New String Amplitudes From Old Fermat (Hyper)surfaces

Abstract

The history of the discovery of bosonic string theory is well documented. This 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 this multidimensional beta function that was developed subsequently is markedly different from that described in physics literature. This work aims to bridge the gap between the mathematical and physical treatments. Using some results of recent publications (e.g. J.Geom.Phys.38 (2001) 81 ; ibid 43 (2002) 45) new topological, algebro-geometric, number-theoretic and combinatorial treatment of the multiparticle Veneziano amplitudes is developed. As a result, an entirely new physical meaning of these amplitudes is emerging: they are periods of differential forms associated with homology cycles on Fermat (hyper)surfaces. Such (hyper)surfaces are considered as complex projective varieties of Hodge type. Although the computational formalism developed in this work resembles that used in mirror symmetry calculations, many additional results from mathematics are used along with their suitable physical interpretation. For instance, the Hodge spectrum of the Fermat (hyper)surfaces is in one-to-one correspondence with the possible spectrum of particle masses. The formalism also allows us to obtain correlation functions of both conformal field theory and particle physics using the same type of the Picard-Fuchs equations whose solutions are being interpreted in terms of periods.MSC: 81T30,81T40,81T99,81U99

New strings for old Veneziano amplitudes

“…In our earlier work, Ref. [10], we have indeed obtained such type of equations for the Veneziano-type integral, Eq. (3.5), using ideas from singularity theory.…”

“…The number-theoretic aspects of the Veneziano amplitudes discussed in our earlier publication, Ref. [10], provide a natural link between dynamics, Rieamnn's zeta function and, more general, L-functions. For the sake of space, we refer our readers to just mentioned literature containing, in addition, a large number of relevant references of major importance.…”

“…For readers unfamiliar with number theory a concise summary of relevant results can be found, for instance, in the Appendix A of our earlier work, Ref. [10]. Given this, let q be some prime and consider the finite field F q of q elements.…”

“…Incidentally, in his paper he studies the number of solutions in the field F q for the Fermat hypersurface F a 0 zN 0 + · · · + a n+1 zN n+1 = 0 (1. 10) living in the complex projective space CP n+1 . Such a hypersurface was discussed in Part I (e.g.…”

“…< s λ , s µ >= δ λ,µ 21 . To make a connection with previous discussion, let us consider the following generating function 10) where notations introduced in Section 1 were used. 22 Such an expansion can be considered as some kind of a basis expansion in which the basis vectors belong to the set z λ .…”