and reductions of type IIA supergravity

We construct a consistent reduction of type IIA supergravity on S3, leading to a maximal gauged supergravity in seven dimensions with the full set of massless SO(4) Yang–Mills fields. We do this by starting with the known S4 reduction of eleven-dimensional supergravity, and showing that it is possible to take a singular limit of the resulting standard SO(5)-gauged maximal supergravity in seven dimensions, whose eleven-dimensional interpretation involves taking a limit where the internal 4-sphere degenerates to R×S3. This allows us to reinterpret the limiting SO(4)-gauged theory in seven dimensions as the S3 reduction of type IIA supergravity. We also obtain the consistent S4 reduction of type IIA supergravity, which gives an SO(5)-gauged maximal supergravity in D=6.