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Invariants and divergences in half-maximal supergravity theories
The invariants in half-maximal supergravity theories in D=4,5 are discussed
in detail up to dimension eight (e.g. R^4). In D=4, owing to the anomaly in the
rigid SL(2,R) duality symmetry, the restrictions on divergences need careful
treatment. In pure N=4 supergravity, this anomalous symmetry still implies
duality invariance of candidate counterterms at three loops. Provided one makes
the additional assumption that there exists a full 16-supercharge off-shell
formulation of the theory, counterterms at L>1 loops would also have to be
writable as full-superspace integrals. At the three-loop order such a
duality-invariant full-superspace integral candidate counterterm exists, but
its duality invariance is marginal in the sense that the full-superspace
counter-Lagrangian is not itself duality-invariant. We show that such marginal
invariants are not allowable as counterterms in a 16-supercharge off-shell
formalism. It is not possible to draw the same conclusion when vector
multiplets are present because of the appearance of F^4 terms in the SL(2,R)
anomaly. In D=5 there is no one-loop anomaly in the shift invariance of the
dilaton, and we argue that this implies finiteness at two loops, again subject
to the assumption that 16 supercharges can be preserved off-shell.Comment: 81 page
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