Notes on the derivation of the general supergravity/matter/Yang-Mills Lagrangian for $N=1$ supersymmetry in $d=4$ dimensions using superspace techniques

TL;DR

This paper provides a comprehensive, step-by-step derivation of the general supergravity/matter/Yang-Mills Lagrangian for N=1 supersymmetry in four dimensions using superspace methods, clarifying the geometric and component field structures.

Contribution

It offers a self-contained, systematic derivation of the supergravity/matter/Yang-Mills Lagrangian in N=1 supersymmetry, including Bianchi identities, superfield actions, and component field equations.

Findings

Derived the supergravity multiplet components from superspace torsion constraints.

Formulated the general supergravity/matter/Yang-Mills Lagrangian in superspace.

Connected superspace formulations to component field equations of motion.

Abstract

The coupling of matter to supergravity with $N=1$ supersymmetry in $d=4$ dimensions is described in a geometric manner by K\"ahler superspace. A straightforward way to implement K\"ahler superspace is via $\mathrm{U}(1)$ superspace by identifying the $\mathrm{U}(1)$ pre-potential with the K\"ahler potential, which is a function of the matter (chiral) superfields. In this framework, the components of the supergravity multiplet are contained in the supervielbein and torsion tensor of superspace. Furthermore, interactions with the Yang-Mills (vector) multiplet are formulated by introducing a connection $1$-superform of an additional gauge structure. In these notes, the Bianchi identities in $\mathrm{U}(1)$ superspace are solved for a particular set of torsion constraints which lead to the minimal supergravity multiplet. Moreover, the solution of the Bianchi identities in the gauge sector is derived and K\"ahler superspace is defined. At the superfield level, the general action of the supergravity/matter/Yang-Mills system and supergravity transformations are formulated, and the equations of motion are deduced. Using projection to lowest components in superspace, the corresponding Lagrangian and supergravity transformations at the component field level are calculated, and the equations of motion of the auxiliary fields are determined. Compared to existing literature, these notes provide a self-contained and consistent step by step derivation of the general supergravity/matter/Yang-Mills Lagrangian for $N=1$ supersymmetry in $d=4$ dimensions by means of superspace techniques.