Local Zeta Functions and Koba–Nielsen String Amplitudes

S S symmetry

Review Local Zeta Functions and Koba–Nielsen String Amplitudes

Miriam Bocardo-Gaspar 1, Hugo García-Compeán 2 , Edgar Y. López 2 and Wilson A. Zúñiga-Galindo 3,4,*

1 Departamento de Matemáticas, Universidad de Guadalajara, Cucei, Blvd. Marcelino García Barragán #1421, 44430 Guadalajara, Mexico; [email protected] 2 Centro de Investigación y de Estudios Avanzados del I.P.N., Departamento de Física, Av. Instituto Politécnico Nacional 2508, Col. San Pedro Zacatenco, 07360 Ciudad de México, Mexico; compean@ﬁs.cinvestav.mx (H.G.-C.); elopez@ﬁs.cinvestav.mx (E.Y.L.) 3 University of Texas Rio Grande Valley, School of Mathematical & Statistical Sciences, One West University Blvd, Brownsville, TX 78520, USA 4 Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional Departamento de Matemáticas, Unidad Querétaro Libramiento Norponiente #2000, Fracc, Real de Juriquilla, 76230 Santiago de Querétaro, Mexico * Correspondence: [email protected] or [email protected]

Abstract: 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. p-Adic string amplitudes share desired characteristics with their Archimedean counterparts, such as crossing symmetry and invariance under Möbius transformations. A direct connection between p-adic amplitudes and the Archimedean ones is through the limit p → 1. Gerasimov and Shatashvili studied the limit p → 1 of the p-adic effective action introduced by Brekke, Freund, Olson and Witten. They showed that this limit gives Citation: Bocardo-Gaspar, M.; rise to a boundary string field theory, which was previously proposed by Witten in the context of García-Compeán, H.; López, E.; background independent string theory. Explicit computations in the cases of 4 and 5 points show Zúñiga-Galindo, W.A. Local Zeta that the Feynman amplitudes at the tree level of the Gerasimov–Shatashvili Lagrangian are related to Functions and Koba–Nielsen String → Amplitudes. Symmetry 2021, 13, 967. the limit p 1 of the p-adic Koba–Nielsen amplitudes. At a mathematical level, this phenomenon https://doi.org/10.3390/ is deeply connected with the topological zeta functions introduced by Denef and Loeser. A Koba– sym13060967 Nielsen amplitude is just a new type of local zeta function, which can be studied using embedded resolution of singularities. In this way, one shows the existence of a meromorphic continuations for Academic Editor: Vladimir Al. the Koba–Nielsen amplitudes as functions of the kinematic parameters. The Koba–Nielsen local zeta Osipov functions are algebraic-geometric integrals that can be defined over arbitrary local fields (for instance R, C, Qp, Fp((T))), and it is completely natural to expect connections between these objects. The limit Received: 1 May 2021 p tends to one of the Koba–Nielsen amplitudes give rise to new amplitudes which we have called Accepted: 24 May 2021 Denef–Loeser amplitudes. Throughout the article, we have emphasized the explicit calculations in Published: 29 May 2021 the cases of 4 and 5 points.

Publisher’s Note: MDPI stays neutral Keywords: string amplitudes; Koba–Nielsen amplitudes; Virasoro–Shapiro amplitudes; regulariza- with regard to jurisdictional claims in tion; p-adic string amplitudes; local zeta functions; resolution of singularities published maps and institutional afﬁl- iations.

1. Introduction In the recent years, the connections between string amplitudes and arithmetic geome- Copyright: © 2021 by the authors. p Licensee MDPI, Basel, Switzerland. try, -adic analysis, combinatorics, etc. have been studied extensively, see e.g., [1–9] and This article is an open access article the references therein. distributed under the terms and The string amplitudes were introduced by Veneziano in the 60s, In [10], further conditions of the Creative Commons generalizations were obtained by Virasoro [11], Koba and Nielsen [12], among others. Attribution (CC BY) license (https:// In the 80s, Freund, Witten and Volovich, among others, studied string amplitudes at the creativecommons.org/licenses/by/ tree level over different number ﬁelds, and suggested the existence of connections between 4.0/). these amplitudes, see e.g., [4,13,14]. In this framework the connection with local zeta

Symmetry 2021, 13, 967. https://doi.org/10.3390/sym13060967 https://www.mdpi.com/journal/symmetry Symmetry 2021, 13, 967 2 of 53

