Supersymmetric Solutions with Fluxes from Algebraic Killing Spinors

We give a general framework for constructing supersymmetric in the presence of non-trivial fluxes of tensor gauge fields. technique involves making a general Ansatz for the metric and then the Killing spinors in terms of very simple projectors on the spinor. These projectors and, through them, the spinors, are determined in terms of the metric Ansatz. The Killing spinor equations fix the tensor gauge fields algebraically, and, with the Bianchi identities, a system of equations for all the metric functions. illustrate this by constructing an infinite family of massive flows that eight supersymmetries in M-theory. This family constitutes all the symmetric Coulomb branch flows of the softly broken, large N scalar-fermion on M2-branes. We reduce the problem to the solution of a single,-linear partial differential equation in two variables. This governs the flow of the fermion mass, and the function that it then generates the entire M-theory solution algebraically in terms of function and its first derivatives. the governing equation is non-linear, it has a very simple perturbation theory which one can see how the Coulomb branch is encoded.