Abstraer. The complex numbers are naturally related to rotations and dilatations in the plane. In this paper we present the function theory associate to the (universM) Clifford algebra Ÿ 1'~ [1], the so called hyperbolic numbers [2,3,4], which can be related to Lorentz transformations and dilatations in the two dimensional Minkowski space-time. After some brief algebraic interpretations (part 1), we present a "Hyperbolic Calcu[us" analogous to the "Calculus of one Complex Variable". The hyperbolic Cauchy-Riemann conditions, hyperbolic derivatives and hyperbolic integrals are introduced on parts 2 and 3. Then special emphasis is given in parts 4 and 5 to conformal hyperbolic transformations which preserve the wave equation, and hyperbo]ic Riemann surfaces which ate naturally associated to classical string motions.

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In substitution for Dirac monopoles with string (and for topological monopoles), "monopoles without string" have recently been introduced on the basis of a generalized potential, the sum of a vector A, and a pseudovector rs B potential. By making recourse to the Clifford bundle '6'(rM,g) [(TxM,g) = R I • 3 ; '6'(T x M,g) = RI.3]' which just allows adding together for each xEM tensors of different ranks, in a previous paper a Lagrangian and Hamiltonian formalism was constructed for interacting monopoles and charges that can be regarded as satisfactory from various points of view. In the present article, after having "completed" the formalism, a purely geometrical interpretation of it is put forth within the Kahler-Clifford bundle %(r* M,g) of differential forms, essential ingredients being a generalized curvature and the Hodge decomposition theorem. Thus the way is paved for the extension of our "monopoles without string" to non-Abelian gauge groups. The analogy with supersymmetric theories is apparent.

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