functions appears naturally. This article is intended as a survey of our recent work on the connections between Koba–Nielsen amplitudes and local zeta functions [6–9] in the sense of Gel’fand, Weil, Igusa, Sato, Bernstein, Denef, Loeser, etc. The p-adic string theories have been studied over the time with some periodic fluctu- ations in their interest (for some reviews, see [3–5,15]). Recently aconsiderable amount of work has been performed on this topic in the context of the AdS/CFT correspondence [16–19]. String theory with a p-adic worldsheet was proposed and studied for the first time in [20]. Later this theory was formally known as p-adic string theory. The adelic scattering amplitudes which are related to the Archimedean ones were studied in [21]. The tree-level string amplitudes were explicitly computed in the case of p-adic string worldsheet in [14,22]. One can obtain these amplitude, in a formal way, from a suitable action using general principles [23]. In [24], it was established that the tree-level string amplitudes may be obtained starting with a discrete field theory on a Bruhat–Tits tree. Determining the convergence of the amplitudes in momentum space is a difficult task, both in the standard and p-adic case; however, for the latter, this was precisely done for the N-point tree amplitudes in [8]. In this article we show (in a rigorous mathematical way) that the p-adic open string N-point tree amplitudes are bona fide integrals that admit meromorphic continuations as rational functions, this is done by associating to them multivariate local zeta functions (also called multivariate Igusa local zeta functions) [25–28]. In [6] we establish in a rigorous mathematical way that Koba–Nielsen amplitudes defined on any local field of characteristic zero (for instance R, C, Qp) are bona fide integrals that admit meromorphic continuations in the kinematic parameters. In the regularization process we use techniques of local zeta functions and embedded resolution of singularities. The p-adic strings are related to ordinary strings at least in two different ways. First, connections through the adelic relations [21] and second, through the limit when p → 1 [23,29]. In [29] was showed that the limit p → 1 of the effective action gives rise to a boundary string field theory, this was previously proposed by Witten in the context of background independent string theory [30,31]. The limit p → 1 in the effective theory can be performed without any problem, since one can consider p as a real parameter and take formally the limit p → 1. The resulting theory is related to a field theory describing an open string tachyon [32]. In the limit p → 1 also there are exact noncommutative solitons, some of these solutions were found in [33]. In [34] a very interesting physical interpretation of this limit was given in terms of a lattice discretization of ordinary string worldsheet. In the worldsheet theory we cannot forget the nature of p as a prime number, thus the analysis of the limit is more subtle. The correct way of taking the limit p → 1 involves the introduction of finite extensions of the p-adic field Qp. The totally ramified extensions give rise to a finer discretization of the worldsheet following the rules of the renormalization group [34]. In [35] Denef and Loeser showed that the limit p approaches to one of a local zeta function give rise a new object called a topological zeta function. By using Denef–Loeser’s theory of topological zeta functions, in [9], we show that the limit p → 1 of tree-level p-adic string amplitudes give rise to new amplitudes, that we have named Denef–Loeser string amplitudes. Gerasimov and Shatashvili showed that in the limit p → 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 [29]. In [9], we conjecture that the Feynman amplitudes of this last Lagrangian are related to the Denef–Loser amplitudes. In [7] we establish rigorously the regularization of the p-adic open string amplitudes, with Chan–Paton rules and a constant B-ﬁeld. These amplitudes were originally introduced by Ghoshal and Kawano. To study these amplitudes, we introduce new multivariate local zeta functions depending on multiplicative characters (Chan–Paton factors) and a phase factor (the B-ﬁeld) which involves an antisymmetric bilinear form. We show that these integrals admit meromorphic continuations in the kinematic parameters, this result allows us to regularize the Ghoshal–Kawano amplitudes, the regularized amplitudes do not have ultraviolet divergencies. The theory is only valid for p congruent to 3 mod 4, this is to preserve a certain symmetry. The limit p → 1 is also considered for the noncommutative Symmetry 2021, 13, 967 3 of 53

effective field theory and the Ghoshal–Kawano amplitudes. We show that in the case of four points, the limit p → 1 of the regularized Ghoshal–Kawano amplitudes coincides with the Feynman amplitudes attached to the limit p → 1 of the noncommutative Gerasimov– Shatashvili Lagrangian. We denote by K a local field of characteristic zero (for instance R, C, Qp), and set m f := ( f1,..., fm) and s := (s1,..., sm) ∈ C , where the fi(x) are non-constant polynomials in the variables x := (x1, ... , xn) with coefficients in K. The multivariate local zeta function attached to ( f, Θ), where Θ is a test function, is defined as

Z m n Z ( f, s) = Θ(x) | f (x)|si dx , when Re(s ) > 0 for all i, Θ ∏ i K∏ i i i=1 i=1 Kn n n and where ∏i=1dxi is the normalized Haar measure of (K , +). These integrals admit meromorphic continuations to the whole Cm,[25,26,36], see also [37,38]. The study of Archimedean and non-Archimedean local zeta functions was started by Weil in the 60 s, in connection to the Poisson-Siegel formula. In the 70 s, Igusa developed a uniform theory for local zeta functions in characteristic zero [25,26], see also [36,39,40]. In the p-adic setting, the local zeta functions are connected with the number of solutions of polynomial congruences mod pl and with exponential sums mod pl [28]. Recently, Denef and Loeser introduced the motivic zeta functions which constitute a vast generalization of p-adic local zeta functions [41,42]. In the 50 s, Gel’fand and Shilov studied the local zeta functions attached to real polynomials in connection with fundamental solutions, indeed, the meromorphic continuation of Archimedean local zeta functions implies the existence of fundamental solutions (i.e., Green functions) for differential operators with constant coefﬁcients. This fact was established, independently, by Atiyah [43] and Bernstein [44]. It is relevant to mention here that in the p-adic framework, the existence of fundamental solutions for pseudodifferential operators is also a consequence of the fact that the Igusa local zeta functions admit a meromorphic continuation, see ([45], Chapter 5), ([46], Chapter 10). This analogy turns out to be very important in the rigorous construction of quantum scalar ﬁelds in the p-adic setting, see [47] and the references therein. Take N ≥ 4, and complex variables s1j and s(N−1)j for 2 ≤ j ≤ N − 2 and sij for D N(N−3) 2 ≤ i < j ≤ N − 2. Put s := sij ∈ C , where D = 2 denotes the total number of indices ij. In [6] we introduce the multivariate local zeta function

Z N−2 N−2 (N) s1j s(N−1)j sij Z (s) := x 1 − x x − x dx , (1) K ∏ j K j K ∏ i j K ∏ i i=2 2≤i

N−2 N−3 where ∏i=2 dxi is the normalized Haar measure on K . We called these integrals Koba- Nielsen local zeta functions. In ([6], Theorems 4.1 and 6.1), we show that these functions are bona ﬁde integrals, which are holomorphic in an open part of CD, containing the set given by −2 −2 < Re(s ) < for all ij. (2) N − 2 ij N Furthermore, they admit meromorphic continuations to the whole CD. The Koba–Nielsen open string amplitudes for N-points over K are formally deﬁned as

