It has been shown recently that the toroidally compactified type IIB string effective action possesses an SL (2, R) invariance as a consequence of the corresponding symmetry in ten dimensions when the self-dual five-form field strength is set to zero. By working in the string frame we clarify how a Z2 subgroup of this SL (2, R) group responsible for producing the strong–weak coupling duality in the ten-dimensional theory produces the same symmetry for the reduced theory. In the absence of the full covariant action of type IIB supergravity theory, we show that the T-dual version of type IIA string effective action (including the R–R terms) in D = 9 also possesses the SL (2, R) invariance, indicating that this symmetry is present for the full type IIB string effective action compactified on the torus. By making use of this symmetry we briefly indicate how, from the known classical solutions of string theory, an SL (2, Z) multiplet of solutions can be constructed. This, in turn, provides strong evidence in favor of the conjecture that SL (2, Z) is an exact symmetry group of the lower-dimensional quantum type II string theory.
Exact classical solutions to the two-dimensional "sigma" model; an unusual application of inverse scattering techniques