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# Advances in Theoretical and Mathematical Physics

## Volume 7 (2003)

### Number 5

### Supersymmetric Solutions with Fluxes from Algebraic Killing Spinors

Pages: 787 – 806

DOI: http://dx.doi.org/10.4310/ATMP.2003.v7.n5.a2

#### Authors

#### Abstract

We give a general framework for constructing supersymmetric solutions in the presence of non-trivial fluxes of tensor gauge fields. This technique involves making a general Ansatz for the metric and then defining the Killing spinors in terms of very simple projectors on the spinor fields. These projectors and, through them, the spinors, are determined *algebraically* in terms of the metric Ansatz. The Killing spinor equations then fix the tensor gauge fields algebraically, and, with the Bianchi identities, provide a system of equations for all the metric functions. We illustrate this by constructing an infinite family of massive flows that preserve eight supersymmetries in *M*-theory. This family constitutes all the radially symmetric Coulomb branch flows of the softly broken, large *N* scalar-fermion theory on *M*2-branes. We reduce the problem to the solution of a single, non-linear partial differential equation in two variables. This equation governs the flow of the fermion mass, and the function that solves it then generates the entire *M*-theory solution algebraically in terms of the function and its first derivatives. While the governing equation is non-linear, it has a very simple perturbation theory from which one can see how the Coulomb branch is encoded.