Z N−2 N−2 (N) k1kj kN−1kj kikj A (k) := x 1 − x x − x dx , K ∏ j K j K ∏ i j K ∏ i i=2 2≤i

N ∑ ki = 0, kiki = 2 for i = 1, . . . , N. i=1 The parameter l is an arbitrary positive integer. Typically, l is taken to be 25 for bosonic strings. However, we do not require using the critical dimension. We choose units such that the tachyon mass is m2 = −2. (N) In the real case, A (k) is (up to multiplication by a positive constant) the open R ( ) Koba–Nielsen amplitude of N-points, see ([4], Section 8), ([48], Section 2). If N = 4, A 4 (k) R is the Veneziano amplitude [10]. In the complex case, by using the results of ([48], Section 2), (N) in [49], the N-point, closed string amplitude at the tree level is the product of A (k) times C a polynomial function in the momenta k. This fact implies that the techniques and results presented in [6] are applicable to classical closed string amplitudes at the tree level. (N) (N) We use the integrals Z (s) as regularizations of the amplitudes A (k). More K K precisely, we redefine (N) (N) A (k) = Z (s) | , (3) K K sij=kikj (N) where Z (s) now denotes the meromorphic continuation of (1) to the whole D, see ([6], K C (N) Theorem 6.1). Furthermore, A (k) extends to a meromorphic function to the whole K ( + ) CN l 1 , see ([6], Theorem 7.1). The articles [6,8] deal with the meromorphic continuation of local zeta functions of (N) type Z (s), but their results are complementary. The main result of [8] is the existence of K (N) a meromorphic continuation for Z (s) in the case in which is a non-Archimedean local K K field of arbitrary characteristic. For instance Qp or Fq((T)), the field of formal Laurent series with coefficients in a finite field Fq. The methods used are based on the stationary phase (N) formula, which allows the explicit calculation of Z (s). On the other hand, the results K presented in [6] depend completely on Hironaka’s resolution of singularities theorem [50], which is available only for fields of characteristic zero. In this framework, the results of [6] (N) show that for any field of characteristic zero , all the zeta functions Z (s) converge on a K K common domain, see (2), and that the possible poles can be described in a geometric form. We propose the following conjecture:

Conjecture 1. There is a meromorphic function Γ(s) in CD such that

(N) (N) Z (s) = Γ(s) Z (s) for any s ∈ D. R ∏ Qp C p<∞

(N) This conjecture and ([6], Theorem 6.1) imply that ∏ Z (s) admits a meromorphic p<∞ Qp continuation to the whole CD. By using (3), this conjecture implies that

(N) (N) A (s) = Γ(s) A (s) R ∏ Qp p<∞

is meromorphic function in CD. Freund and Witten established (formally) that the ordinary Veneziano and Virasoro–Shapiro four-particle scattering amplitudes can be factored in terms of an inﬁnite product of non-Archimedean string amplitudes [21]. More precisely, they established that the product of the p-adic string amplitudes multiplied by the crossing- symmetric Veneziano amplitude is equal to one. Later Aref’eva, Dragovich and Volovich noted that the inﬁnite product used by Freund and Witten is divergent, and established that the product of the p-adic string amplitudes multiplied by the crossing-symmetric Veneziano amplitude is equal to an expression containing the Riemann zeta function [51]. Symmetry 2021, 13, 967 5 of 53

(N) We already mentioned that the Denef–Loeser amplitudes Atop (s) are the limit of the (N) Koba–Nielsen amplitudes A (s) when p tends to one. This is a consequence of the fact Qp (N) (N) that Z (s) = lim → Z (s), which in turn is a consequence of the motivic integration top p 1 Qp theory, see e.g., [41,52]. Motivic integration is a generalization of the p-adic integration and the integration with respect to the Euler characteristic. By using this theory one can (N) construct a function Z (L) such that C((T)) (N) (N) (N) (N) Z (s) = Z (L) |L=p and Z (s) = Z (L) |L= , Qp C((T)) top C((T)) 1 which in turn implies the existence of new type of ‘motivic string amplitude’ at the three level (N) A (L) which is a generalization of the p-adic Koba–Nielsen and Denef–Loeser amplitudes. C((T))

( ) Problem 1. Does the motivic string amplitudes A N (L) have a physical meaning? Is there an C((T)) ( ) effective action such that the Feynman integrals at the three level are exactly the A N (L)? C((T))

In [9], we veriﬁed that the four and ﬁve-point Feynman amplitudes of the (N) Gerasimov-Shatashvili Lagrangian are related by lim → A (s). Then we propose the p 1 Qp following conjecture:

Conjecture 2. The Feynman amplitudes at the three level of the Gerasimov–Shatashvili Lagrangian (N) are by related lim → A (s). p 1 Qp

