# Meromorphic Continuation of Koba-Nielsen String Amplitudes

## TL;DR

This paper rigorously proves that Koba-Nielsen string amplitudes over local fields are valid integrals with meromorphic continuations, connecting string theory amplitudes with local zeta functions and statistical mechanics.

## Contribution

It provides a rigorous mathematical framework for the meromorphic continuation of Koba-Nielsen amplitudes over arbitrary local fields of characteristic zero, unifying open and closed string cases.

## Key findings

- Koba-Nielsen amplitudes are proven to be genuine integrals with meromorphic continuations.
- Regularization of p-adic open string amplitudes with additional factors is achieved.
- Local zeta functions related to these amplitudes are shown to be partition functions of 1D log-Coulomb gases.

## Abstract

In this article, we establish in a rigorous mathematical way that Koba-Nielsen amplitudes defined on any local field of characteristic zero are bona fide integrals that admit meromorphic continuations in the kinematic parameters. Our approach allows us to study in a uniform way open and closed Koba-Nielsen amplitudes over arbitrary local fields of characteristic zero. In the regularization process we use techniques of local zeta functions and embedded resolution of singularities. As an application we present the regularization of p-adic open string amplitudes with Chan-Paton factors and constant B-field. Finally, all the local zeta functions studied here are partition functions of certain 1D log-Coulomb gases, which shows an interesting connection between Koba-Nielsen amplitudes and statistical mechanics.