A 10d view on the KKLT AdS vacuum and uplifting

We analyse the ten-dimensional Einstein equations in the KKLT setting. We verify that the quartic gaugino term is needed to remove singularities in the on-shell action as suggested by Hamada et. al. We contrast two approaches that have been taken in the literature when employing the effect of gaugino condensation in the ten-dimensional equations of motion. Here we follow the proposal to insert explicit non-zero fermion bilinar vev into the localised energy-momentum tensor of the 7-branes obtained from varying the 10d on-shell action with respect to the 10d metric. Our procedure is common in semi-classical physics and is manifestly local in 10d. However, it does not lead to the KKLT effective field theory. The alternative procedure of deriving the energy momentum tensor after replacing fermion bilinears by the gaugino vev, might be less well motivated in 10d, but reproduces the results of the KKLT effective field theory.