N=(4,2) chiral supergravity in six dimensions and solvable Lie algebras

Decomposition of the solvable Lie algebras of maximal supergravities in D=4, 5 and 6 indicates, at least at the geometrical level, the existence of an N=(4,2) chiral supergravity theory in D=6 dimensions. This theory, with 24 supercharges, reduces to the known N=6 supergravity after a toroidal compactification to D=5 and D=4. Evidence for this theory was given long ago by B. Julia. We show that this theory suffers from a gravitational anomaly equal to 4/7 of the pure N=(4,0) supergravity anomaly. However, unlike the latter, the absence of N=(4,2) matter to cancel the anomaly presumably makes this theory inconsistent. We discuss the obstruction in defining this theory in D=6, starting from an N=6 five-dimensional string model in the decompactification limit. The set of massless states necessary for the anomaly cancellation appears in this limit; as a result the N=(4,2) supergravity in D=6 is extended to N=(4,4) maximal supergravity theory.