A precise formulation of this conjecture is given at the end of Section6. This survey is organized as follows. In Section2, we review briefly basic aspects of string amplitudes. Section3 aims to provide an introduction to p-adic analysis. Section4 is dedicated to the p-adic Koba–Nielsen amplitudes. We compute explicitly the amplitudes for 4 and 5 points. At the end of the section we provide an outline of the proof that the p-adic Koba–Nielsen amplitudes admit meromorphic continuations as rational functions −k k in p i j . The Koba–Nielsen amplitudes can be defined over any local field (i.e., R, C, Qp, Fp((T)), etc.). Section5 is dedicated to amplitudes defined on local fields such as Qp, Fp((T)). A central idea is that the limit p tends to one of the p-adic Koba–Nielsen amplitudes gives a new amplitude, the Denef–Loeser amplitude. We compute the Denef– Loeser amplitudes for 4 and 5 points. Section6 deals with the physical interpretation of the limit p → 1 of p-adic string amplitudes. In this section we show, in the case of 4 and 5 particles, that the Feynman integrals at the tree level of the Gerasimov–Shatashvili Lagrangian are related Denef–Loeser amplitudes. We also give a precise formulation of Conjecture2 . Section7 deals with the N-point p-adic string amplitudes, with Chan–Paton rules and a constant B-field. We emphasize the explicit calculations of these amplitudes in the cases N = 4, 5. In Section8, we briefly review some basic definitions and some central results about analytic manifolds on local fields of characteristic zero, Hironaka’s resolution of singularities theorem and multivariate Igusa zeta functions. In Section 10, we give an outline of the proof of the fact that Koba–Nielsen amplitudes defined on local fields of characteristic zero admit meromorphic continuations in kikj ∈ C. In the cases N = 4, 5, we show explicitly the existence of a meromorphic continuations for amplitudes by outlining the construction of certain embedded resolution of singularities.

2. Generalities of String Amplitudes String theory is the theory of tiny one-dimensional extended objects propagating in an underlying Riemannian or pseudo-Riemannian spacetime manifold M. The theory is deﬁned as a theory for a dynamical embedding map X : Σg,b,N → M, where Σg,b,N is a compact and oriented Riemann surface with a possible nonempty boundary (∂Σ 6= 0) Symmetry 2021, 13, 967 6 of 53

characterized by the genus g, their number of boundaries b and N marked points. Σg,b,N has local coordinates σ = (σ0, σ1) (σa with a = 0, 1). Under a Wick rotation σ1 → σ1 and σ2 = iσ0, the metric written in terms of coordinates (σ1, σ2) becomes a metric with Euclidean signature. In addition to X there is a non-dynamical filed hab on Σg,N, which is the intrinsic metric. The string may oscillate in the target space M and it has an infinite number of quantum modes of oscillation carrying a representation of the Poincaré group of the target space ( In the present review will be enough to consider M to be the flat Minkowskian spacetime) and characterized by a mass and a spin. In the bosonic string the spectrum consists of a tachyonic mode with negative square mass. There are a finite number of massless modes described by massless fields on M. These modes are the target space metric gµν with Lorentzian signature (−, +, ... , +), the antisymmetric field Bµν, and the dilaton field Φ, where µ, ν = 0, ... , D − 1. M is the target space which is a Riemannian manifold of 26 dimensions in the case of the bosonic string and 10 dimensions for the superstring [53–55]. The Euclidean worldsheet action for the bosonic open string is given by I = S + S0, (4) where Z √ T 2 ab µ ν 0 ab µ S = d σ h h ∂aX ∂bX gµν(X) − 2πiα Bµνe ∂aX ∂bXν 2 Σ Z √ Z (5) T 2 ab µ ν i µ ν = d σ hh ∂aX ∂bX gµν(X) − dlBµνX ∂tX , 2 Σ 2 ∂Σ = 1 0 = `2 ` where T 2πα0 is the string tension, here α s and s stands for the string length. In the µ above action the physical degrees of freedom are X and hab, ∂t is the tangential derivative 0 along ∂Σ and Bµν is an antisymmetric field. Moreover, the action S is given by

0 S = Φ0χ(Σ), √ 1 R 2 where χ(Σ) = 4π d σ hR + BT is the Euler characteristic of Σ and BT is the Gibbons- Hawking boundary term that we encode in χ(Σ) with the number of boundaries b. Notice that the vacuum expectation value of the dilaton Φ0 can also be incorporated in the action through this topological term. It is also worth noting that this is the most general action that has worldsheet reparametrization invariance, in particular Weyl invariance (hab → Λ(σ)hab), and is renormalizable by power counting (this implies that there must be two worldsheet derivatives in each term). The second term is analogous to placing an electromagnetic field in the background with which the string interacts; it is a gauge field in that sense. Mathematically it is the integration of the pullback of the B-field to the string ν worldsheet. If we define Aµ := BµνX , the second term of the second line in (5) may be written as Z i µ Aµ∂tX , (6) 2 ∂Σ which is exactly the term considered in [56] where the partition function of the action was obtained explicitly at tree level. Strings propagating in non-trivial backgrounds have been considered in other works [57]. We now consider flat space as well as a constant B-field. The open string action (5) determines the Neumann boundary conditions

ν 0 ν gµν∂σX + 2πiα Bµν∂tX |∂Σ = 0,

where ∂σ is the normal derivative to ∂Σ. We should note that these conditions are in general complex, because we have taken a Wick rotation to a Euclidean signature. We will be concerned about the case when the string worldsheet Σ is a disk, which corresponds to the open string in the classical limit. It is known that the disk can be transformed into the Symmetry 2021, 13, 967 7 of 53

upper half plane via a conformal transformation, whose boundary is the real line. In these variables the Neumann boundary conditions are [58]

ν 0 ν gµν(∂ − ∂¯)X + 2πα Bµν(∂ + ∂¯)X |z=z¯ = 0, where ∂ = ∂/∂z and ∂¯ = ∂/∂z¯, with Im z ≥ 0. The perturbative scattering amplitudes of N particles are deﬁned as the correlation V ( ) function of N vertex operators Λi ki . Furthermore, these amplitudes admit an expansion over the genus g of χ(Σ) = 2 − 2g − b as follows:

