  The (4,0) supermultiplet in 6 dimensions contains a 4th 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 5D N=8
supergravity multiplet, with the 4th 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 2-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 5D
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 5D 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
6D 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.
