An ExFT-based algorithm reduces the uplift problem to a PDE on the base of the internal manifold and classifies the type IIB uplifts of pure SO(4)-gauged N=4 supergravity by two harmonic functions on a Riemann surface.
Coleman,Aspects of symmetry, vol
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How to uplift non-maximal gauged supergravities
An ExFT-based algorithm reduces the uplift problem to a PDE on the base of the internal manifold and classifies the type IIB uplifts of pure SO(4)-gauged N=4 supergravity by two harmonic functions on a Riemann surface.