∞ 2g−2+b A(Λ1, k1,..., ΛN, kN) = ∑ λs Ag(Λ1, k1,..., ΛN, kN), g=0

Φ where λs := e 0 is the coupling constant of string and Ag is the Feynman functional integral

1 Z N ( k k ) = (− ) V (k ) Ag Λ1, 1,..., ΛN, N DXDh exp S ∏ Λi i , Vol Mg,b,N i=1 here Vol stands for the volume of the symmetry groups leaving invariant the action (4). Here the integration is carried out over Mg,b,N the moduli space of Riemann surfaces of genus g, b boundaries and N marked points inserted on the boundary. The difﬁculty of integration increases with higher values of g, b and N. Fortunately it can be carried out in an explicit form for a few simple cases with lower values of g, b and N. Thus, for instance, the case for closed strings with genus g = 0 and N = 4, or open strings with g = 0, = = V ( ) b 1 and N 4, are explicitly performed. Vertex operators Λi ki are functionals of the embedding ﬁelds X and their derivatives. They are given by Z √ 2 ikX(σ) VΛ(k) = d σ hWΛ(σ)e ,

where WΛ(σ) represents a functional of X and its derivatives associated with the species of ﬁeld in the string spectrum, X(σ) = (X0, ... , X25), k = (k0, ... , k25) are the position and momentum vectors in target space M and

25 µ ν kX(σ) := ∑ k ηµνX (σ) µ,ν=0

is the Minkowskian inner product. For instance for tachyons Λ = t, Wt(σ) = 1. For gauge µ fields Λ = A and WA(σ) = εµ∂tX , where εµ is the polarization vector. The graviton is ab µ ν WG(σ) = εµνh ∂aX ∂bX , where εµν is the polarization tensor. For bosonic closed strings, 2 = 4 2 = the tachyon is the lowest state and it has a k α0 , where k : kk is the Minkowskian 2 = 1 0 = 1 norm. The open string tachyon has k α0 . In units where the Regge slope α 2 or 1 2 where the string tension T = π we have that for the closed string tachyon k = 8 and for the open string k2 = 2, which will be used in the following sections of the present survey. The massless open string gauge field and the closed string graviton have k2 = 0. For open strings one can see that in the functional integral with action (5) the expansion in Taylor’s series of exponential of the second term of (5), leads to a superposition of amplitudes with different powers of the vertex operator of the gauge field (6) inserted in the boundary ∂Σ. From now on, we restrict ourselves to the case of bosonic closed strings with g = 0 and bosonic open strings with g = 0 and b = 1, i.e., the 2-sphere and the disk. For closed strings, by worldsheet diffeomorphisms and Weyl transformations of Σ, allow us to fix the worldsheet metric hab to the Euclidean flat metric δab. This is the called the conformal gauge. In the absence of a B-field, there is a remnant SL(2, C) symmetry which cannot be fixed by the local Weyl symmetry. This is called the conformal Killing symmetry. This symmetry does act on the complex worldsheet coordinates (z, z) of the 2-sphere separately. In the holomorphic sector (for the anti-holomorphic sector a similar argument is applied) Symmetry 2021, 13, 967 8 of 53

az + b z 7→ , cz + d where ad − bc = 1 and a, b, c, d ∈ R. This symmetry allows the choice of three points on the sphere which could be at 0, 1 and ∞. For open strings the remnant symmetry preserves the boundary is SL(2, R). In the open string case the vertex operators are inserted on the boundary of the disk and this symmetry can be used to fix three points on the boundary that also may be 0, 1 and ∞. As with the case of closed strings, for open strings the presence of a B-field spoils the SL(2, R) symmetry avoiding the possibility to fix three points of the positions of the vertex operators on the boundary ∂Σ. The two-point function with these boundary conditions is given by h hXµ(z)Xν(z0)i = −α0 gµν log |z − z0| − gµν log |z − z¯0| (7) 1 z − z¯0 i + Gµν log |z − z¯0|2 + θµν log + Dµν 2πα0 z¯ − z0 with 1 1 µν Gµν = g , (8) g + 2πα0B g − 2πα0B 1 1 µν θµν = −(2πα0)2 B , g + 2πα0B g − 2πα0B

and where Dµν are quantities independent on z. Basically, what we want are the N-tachyon scattering amplitudes at tree level for the action (5). This is done by inserting the following N tachyon vertex operators V(k, z) = eikX(z) at different points of the boundary of the open string worldsheet ∂Σ and obtaining the correlation functions of these operators. To obtain it we must realize that the correlators in the path integral formalism can be computed as Gaussian integrals [54]. Consider the following integral

Z Z 1 Z DX exp d2z(X∆X + iJX) ∼ exp d2zd2z0 J(z)K(z, z0)J(z0) , (9) 2

where ∆ is a differential operator. The symbol ∼ has been used to indicate that there are some proportional factors not relevant to the analysis. Here K is the inverse operator, or Green’s function, that satisﬁes

∆K(z, z0) = δ(z − z0). We can see that we can use this result to obtain the scattering amplitudes by choosing the appropriate source J(z).

hV(k1, z1)V(k2, z2) ... V(kN, zN)i Z ∼ DX exp{−S + i(k1X1(σ) + ··· + kN XN(σ))}.

