Chapter 6: Explicit computation of the supergravity action

In this chapter, we will expand the Lagrangian (5.35) for a generic supersymmetric theory. We will hence derive the most general expression of the supergravity Lagrangian in terms of the components of the gravitation multiplet and of those of the chiral and vector superfield of the theory. We specify the matter sector of the model by a set of chiral superfields denoted by Φⁱ which are assumed to lie in a representation ℜ of a gauge group G. Subsequently, the antichiral superfields ${\Phi }_{i^{*}}^{†}$ lie in the $\overline{\Re }$ representation of G (the complex conjugate representation). To the gauge group we associate a vector superfield V, lying in the adjoint representation of G, and we adopt the Wess-Zumino gauge (4.40). In order to facilitate the computation of the different terms appearing in the Lagrangian, we decompose the latter into different sectors, i.e., the pure gravitation sector, the gravitational and gauge interactions of the matter content and those of the gauge superfields. This approach is complementary to historical approaches obtained by means of the tensor calculus [Cremmer et al. (1979, 1982, 1978b, 1983b)].