We investigate possible existence of duality symmetries which exchange the Kaluza–Klein modes with the wrapping modes of a BPS saturated p-brane on a torus. Assuming the validity of the conjectured U-duality symmetries of type-II and heterotic string theories and M-theory, we show that for a BPS saturated p-brane there is an SL (2, Z) symmetry that mixes the Kaluza–Klein modes on a (p+1)-dimensional torus T(p+1) with the wrapping modes of the p-brane on T(p+1). The field that transforms as a modular parameter under this SL (2, Z) transformation has as its real part the component of the (p+1)-form gauge field on T(p+1), and as its imaginary part the volume of T(p+1), measured in the metric that couples naturally to the p-brane.
T-DUALITY OF p-BRANES