By Janssen B.

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A c + d ; Fi ! 62 these two equations get mapped one into another. The equations of motion of all other elds are left invariant. If we consider the particular SL2; R transformation where the group parameters have the values a = d = 0 and b = ,c = 1, we nd the transformation  ! ,1 ; F i ! 64  1   51 2 j  For 1 = 0 this corresponds to a strong weak coupling symmetry, where the electric i get interchanged with the magnetic ones of  F i , together with an inversion elds of F  of the string coupling constant e !

If the elds of the uncompacti ed theory depend on the compact coordinates, then extra massive states appear in the lower-dimensional theory. 12 Since the eld ^ ^x^  depends on all coordinates x^^ , in particular also on the compact coordinate x, we can perform a separation of variables and do a Fourier expansion of eld in the x-coordinate. 13 n where R is the radius of the compact dimension. The equation of motion for the 2 It turns out that only the compacti cations that preserve some amount of supersymmetry are consistent.

Under this symmetry the strong coupling regime of a theory gets mapped to the weak coupling regime of another theory and vice versa. 2: Duality relations between the various string theories in ten dimensions and M -theory in eleven dimensions: the arrows indicate dimensional reduction from D to D , 1 dimensions, the dotted lines represent an S -duality and a straight lines T -duality. Strong and weak coupling regimes therefore turn out to be di erent, but equivalent formulations of the same underlying theory.

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