Since we are interested in the vertex operators inserted at the boundary, we should restrict the propagator to it. By taking z = τ and z0 = τ0 for real τ, τ0 in (7) we get

0 0 0 i 0 hXµ(τ)Xν(τ )i = −α Gµν log τ − τ + θµνsgn(τ − τ ), (10) 2 where sgn is the sign function. Now we can see that it sufﬁces to choose the current as

J(τ) = ∑ δ(τ − τl)kl l Symmetry 2021, 13, 967 9 of 53

in (9). From this we can obtain the scattering amplitudes by finally integrating the expected values of the vertex operators over the entire boundary of the worldsheet, which now is just integrating over the real variable τ. The result is

Z Z ! N N −i l m d τhV(k1, τ1)V(k2, τ2) ···V(kN, τN )i ∼ d τ exp ∑ k θk sgn(τl − τm) 2 l>m ! 0 l m × exp α ∑ k Gk log|τl − τm| (11) l,m Z ! N −i l m α0kl Gkm = d τ exp ∑ k θk sgn(τl − τm) ∏ |τl − τm| , 2 l>m l,m

l m l µν m l m l µν m where k θk := kµθ kν and k Gk := kµG kν . These amplitudes are of the Koba– Nielsen type. The factor involving the sign function in the last line is actually more general, because it only depends on the variables through the sign functions, and any derivatives of it are 0. This allows us to write

* N + 2 ikl X(τl ) ∏ Pl(∂X(τl), ∂ X(τl) ... )e l=1 G,θ * N + i l µν m ∑l>m kµθ kν sgn(τl −τm) 2 ikl X(τl ) = e 2 ∏ Pl(∂X(τl), ∂ X(τl) ... )e . l=1 G,θ=0

The symbol h· · · iG,θ = 0 means that the expectation value taken on the second line is using the propagator (10) without the second term. This is achieved by writing the polynomial as an exponential and keeping only the corresponding linear terms. For example [48]:

µ µ ikX(z) µ µ V(ζ, k, z) = iζ ∂X e = [exp(ikX + iζ ∂X )]linear,

where [··· ]linear indicates to only keep linear terms in ζ. So, we can see two direct effects of the B-field on the action (5). The first is that the effective metric on spacetime becomes Gµν given in (8). The second is that at tree level (Remember, this came from the fact that we are in the full upper half plane with the simplest zero genus topology) the scattering amplitudes acquire a phase factor that depends on the cyclic ordering of the momenta kµ and the matrix θ. But the overall form of the amplitudes is the same as without the B-field. In the case of open strings for N points with vanishing B-field and with flat metric 0 Gµν = ηµν and units with α = 1 and for a fixed ordering of the external momenta ki, from (11) we obtain the Koba–Nielsen amplitude, with the three fixed points in x1 = 0, xN−1 = 1 and xN = ∞,

N−2 (N) Z A (k) = dx dx ··· dx |x |k1kj |1 − x |kN−1kj eR 2 3 N−2 ∏ j j j=2 0

kikj × ∏ |xi − xj| , 2≤i

where we have changed the variables τ to x’s variables in (11), In [48,53–55]. The four-point amplitude is Veneziano’s amplitude

Z 1 (4) Ae (k) = dx|x|k1k2 |1 − x|k2k3 . (12) R 0 It is known that this deﬁnition of Veneziano’s amplitude describes only one of the three decay channels s, t and u corresponding to the different inequivalent permutations of Symmetry 2021, 13, 967 10 of 53

the momenta. If we consider the sum over the three channels and performing a suitable change of variables [59] is possible to write the total amplitude as Z (4) A (k) = dx|x|k1k2 |1 − x|k2k3 . R R In terms of the gamma function the above amplitude writes

0 0 0 0 ( ) Γ(−1 − α s)Γ(−1 − α t) Γ(−1 − α s)Γ(−1 − α u) A 4 (k) = + R Γ(−2 − α0(s + t)) Γ(−2 − α0(s + u)) Γ(−1 − α0t)Γ(−1 − α0u) + Γ(−2 − α0(t + u))

2 where s, t and u are the Mandelstam variables and they are deﬁned as s := −(p1 + p2) , 2 2 2 2 t := −(p1 + p3) and u := −(p1 + p4) and they satisfy s + t + u = − ∑i ki = ∑i Mi , and Z ∞ Γ(u) = dt tu−1e−t 0 is the gamma function. Thus, the 4-point amplitudes whose integration is R it already includes the three channels. Moreover, this procedure also works for N point amplitudes [4,48] and it yields

Z N−2 (N) k1kj kN−1kj A (k) = dx2dx3 ··· dxN−2 |xj| |1 − xj| R N−3 ∏ R j=2

kikj × ∏ |xi − xj| . 2≤i

For completeness we write the form of closed strings in absence of B-ﬁeld the N-point amplitude is given by Z (N) 2 kikj/2 A (k) = d zi |zi − zj| . C N−2 ∏ ∏ C i

The four-point amplitude is the Virasoro–Shapiro amplitude Z (4) A (k) = d2z|z|k1k4/2|1 − z|k2k4/2 C C2 0 0 0 Γ(−1 − α s )Γ(−1 − α t )Γ(−1 − α u ) = 4 4 4 . α0s α0t α0u Γ(2 + 4 )Γ(2 + 4 )Γ(2 + 4 ) It is worth noting that the Veneziano four-point amplitude (12), describing only one channel, can be written also as an integral over the whole R if we introduce some multiplicative characters as sgn(x):

Z Z (4) 1 1 A (k) = dx|x|k1k2 |1 − x|k2k3 + dx|x|k1k2 |1 − x|k2k3 sgn(x) sgn(1 − x), eR 2 R 2 R where the function sign(x) is +1 of x > 0 and is −1 if x < 0 [14]. The generalization for N-point amplitudes is given by

Z N−2 1 − sgn − x (1 − x ) (4) k1kj kN−1kj j j A˜ (k) = dx2dx3 ··· dxN−2 |xj| |1 − xj| R N−3 ∏ R j=2 2 Symmetry 2021, 13, 967 11 of 53

kikj 1 × ∏ |xi − xj| 1 − sgn (−1)(xi − xj) , 2≤i

3. Essential Ideas of p-Adic Analysis In this section, we review some ideas and results on p-adic analysis that we will use throughout this article. For an in-depth exposition, the reader may consult [3,61,62].

