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

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