  An anti-D3-brane plays a crucial role in the construction of semi-realistic
cosmological models in string theory. Part of its action provides an uplift
term that has been used to lift AdS solutions to phenomenologically viable dS
vacua in the KKLT and LVS setups. In the last few years it has been shown that
this uplift breaks supersymmetry spontaneously and can be described in the 4d
$\mathcal{N}=1$ supergravity language by using constrained supermultiplets.
Here we derive the complete 4d $\mathcal{N}=1$ supergravity action for an
anti-D3-brane coupled to all closed string background fields. In particular we
include the vector field, the scalar fields and all fermions that live on the
anti-D3-brane.
