We construct local solutions to 11-dimensional supergravity (or M-theory), which are invariant under the superalgebra D(2,1;c′;0)⊕D(2,1;c′;0) for all values of the parameter c′. The BPS constraints are reduced to a single linear PDE on a complex function G. The PDE was solved in 0806.0605 modulo application of boundary and regularity conditions. The physical fields of the solutions are determined by c′, a harmonic function h, and the complex function G. h and G are both functions on a 2-dimensional compact Riemannian manifold. The harmonic function h is freely chosen. We obtain the expressions for the metric and the field strength in terms of G, h, and c′ and show that these are indeed valid solutions of the Einstein, Maxwell, and Bianchi equations. Finally we give a construction of one parameter deformations of AdS7×S4 and AdS4×S7 as a function of c′.
@article{arxiv.1209.1845,
title = {Exact Half-BPS Flux Solutions in M-theory with $D(2, 1; c'; 0)^2$ Symmetry: Local Solutions},
author = {John Estes and Roman Feldman and Darya Krym},
journal= {arXiv preprint arXiv:1209.1845},
year = {2013}
}