S 3 and S 4 Reductions of Type IIA Supergravity

We construct a consistent reduction of type IIA supergravity on S 3 , 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 S 4 reduction of eleven-dimensional super-gravity, 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 IR × S 3. This allows us to reinterpret the limiting SO(4)-gauged theory in seven dimensions as the S 3 reduction of type IIA supergravity. We also obtain the consistent S 4 reduction of type IIA supergravity, which gives an SO(5)-gauged maximal supergravity in D = 6.