This article is related to construction of zeta strings from $p$-adic ones. In addition to investigation of $p$-adic string for a particular prime number $p$, it is also interesting to study collective effects taking into account all primes $p$. An idea behind this approach is that a zeta string is a whole thing with infinitely many faces which we see as $p$-adic strings. The name zeta string has origin in the Riemann zeta function contained in related Lagrangian. The starting point in construction a zeta string is Lagrangian for a $p$-adic open string. There are two types of approaches to get a Lagrangian for zeta string from Lagrangian for $p$-adic strings: additive and multiplicative approaches, that are related to two forms of the definition of the Riemann zeta function. As a result of differences in approaches, one obtains several different Lagrangians for zeta strings. We briefly discuss some properties of these Lagrangians, related potentials, equations of motion, mass spectrum and possible connection with ordinary strings. This is a review of published papers with some new views.
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 …
After a brief review of the idea and main results of the original p-adic string work, I describe the recent interest in p-adic strings in the context of AdS/CFT duality
Solutions of Calabi-Yau Differential Operators as Truncated p-adic Series and Efficient Computation of Zeta Functions
University of Sheffield（谢菲尔德大学）；PRISMA+ Cluster of Excellence & Mainz Institute for Theoretical Physics Johannes Gutenberg-Universität Mainz（PRISMA卓越集群与美因茨理论物理研究所 约翰内斯·古腾堡美因茨大学）；Okinawa Institute of Science and Technology（冲绳科学技术大学院大学）；Cornell University（康奈尔大学）
We construct a $p$-adic analog to AdS/CFT, where an unramified extension of the $p$-adic numbers replaces Euclidean space as the boundary and a version of the Bruhat-Tits tree replaces the bulk. Correlation functions are computed in the simple case of a single massive scalar in the bulk, with results that are strikingly similar to ordinary holographic correlation functions when expressed in terms of local zeta functions. We give some brief discussion of the geometry of $p$-adic chordal distance and of Wilson loops. Our presentation includes an introduction to $p$-adic numbers.
We consider a generalization of elliptic multiple zeta values, which we call twisted elliptic multiple zeta values. These arise as iterated integrals on an elliptic curve from which a rational lattice has been removed. At the cusp, twisted elliptic multiple zeta values are shown to degenerate to cyclotomic multiple zeta values in the same way as elliptic multiple zeta values degenerate to classical multiple zeta values. We investigate properties of twisted elliptic multiple zeta values and consider them in the context of the non-planar part of the four-point one-loop open-string amplitude.
We relate one-loop scattering amplitudes of massless open- and closed-string states at the level of their low-energy expansion. The modular graph functions resulting from integration over closed-string punctures are observed to follow from symmetrized open-string integrals through a tentative generalization of the single-valued projection known from genus zero.
For an arbitrary prime number $p$, we propose an action for bosonic $p$-adic strings in curved target spacetime, and show that the vacuum Einstein equations of the target are a consequence of worldsheet scaling symmetry of the quantum $p$-adic strings, similar to the ordinary bosonic strings case. It turns out that certain $p$-adic automorphic forms are the plane wave modes of the bosonic fields on $p$-adic strings, and that the regularized normalization of these modes on the $p$-adic worldsheet presents peculiar features which reduce part of the computations to familiar setups in quantum fiel…
We study in a rigorous mathematical way p-adic quantum field theories whose N-point amplitudes are the expectation of products of vertex operators. We show that this type of amplitudes admit a series expansion where each term is an Igusa's local zeta function. The lowest term in this series is a regularized version of the p-adic open Koba-Nielsen string amplitude.
We establish rigorously the regularization of the p-adic open string amplitudes, with Chan-Paton rules and a constant B-field, introduced by Ghoshal and Kawano. In this study we use techniques of multivariate local zeta functions depending on multiplicative characters and a phase factor which involves an antisymmetric bilinear form. These local zeta functions are new mathematical objects. We attach to each amplitude a multivariate local zeta function depending on the kinematics parameters, the B-field and the Chan-Paton factors. We show that these integrals admit meromorphic continuations in t…
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 give…
In this paper we show that in perturbative string theory the genus-one contribution to formal 2-point amplitudes can be related to the genus-zero contribution to 4-point amplitudes. This is achieved by studying special linear combinations of multiple zeta values that appear as coefficients of the amplitudes. We also exploit our results to relate closed strings to open strings at genus one using Brown's single-valued projection, proving a conjecture of Broedel, Schlotterer and the second author.
Nonlinear equations for p-adic open, closed, and open-closed strings
我们研究了一维非线性伪微分方程组边值问题的解的结构，该方程组描述了标量快子场的p-adic开弦、闭弦和开闭弦的动力学（滚动）。对于开闭弦，我们证明了该方法收敛的形式p= 4 n +1的奇值p的条件下，封闭字符串的解决方案是已知的。当p=2时，讨论了边值问题解的存在性和不存在性，并指出了出现间断解的可能性.
