Exact Half-BPS Flux Solutions in M-theory with $D(2,1;c^{\prime };0{)}^2$ Symmetry: Local Solutions

Abstract

We construct local solutions to 11-dimensional supergravity (or M-theory), which are invariant under the superalgebra $D(2,1;c^{\prime };0)\oplus D(2,1;c^{\prime };0)$ for all values of the parameter $c^{\prime }$. 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 $c^{\prime }$, 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 $c^{\prime }$ 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 $Ad{S}_7\times S^4$ and $Ad{S}_4\times S^7$ as a function of $c^{\prime }$.