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Anti-D3 branes and non-perturbative effects in flux compactificationsspontaneously break supersymmetry and stabilise moduli in a metastable deSitter vacua. The low energy 4D effective field theory description for suchmodels would be a supergravity theory with non-linearly realised supersymmetry.Guided by string theory modular symmetry, we compute this non-linearsupergravity theory, including dependence on all bulk moduli. Using either aconstrained chiral superfield or a constrained vector field, the upliftingcontribution to the scalar potential from the anti-D3 brane can beparameterised either as an F-term or Fayet-Iliopoulos D-term. Using again themodular symmetry, we show that 4D non-linear supergravities that descend fromstring theory have an enhanced protection from quantum corrections bynon-renormalisation theorems. The superpotential giving rise to metastable deSitter vacua is robust against perturbative string-loop and $\alpha'$corrections.

Anti-D3 branes and moduli in non-linear supergravity