Flowing with eight supersymmetries inM-theory andF-theory

We consider holographic RG flow solutions with eight supersymmetries andstudy the geometry transverse to the brane. For both M2-branes and forD3-branes in F-theory this leads to an eight-manifold with only a four-formflux. In both settings there is a natural four-dimensional hyper-Kahler slicethat appears on the Coulomb branch. In the IIB theory this hyper-Kahlermanifold 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. Thissolution is first obtained using gauged supergravity and then lifted to elevendimensions. In this new solution, the brane probes have an Eguchi-Hanson modulispace with the M2-branes spread over the non-trivial 2-sphere. It is also shownthat the new solution is valid for a class of orbifold theories. We discuss howthe hyper-Kahler structure on the slice extends to some form of G-structure inthe eight-manifold, and describe how this can be computed.Comment: 29 pages, 1 figure, harvma