler than earlier treatments, because we use classical background values of the auxiliary fields in the supergravity multiplet. We demonstrate our procedure using several examples. For $\mathcal{M}=AdS_4$ we reproduce the known results in the literature. A supersymmetric Lagrangian for $\mathcal{M}=\mathbb{S}^4$ exists, but unless the field theory is conformal, it is not reflection positive. We derive the Lagrangian for $\mathcal{M}=\mathbb{S}^3\times \mathbb{R}$ and note that the time direction $\mathbb{R}$ can be rotated to Euclidean signature and be compactified to $\mathbb{S}^1$ only when the theory has a continuous R-symmetry. The partition function on $\mathcal{M}=\mathbb{S}^3\times \mathbb{S}^1$ is independent of the parameters of the flat space theory and depends holomorphically on some complex background gauge fields. We also consider R-invariant $\mathcal{N}=2$ theories on $\mathbb{S}^3$ and clarify a few points about them.

• The paper presents a novel approach using classical background values from the supergravity multiplet to simplify rigid supersymmetry in curved spacetimes.
• It demonstrates consistent supersymmetry embeddings in geometries like AdS4, S4, and S3 x R, emphasizing the critical role of R-symmetry and the Ferrara-Zumino multiplet.
• The findings on holomorphic partition functions and FZ-multiplet constraints provide a robust framework for advanced studies in supersymmetric dualities and quantum field embeddings.

An Expert Review of "Rigid Supersymmetric Theories in Curved Superspace"

The paper presented by Guido Festuccia and Nathan Seiberg offers a comprehensive examination of rigid supersymmetric field theories in curved spacetime, particularly within four-dimensional theories characterized by four supercharges. This work leverages classical background values from the supergravity multiplet's auxiliary fields, which substantially simplifies their approach compared to previous techniques.

Key Contributions

The authors provide clear methodologies for embedding rigid supersymmetric theories in curved spacetimes. They focus on several specific spacetime configurations including $AdS_4$, $S^4$, and $S^3 \times \mathbb{R}$, revealing nuanced dependencies and conditions required for maintaining supersymmetry across these spaces:

1. Use of Classical Background Values: The elegant application of classical background values within the supergravity multiplet reduces the complexity previously encountered by others. Their methodology revives particular interest in using new minimal supergravity as an alternative to earlier approaches.
2. Scenarios in Various Geometries:
   • For $AdS_4$, the paper aligns with existing literature by showcasing that supersymmetric Lagrangians remain invariant and conformal fields can be placed naturally on this space.
   • On $S^4$, although similar embeddings are feasible, they identify that reflection positivity fails unless the field theory exhibits conformal symmetry.
   • The analysis around $S^3 \times \mathbb{R}$ introduces the requirement of continuous R-symmetry for compactifications that retain supersymmetry, a result achieved through insightful deployment of the Ferrara-Zumino supercurrent multiplet.
3. Implications of R-symmetry: The intricacies of maintaining R-symmetry when rotating spacetimes into Euclidean signatures or compactifying them highlight potential theoretical misalignments in lacking such symmetry.
4. Holomorphic Partition Functions: Perhaps one of the most striking revelations is that on $S^3 \times S^1$, the partition function showcases holomorphic dependence on complex background gauge fields and becomes independent of flat space theory parameters.
5. Absence of FZ-multiplet Theories: The analysis concerning theories that cannot possess an FZ-multiplet, emphasizing the need for R-symmetry or additional dynamical fields for AdS embeddings, holds considerable theoretical gravitas.

Numerical and Theoretical Implications

The numerical observations in this paper, particularly regarding holomorphy implications in various spacetimes, establish significant footing for theoretical predictions within the field of supersymmetric field theories. Theoretical implications extend these findings towards their application in assessing dualities, partition function computations, and the general behavior of 4D superconformal indices when embedded in non-trivial geometries.

Speculation on Future Developments

Future explorations may extend these results by probing beyond the confines of R-symmetries and FZ-multiplets into newer classes of symmetries or multiplet definitions that could facilitate novel embeddings. Further, the compelling handling of complexities in supergravity to derive low-energy effective actions is ripe for broader application across quantum fields encased within exotic backgrounds or topologies.

Overall, the paper not only resolves several outstanding questions about rigid supersymmetric theories in curved superspace but also sets a framework for advanced research in supersymmetry, potentially fostering deeper insights into fundamental physics and symmetric relationships between quantum fields and geometry.