Multidimensional analogues of the geometric $s\leftrightarrow t$ duality

Customarily, the $s\leftrightarrow t$ duality property for scattering amplitudes, e.g., for the Veneziano amplitude, is naturally related to two-dimensional geometry. Saito and the author previously proposed a simple geometric construction of such amplitudes. Here, we construct analogues of one such amplitude related to multidimensional Euclidean spaces; the three-dimensional case is discussed in detail. The result is a variant of the Regge calculus closely related to integrable models.