- The paper constructs 3D N=2 supersymmetric field theories using the rigid limit of new minimal supergravity, establishing integrability conditions for almost contact metric structures.
- It demonstrates that a single supercharge is always locally achievable, with additional supercharges emerging on manifolds like Seifert structures.
- The research applies localization on a squashed sphere to compute energy-momentum tensor correlators, linking geometric deformations with SCFT behavior.
Supersymmetric Field Theories on Three-Manifolds
The paper "Supersymmetric Field Theories on Three-Manifolds" by Cyril Closset, Thomas T. Dumitrescu, Guido Festuccia, and Zohar Komargodski, addresses the construction of supersymmetric field theories on Riemannian three-manifolds, focusing on three-dimensional N=2 theories with a U(1)R​ symmetry. The authors explore the conditions under which these theories possess rigid supersymmetry by employing the rigid limit of new minimal supergravity in three dimensions.
The core finding of this work is that the constructed field theory on a manifold M possesses one or more rigid supersymmetries if M admits an almost contact metric structure satisfying certain integrability conditions. This elegant result imposes global constraints on potential three-manifold candidates but guarantees that a single supercharge can always be constructed locally. Notably, the existence of additional supercharges is linked to specific geometric structures, such as Seifert manifolds, which universally support two supercharges of opposite R-charge.
Key implications of their formulation are evident when considering the flat-space limits of these theories. For instance, the two-point function of the energy-momentum tensor in flat space is computable via localization techniques on a squashed sphere, thereby demonstrating the interplay between geometry and supersymmetric theory localization.
The analysis is divided into several substantive sections. Initially, the authors set out the framework for coupling three-dimensional quantum field theories to a Riemannian geometry, highlighting the role of rigid supersymmetry in simplifying supergravity analysis. The supersymmetry transformations associated with both supergravity fields and matter fields are detailed, laying the groundwork for constructing supersymmetric Lagrangians.
In a robust treatment of mathematical structures, the paper defines the conditions necessary for various classes of three-manifolds to support supersymmetric theories. An elegant comparison is made between two-dimensional complex structures and the almost contact structures required for three manifolds. This approach underscores the broader mathematical implications and connections to transversely holomorphic foliations.
The paper's results lead to significant theoretical insights, especially in the context of N=2 superconformal field theories (SCFTs). On a global scale, the authors demonstrate how subtle changes to the geometric background, such as squashing the three-sphere, affect the theory's partition function and correlation functions. This is a pivotal aspect that ties together localization results and computational techniques applicable to SCFTs.
This research marks a vital step in understanding the geometric and physical intricacies of placing supersymmetric field theories on non-trivial manifolds. It lays the foundation for further exploration into both mathematical physics and the practical applications of these theoretical constructs. Future work could explore more complex three-manifolds or explore the extension of these techniques to higher dimensions or non-Abelian gauge symmetries, thus expanding our ability to understand quantum field theories in diverse and complex geometric settings.
