arXiv subjects

M. Bocardo-Gaspar

Publications and source records attributed to M. Bocardo-Gaspar.

4 recordsLinked to original sources

arXiv2021

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.

hep-th↗

arXiv2020

Meromorphic Continuation of Koba-Nielsen String Amplitudes

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.

math-ph↗

arXiv2020

Poles of Non-Archimedean Zeta Functions for Non-degenerate Rational Functions

In this article, we study local zeta functions over non-Archimedean locals fields of arbitrary characteristic attached to rational functions and characters $χ$ of the units of the ring of integers $\mathcal{O}_{K}$, by using an approach based on the multivariate $π$-adic stationary phase formula and Newton polyhedra. When the rational function is non-degenerate with respect to its Newton polyhedron, we give an explicit formula for the local zeta function and a list of the possible poles in terms of the normal vectors of the supporting hyperplanes of the Newton polyhedron attached to the rational function and their expected multiplicities. Furthermore, we obtain some conditions under which the local zeta function attached to the trivial character has at least one real pole by describing the largest negative real pole and the smallest positive one.

math.NT↗

arXiv2018

On $p$-adic string amplitudes in the limit $p$ approaches to one

In this article we discuss the limit $p$ approaches to one of tree-level $p$-adic open string amplitudes and its connections with the topological zeta functions. There is empirical evidence that $p$-adic strings are related to the ordinary strings in the $p \to 1$ limit. Previously, we established that $p$-adic Koba-Nielsen string amplitudes are finite sums of multivariate Igusa's local zeta functions, consequently, they are convergent integrals that admit meromorphic continuations as rational functions. The meromorphic continuation of local zeta functions has been used for several authors to regularize parametric Feynman amplitudes in field and string theories. Denef and Loeser established that the limit $p \to 1$ of a Igusa's local zeta function gives rise to an object called topological zeta function. By using Denef-Loeser's theory of topological zeta functions, we show that limit $p \to 1$ of tree-level $p$-adic string amplitudes give rise to certain amplitudes, that we have named Denef-Loeser string amplitudes. Gerasimov and Shatashvili showed that in limit $p \to 1$ the well-known non-local effective Lagrangian (reproducing the tree-level $p$-adic string amplitudes) gives rise to a simple Lagrangian with a logarithmic potential. We show that the Feynman amplitudes of this last Lagrangian are precisely the amplitudes introduced here. Finally, the amplitudes for four and five points are computed explicitly.

hep-th↗