11D supergravity on AdS₄  × S⁷ versus AdS₇× S⁴

The maximally supersymmetric Freund–Rubin vacua for 11 dimensional supergravity, namely AdS ₄ × S ⁷ and AdS ₇ × S ⁴, admit an analytic continuation to S ⁴ × S ⁷. From the full harmonic expansions on S ⁴ × S ⁷, it is shown that by analytical continuation to either AdS ₄, or to AdS ₇, the detailed structure of the Kaluza–Klein spectrum can be obtained for both vacua in a unified manner. The results are shown to be related by a simple rule which interchanges the spacetime and internal space representations. We also obtain the linearized field equations for the singletons and doubletons but they can be gauged away by fixing certain Stuckelberg shift symmetries inherited from the Kaluza–Klein reduction. (paper)