Gravity, duality and conformal symmetry

The (4,0) supermultiplet in six dimensions contains a fourth-rank tensor gauge field with the symmetries of the Riemann tensor and is superconformal, with 32+32 supersymmetries. Dimensional reduction on a circle gives the five dimensions N=8 supergravity multiplet, with the fourth-rank tensor reducing to the graviton. If there is an interacting (4,0) theory, it should reduce to the full N=8 supergravity theory and so would give a conformal theory of gravity that would reduce to conventional gravity with the usual two-derivative action at low energies. This paper revisits the conjecture that a non-lagrangian interacting (4,0) superconformal theory arises from a strong coupling limit of five-dimensional supergravity (suitably embedded in M-theory) describing M-theory at energies beyond the Planck scale. A key test for this is identified: M-theory toroidally compactified to five dimensions should have certain BPS states carrying a singlet central charge. These 1/2 BPS states are not related to any of the standard BPS states by dualities and do not correspond to non-singular soliton solutions-they appear to correspond to singular solutions related to gravitational instantons. Such states are needed to provide the Kaluza-Klein modes for the compactified six-dimensional theory. If there are no such states, then the conjecture is false, while the presence of such states would be strong indication that the conjecture could be true.