  We present warped compactification solutions of six-dimensional supergravity,
which are generalizations of the Randall-Sundrum warped brane world to
codimension two and to a supersymmetric context. In these solutions the dilaton
varies over the extra dimensions, and this makes the electroweak hierarchy only
power-law sensitive to the proper radius of the extra dimensions (as opposed to
being exponentially sensitive as in the RS model). Warping changes the
phenomenology of these models because the Kaluza-Klein gap can be much larger
than the internal space's inverse proper radius. We provide examples both for
Romans' nonchiral supergravity and Salam-Sezgin chiral supergravity, and in
both cases the solutions break all of the supersymmetries of the models. We
interpret the solution as describing the fields sourced by a 3-brane and a
boundary 4-brane (Romans' supergravity) or by one or two 3-branes (Salam-Sezgin
supergravity), and we identify the topological constraints which are required
by this interpretation. For both types of solutions the 3-branes are flat for
all topologically-allowed values of the brane tensions. We identify the general
mechanism for and limitations of the self-tuning of the effective 4D
cosmological constant in higher-dimensional supergravity which these models
illustrate.
