Matter-coupled de Sitter supergravity

The de Sitter supergravity describes the interaction of supergravity with general chiral and vector multiplets and also one nilpotent chiral multiplet. The extra universal positive term in the potential, generated by the nilpotent multiplet and corresponding to the anti-D3 brane in string theory, is responsible for the de Sitter vacuum stability in these supergravity models. In the flat-space limit, these supergravity models include the Volkov–Akulov model with a nonlinearly realized supersymmetry. We generalize the rules for constructing the pure de Sitter supergravity action to the case of models containing other matter multiplets. We describe a method for deriving the closed-form general supergravity action with a given potential $K$, superpotential $W$, and vector matrix $f_{AB}$ interacting with a nilpotent chiral multiplet. It has the potential $V=e^K(|F^2|+|DW|^2-3|W|^2)$, where $F$ is the auxiliary field of the nilpotent multiplet and is necessarily nonzero. The de Sitter vacuums are present under the simple condition that $|F^2|-3|W|^2>0$. We present an explicit form of the complete action in the unitary gauge.