Flowing with Eight Supersymmetries in M-Theory and F-theory

We consider holographic RG flow solutions with eight supersymmetries and study the geometry transverse to the brane. For both M2-branes and for D3-branes in F-theory this leads to an eight-manifold with only a four-form flux. In both settings there is a natural four-dimensional hyper-Kahler slice that appears on the Coulomb branch. In the IIB theory this hyper-Kahler manifold encodes the Seiberg-Witten coupling over the Coulomb branch of a U(1) probe theory. We focus primarily upon a new flow solution in M-theory. This solution is first obtained using gauged supergravity and then lifted to eleven dimensions. In this new solution, the brane probes have an Eguchi-Hanson moduli space with the M2-branes spread over the non-trivial 2-sphere. It is also shown that the new solution is valid for a class of orbifold theories. We discuss how the hyper-Kahler structure on the slice extends to some form of G-structure in the eight-manifold, and describe how this can be computed.