3.1. The Field of p-Adic Numbers

Throughout this article p will denote a prime number. The field of p-adic numbers Qp is defined as the completion of the field of rational numbers Q with respect to the p-adic norm | · |p, which is defined as

 =  0 if x 0 |x|p =  −γ γ a p if x = p b , where a and b are integers coprime with p. The integer γ := ord(x), with ord(0) := +∞, is n called the p-adic order of x. We extend the p-adic norm to Qp by taking

n ||x||p := max |xi|p, for x = (x1,..., xn) ∈ Qp. 1≤i≤n −ord(x) We deﬁne ord(x) = min1≤i≤n{ord(xi)}, then ||x||p = p . The metric space n Qp, || · ||p is a complete ultrametric space. As a topological space Qp is homeomorphic to a Cantor-like subset of the real line, see e.g., [3,61]. Any p-adic number x 6= 0 has a unique expansion of the form

∞ ord(x) i x = p ∑ xi p , i=0

where xi ∈ {0, 1, 2, . . . , p − 1} and x0 6= 0. n n r r For r ∈ Z, denote by Br (a) = {x ∈ Qp; ||x − a||p ≤ p } the ball of radius p with center at n n n n a = (a1, ... , an) ∈ Qp, and take Br (0) := Br . Please note that Br (a) = Br(a1) × · · · × Br(an), r r where Br(ai) := {x ∈ Qp; |xi − ai|p ≤ p } is the one-dimensional ball of radius p with center n at ai ∈ Qp. The ball B0 equals the product of n copies of B0 = Zp, the ring of p-adic integers. n −r n n n r In addition, Br (a) = a + p Zp . We also denote by Sr (a) = {x ∈ Qp; ||x − a||p = p } r n n n 1 × the sphere of radius p with center at a ∈ Qp, and take Sr (0) := Sr . We notice that S0 = Zp n × n (the group of units of Zp), but Zp ( S0 . The balls and spheres are both open and closed n n subsets in Qp. In addition, two balls in Qp are either disjoint or one is contained in the other. n As a topological space Qp, || · ||p is totally disconnected, i.e., the only connected n n subsets of Qp are the empty set and the points. A subset of Qp is compact if and only if it is n closed and bounded in Qp, see e.g., ([3], Section 1.3), or ([61], Section 1.8). The balls and n spheres are compact subsets. Thus, Qp, || · ||p is a locally compact topological space.

Remark 1. There is a natural map, called the reduction mod p and denoted as ·, from Zp onto ∞ j Fp, the ﬁnite ﬁeld with p elements. More precisely, if x = ∑j=0 xj p ∈ Zp, then x = x0 ∈ Fp = n {0, 1, . . . , p − 1}. If a = (a1,..., an) ∈ Zp, then a = (a1,..., an). Symmetry 2021, 13, 967 12 of 53

n 3.2. Integration on Qp Since (Qp, +) is a locally compact topological group, there exists a Borel measure dx, called the Haar measure of (Qp, +), unique up to multiplication by a positive constant, such R R R that U dx > 0 for every nonempty Borel open set U ⊂ Qp, and satisfying E+z dx = E dx for every Borel set E ⊂ Qp, see e.g., ([63], Chapter XI). If we normalize this measure by the condition R dx = 1, then dx is unique. From now on we denote by dx the normalized Zp n n Haar measure of (Qp, +) and by d x the product measure on (Qp, +). n n A function ϕ : Qp → C is said to be locally constant if for every x ∈ Qp there exists an open compact subset U, x ∈ U, such that ϕ(x) = ϕ(u) for all u ∈ U. Any locally n constant function ϕ : Qp → C can be expressed as a linear combination of characteristic ( ) = ∞ ( ) ∈ ( ) functions of the form ϕ x ∑k=1 ck1Uk x , where ck C and 1Uk x is the characteristic n function of Uk, an open compact subset of Qp, for every k. If ϕ has compact support, then ( ) = L ( ) ϕ x ∑k=1 ck1Uk x and in this case Z Z Z n n n ϕ(x)d x = c1 d x + ... + cL d x. n Qp U1 UL A locally constant function with compact support is called a Bruhat-Schwartz function. n n These functions form a C-vector space denoted as D Qp . By using the fact that D Qp n n is a dense subspace of Cc Qp , the C-space of continuous functions on Qp with compact R n support, with the topology of the uniform convergence, the functional ϕ → n ϕ(x)d x, Qp n n ϕ ∈ D Qp has a unique continuous extension to Cc Qp , as an unbounded linear functional. For integrating more general functions, say locally integrable functions, the following notion of improper integral will be used.

