NEW STRING AMPLITUDES FROM OLD FERMAT (HYPER)SURFACES

Abstract

The bosonic string theory evolved as an attempt to find a multidimensional analogue of Euler's beta function to describe the multiparticle Veneziano amplitudes. Such an analogue had in fact been known in mathematics at least in 1922. Its mathematical meaning was studied subsequently from different angles by mathematicians such as Selberg, Weil and Deligne among others. The mathematical interpretation of multidimensional beta function that was developed subsequently is markedly different from that described in physics literature. This paper aims to bridge the gap between the mathematical and physical treatments of such beta functions thus providing new topological, algebro-geometric, number-theoretic and combinatorial treatments of the multiparticle Veneziano and Veneziano-like amplitudes. As a result, the entirely new physical meaning of these amplitudes is emerging: they are periods connected with differential forms living on Fermat (hyper)surfaces. Such surfaces are considered as complex projective varieties of Hodge type. Obtained results allow to interpret the particle mass spectrum in terms of the Hodge spectrum of underlying Fermat (hyper)surface. The computational formalism although resembles that used in mirror symmetry calculations employs many additional results from topology, number theory, theory of singularities, etc. It allows to obtain the correlation functions in particle physics and conformal field theories using the same type of the Picard–Fuchs equations.