Regularization of p-adic String Amplitudes, and Multivariate Local Zeta Functions

TL;DR

This paper proves p-adic string amplitudes are well-defined integrals, introduces Igusa-type integrals for regularization, and explores their meromorphic continuations and connections with local zeta functions, providing a new approach to p-adic string theory.

Contribution

It establishes the integrality of p-adic string amplitudes and links them to multivariate local zeta functions, offering a novel regularization method.

Findings

p-adic Koba-Nielsen amplitudes are genuine integrals

Igusa-type integrals admit meromorphic continuations

Connections between divergencies and local zeta functions

Abstract

We prove that the p-adic Koba-Nielsen type string amplitudes are bona fide integrals. We attach to these amplitudes Igusa-type integrals depending on several complex parameters and show that these integrals admit meromorphic continuations as rational functions. Then we use these functions to regularize the Koba-Nielsen amplitudes. As far as we know, there is no a similar result for the Archimedean Koba-Nielsen amplitudes. We also discuss the existence of divergencies and the connections with multivariate Igusa's local zeta functions.