# The Supercurrent and Einstein equations in the Superconformal formulation

## TL;DR

This paper presents a new formulation of the supercurrent and Einstein equations in curved superspace using superconformal methods, clarifying their structure and couplings in supergravity theories.

## Contribution

It introduces a novel expression for the supercurrent in curved superspace and explores its implications for Einstein equations and matter couplings in superconformal supergravity.

## Key findings

- Derived a new supercurrent expression in curved superspace.
- Connected the supercurrent to Einstein equations and matter couplings.
- Identified differences in current improvement between conformal and Poincaré gauges.

## Abstract

We give a new expression for the supercurrent and its conservation in curved ${\cal N}=1$, $D=4$ superspace using the superconformal approach. The first component of the superfield, whose lowest component is the vector auxiliary field gives the (super)Einstein equations. Its trace and couplings to conformal and non-conformal matter is presented. In a suitable dilatational gauge, the conformal gauge, we obtain an update of the Callan-Coleman-Jackiw improved currents for conformal matter, containing $R$-symmetry corrections for a new traceless covariantly conserved energy--momentum tensor. We observe that in the Poincar\'e gauge, where standard Poincar\'e supergravity is usually formulated, the currents are not improved and then the higher conformal symmetry of the matter sector is obscured. The curvature multiplets are used to find supersymmetric curved backgrounds and some examples are exhibited in agreement with existing results.