Local Zeta Functions and Koba-Nielsen String Amplitudes

This article is a survey of our recent work on the connections between Koba-Nielsen amplitudes and local zeta functions (in the sense of Gel'fand, Weil, Igusa, Sato, Bernstein, Denef, Loeser, etc.). Our research program is motivated by the fact that the p-adic strings seem to be related in some interesting ways with ordinary strings. For instance, connections through the adelic relations and through the limit when p tends to 1. Gerasimov and Shatashvili studied the limit p tends to 1 of the p-adic effective action introduced by Brekke, Freund, Olson and Witten. They showed that this limit gives rise to a boundary string field theory, which was previously proposed by Witten in the context of background independent string theory. Explicit computations in the cases of 4 and 5 points show that the Feynman amplitudes at the tree level of the Gerasimov-Shatashvili Lagrangian are related with the limit p tends to 1 of the p-adic Koba-Nielsen amplitudes. At a mathematical level, this phenomenon is deeply connected with the topological zeta functions introduced by Denef and Loeser. A Koba-Nielsen amplitude is just a new type of local zeta function, which can be studied by using embedded resolution of singularities. In this way, one shows the existence of a meromorphic continuations for the Koba-Nielsen amplitudes as functions of the kinematic parameters. The Koba-Nielsen local zeta functions are algebraic-geometric integrals that can be defined over arbitrary local fields (for instance R, C, Q_{p}, F_{p}((T)), and it is completely natural to expect connections between these objects. The limit p tends to one of the Koba-Nielsen amplitudes give rise to a new amplitudes which we have called Denef-Loeser amplitudes. Along the article, we have emphasized the explicit calculations in the cases of 4 and 5 points.