By Janssen B.
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3 General Picture In the previous sections we have encountered two kinds of duality transformations: the T -duality which relates di erent string compacti cations with each other and the S - duality that maps the strong coupling limit of a theory to the weak coupling limit of another or, in the case of Type IIB and D = 4; N = 4 Heterotic theory the same theory. These duality symmetries shed new light on the problems that arose in string theory up to the beginning of the nineteen nineties: The wide variety of possible compacti cation manifolds and the di erent degenerate string vacua that follow from them.
We will choose, for later convenience, the z -directions in the case of the wave to be a world volume direction, but in the case of the Kaluza-Klein monopole a transverse direction. 1 gives an overview of the di erent solutions we encountered in the various theories. In Chapter 3 we will see that these theories are related to each other via duality transformations. This means that there also must exist duality relations between the di erent solutions and the world volume actions that describe their dynamics.
46 , is a map from Type IIA to Type IIB and vice versa. In a generalization to reduction over d dimensions, the T -duality group is O d; d; Z and the moduli parametrise the coset O d; d = O d O d . 47 between Solutions In the previous subsection we have seen that some of the string theories may be connected via T -duality, at least at the level of the string e ective action. This implies that also T -duality transformations should exist between the solutions of these actions. 4 we intrinsically made use of the fact we were doing a duality transformation on a string-like solution: only on a two-dimensional world volume can a scalar X be dualized to another scalar X~ .