Superconformal Symmetry, Supergravity and Cosmology

TL;DR

This work develops a complete SU(2,2|1) superconformal formulation of N=1 gauge theories coupled to supergravity, revealing how conformal symmetry and its gauge fixing recover standard Poincaré supergravity and enabling a robust exploration of cosmology in a conformally flat FRW background. The authors derive the full superconformal action, expose the geometric structure via Kähler geometry, and show how Fayet–Iliopoulos terms arise from conformon gauging. They then formulate gravitino dynamics in cosmological settings, including a detailed gravitino–goldstino equivalence in the large $M_P$ limit and the unitary gauge treatment, and they analyze gravitino production after inflation for theories with one or two chiral multiplets, highlighting that helicity-1/2 gravitinos can be produced efficiently even when gravity is weak. The framework clarifies the super-Higgs effect in cosmology and provides a basis for evaluating gravitino yields against cosmological constraints, while also suggesting potential strong-coupling phases and alternative production mechanisms that warrant further study.

Abstract

We introduce the general N=1 gauge theory superconformally coupled to supergravity. The theory has local SU(2,2|1) symmetry and no dimensional parameters. The superconformal origin of the Fayet-Iliopoulos terms is clarified. The phase of this theory with spontaneously broken conformal symmetry gives various formulations of N=1 supergravity interacting with matter, depending on the choice of the R-symmetry fixing. We have found that the locally superconformal theory is useful for describing the physics of the early universe with a conformally flat FRW metric. Few applications of superconformal theory to cosmology include the study of i) particle production after inflation, particularly the non-conformal helicity 1/2 states of gravitino, ii) the super-Higgs effect in cosmology and the derivation of the equations for the gravitino interacting with any number of chiral and vector multiplets in the gravitational background with varying scalar fields, iii) the weak coupling limit of supergravity and gravitino-goldstino equivalence. This explains why gravitino production in the early universe is not suppressed in the limit of weak gravitational coupling. We discuss the possible existence of an unbroken phase of the superconformal theories, interpreted as a strong coupling limit of supergravity.

Figures (4)

• Figure 1: The constraint on the ratio $n_{3/2}/s$ that follows from the theory of nucleosynthesis, according to M. Kawasaki and K. Kohri. The allowed values for $n_{3/2}/s$ are below the full (or broken) curve. The two curves correspond to the two slightly different observational results concerning the cosmological abundance of $\rm{}^4He$. We are grateful to Kawasaki and Kohri for permission to present their results in our paper.
• Figure 2: Effective potential in the Polónyi model, $W= \zeta (\phi + 2-\sqrt3)$, as a function of $\mathop{\rm Re}\nolimits \phi$.
• Figure 3: Effective potential in the theory $W= \zeta (\phi + 2-\sqrt3) + C_1 (\phi+C_2)(1-\tanh(C_3(\phi+ C_2)))$ as a function of $\mathop{\rm Re}\nolimits \phi$.
• Figure 4: Effective potential in the theory $W= \zeta (\phi + 2-\sqrt3) + C_1 (\phi+C_2)(1-\tanh(C_3(\phi+ C_2)))$ as a function of $\mathop{\rm Re}\nolimits \phi$ and $\mathop{\rm Im}\nolimits \phi$.