Towards new membrane flow from de Wit-Nicolai construction

The internal 4-form field strengths with 7-dimensional indices have been constructed by de Wit and Nicolai in 1986. They are determined by the following six quantities: the 56-bein of 4-dimensional N = 8 gauged supergravity, the Killing vectors on the round seven-sphere, the covariant derivative acting on these Killing vectors, the warp factor, the field strengths with 4-dimensional indices and the 7-dimensional metric. In this paper, by projecting out the remaining mixed 4-form field strengths in an SU(8) tensor that appears in the variation of spin-1/2 fermionic sector, we also write down them explicitly in terms of some of the above quantities. For the known critical points, the N = 8 SO(8) point and the nonsupersymmetric SO(7) ⁺ point, we reproduce the corresponding 11-dimensional uplifts by computing the full nonlinear expressions. Moreover, we find out the 11-dimensional lift of the nonsupersymmetric SO(7) ⁺ invariant flow. We decode their implicit formula for the first time and the present work will provide how to obtain the new supersymmetric or nonsupersymmetric membrane flows in 11 dimensions.