Component vs. Superspace in D=4 N=1 Conformal Supergravity

The standard model of particle physics describes the electroweak and strong forces. The supersymmetric standard model suggests the unification of the electroweak and strong forces at certain energy scale. In nature,　gravitational force also exists. If we consider the unification　of four forces ultimately, supersymmetry with gravity - supergravity - is one of the promising candidate for the unified　theory. Formally, the action of supergravity can be obtained from the conformal supergravity by imposing gauge fixing conditions. There are two formulations to deal with it. One is the component formulation called the superconformal tensor calculus, which is practically useful in that we can calculate in the language of ordinary field theory. The other is the conformal superspace formalism,　which manifests the supersymmetry in a geometrically clear way. However, the consistency and correspondence　between them has not been understood so far, and it is important to show such a correspondence explicitly. The aim of this talk is to show the consistency and construct the　general and explicit correspondence between these two　formulations. Firstly we show that the new geometric　formulation　does not have the restriction discussed previously. Next, we construct　the correspondences of the curvatures, conformal multiplets, chiral projections, actions of the theories, and gauge-fixing conditions for Poincare supergravity with the canonical Einstein-Hilbert term.