Construction of N = 8 supergravity theories by dimensional reduction

Abstract

In this paper I ask which N = 8 supergravity theories in four dimensions can be obtained by dimensional reduction of the N = 1 supergravity theory in eleven dimensions. Several years ago Scherk and Schwarz produced a particular class of N = 8 theories by giving a dimensional reduction scheme on the restricted class of coset spaces, G/H, with dim H = 0 (and therefore dim G = 7). I generalize their considerations by looking at arbitrary (seven-dimensional) coset spaces. Also, instead of giving a particular ansatz which happens to work, I set about the distinctly more difficult task of determining all ansatzes which produce N = 8 theories. The basic ingredient of my dimensional reduction scheme is the demand that certain symmetries, including supersymmetry, be truncated consistently. I find the surprising result that the only N = 8 theories obtainable within the contexts of my scheme are those theories already written down by Scherk and Schwarz. In particular dim H = 0 and dim G = 7. Independently of these considerations, I prove that any dimensional reduction scheme which consistently truncates supersymmetry must also be consistent with the equations of motion. I discuss Lorentz-invariant solutions of the theories of Scherk and Schwarz, pointing out that since the ansatz of Scherk and Schwarz consistently truncates supersymmetry, any solution of these theories is also a solution of the N = 1 supergravity theory in eleven dimensions and, hence, in particular that there is a Freund-Rubin-type ansatz for these theories. However I demonstrate that for most gauge groups the ansatz must be trivial which implies that for these theories the cosmological constant of any Lorentz-invariant solution must be zero (classically). Finally, I make some comparisons with work by Manton on dimensional reduction.