An SL(2,Bbb Z) anomaly in IIB supergravity and its F-theory interpretation

The SL(2,Z) duality transformations of type IIB supergravity are shown to beanomalous in generic F-theory backgrounds due to the anomalous transformationof the phase of the chiral fermion determinant. The anomaly is partiallycancelled provided the ten-dimensional type IIB theory lagrangian contains aterm that is a ten-form made out of the composite U(1) field strength and fourpowers of the curvature. A residual anomaly remains uncancelled, and thisimplies a certain topological restriction on consistent backgrounds of theeuclidean theory. A similar, but slightly stronger, restriction is also derivedfrom an explicit F-theory compactification on K3 x M8 (where M8 is aneight-manifold with a nowhere vanishing chiral spinor) where the cancellationof tadpoles for Ramond--Ramond fields is only possible if M8 has an Eulercharacter that is a positive multiple of 24. The interpretation of thisrestriction in the dual heterotic theory on T2 x M8 is also given.Comment: Argument has been streamlined and references have been added. 18 pages, harvmac (b