1 n Deﬁnition 1. A function ϕ ∈ Lloc is said to be integrable in Qp if Z m Z lim ϕ(x)dnx = lim ϕ(x)dnx m→+∞ m→+∞ ∑ n j=−∞ n Bm Sj R n exists. If the limit exists, it is denoted as n ϕ(x)d x, and we say that the (improper) integral exists. Qp

3.3. Analytic Change of Variables n A function h : U → Qp is said to be analytic on an open subset U ⊂ Qp, if for every b ∈ U there exists an open subset Ue ⊂ U, with b ∈ Ue, and a convergent power series i i i i1 in ∑i ai(x − b) for x ∈ Ue, such that h(x) = ∑i∈Nn ai(x − b) for x ∈ Ue, with x = x1 ··· xn , ∂ ∂ i i = (i ,..., in). In this case, h(x) = ∑ ∈ n ai (x − b) is a convergent power series. Let 1 ∂xl i N ∂xl n U, V be open subsets of Qp. A mapping h : U → V, h = (h1,..., hn) is called analytic if each hi is analytic. Let ϕ : V → C be a continuous function with compact support, and let h : U → V be an analytic mapping. Then

R n R n ϕ(y)d y = ϕ(h(x))|Jac(h(x))|pd x, (13) V U h i where Jac(h(z)) := det ∂hi (z) , see e.g., ([64], Section 10.1.2). ∂xj 1≤i≤n 1≤j≤n

Example 1. In dimension one, the change of variables Formula (13) implies that Z Z dx = |a|p dy, aU U Symmetry 2021, 13, 967 13 of 53

× for any a ∈ Qp and any Borel set U ⊆ Qp. For example, Z Z Z Z −r r r −r dx = dx = |p |p dy = p dy = p , x = p y (14) −r Br p Zp Zp Zp

and Z Z Z dx = dx − dx = pr − pr−1 = pr(1 − p−1). (15)

Sr Br Br−1

Example 2. We now compute the following integral Z s Z(s) = |x|pdx, s ∈ C, with Re(s) > 0 ∈ C.

Zp × × We use the partition Zp = pZp t Zp , where Zp is the group of units of Zp:

( ∞ ) × i Zp := {x ∈ Zp; |x|p = 1} = ∑ xi p ∈ Zp; x0 6= 0 . i=0 Using (15),

Z Z Z Z s s −1−s s Z(s) = |x|pdx + |x|pdx = p |x|pdx + dx × × pZp Zp Zp Zp Z −1−s s −1 −1−s −1 = p |x|pdx + (1 − p ) = p Z(s) + (1 − p ).

Therefore, 1 − p−1 Z(s) = for Re(s) > −1. (16) 1 − p−1−s

Example 3. We now consider the integral Z s Z(s) := |1 − x|pdx, s ∈ C, with Re(s) > 0 ∈ C. × Zp × × Assuming that = t × a + p p , where = {1, 2, ..., p − 1}, and (16) we have Zp a0∈Fp 0 Z Fp

Z p−1 Z Z s s −1−s s Z(s) = |1 − x|pdx + ∑ |1 − x|pdx =p |y|pdy a0=2 1+pZp a0+pZp Zp − p 1 Z 1 − p−1 +p−1 |1 − (a + py)|s dy = p−1−s + p−1(p − 2). ∑ 0 p 1 − p−1−s a0=2 Zp

3.4. Some p-Adic Integrals In this section, we compute some integrals that we will use later on.

Example 4. We consider the following integral: Z s Z(s) := |x − y|pdxdy, s ∈ C, with Re(s) > 0 ∈ C.

Zp×Zp Symmetry 2021, 13, 967 14 of 53

2 2 2 2 × × × × Using the partition Zp = (pZp) t S0, with S0 = pZp × Zp t Zp × pZp t Zp × Zp , we have

Z Z Z s s s Z(s) = |x − y|pdxdy + |x − y|pdxdy + |x − y|pdxdy 2 × × (pZp) pZp×Zp Zp ×pZp Z Z Z s −2−s −1 −1 + |x − y|pdxdy = p Z(s) + p dxdy + p dxdy × × × × Zp ×Zp Zp×Zp Zp ×Zp Z s + |x − y|pdxdy. × × Zp ×Zp

To compute the last integral, we use the partition

× 2 F (Zp ) = a0 + pZp × a1 + pZp × a0,a1∈Fp as follows:

Hence, 1 − p−1 Z(s) = . (17) 1 − p−1−s

Example 5. We now show that

Z 1 − p−1 Z(s) := |x − y|s dxdy = p−2(p − 1)(p − 2) + p−2−s(p − 1) . (18) p 1 − p−1−s × × Zp ×Zp

Indeed, by changing variables as x = uv, y = v, Z Z s −1 s Z(s) = |u − 1|pdudv = 1 − p |u − 1|pdu. × × × Zp ×Zp Zp Now the result follows from Example3.

Example 6. We now compute the following integral Z s1 s2 s3 Z(s1, s2, s3) := |1 − x|p |1 − y|p |x − y|p dxdy, × × Zp ×Zp

where s1, s2, s3 ∈ C satisfying Re(s1), Re(s2), Re(s3) > 0. By using the partition

× 2 F (Zp ) = a0 + pZp × a1 + pZp , × a0,a1∈Fp × where Fp = {1, 2, ..., p − 1}, Symmetry 2021, 13, 967 15 of 53

p−1 p−1 Z s1 s2 s3 Z(s1, s2, s3) = ∑ ∑ |1 − x|p |1 − y|p |x − y|p dxdy a0=1 a1=1 (a0+pZp)×(a1+pZp) p−1 p−1 = ( ) : ∑ ∑ Ja0,a1 s1, s2, s3 . a0=1 a1=1

By changing variables as x = a0 + pu, y = a1 + pv we have

Z −2 s1 s2 s3 Ja0,a1 (s1, s2, s3) = p |1 − (a0 + pu)|p |1 − (a1 + pv)|p |(a0 + pu) − (a1 + pv)|p dudv. 2 Zp

Now, we compute the integrals Ja0,a1 . This calculation involves several cases. Case 1. If a06=1, a16=1 a0 6= a1, then |1 − (a0 + pu)|p = 1, |1 − (a1 + pv)|p = 1, and |(a0 + pu) − (a1 + pv)|p = 1. and Z −2 −2 Ja0,a1 (s1, s2, s3) = p dudv = p . 2 Zp There are (p − 2)(p − 3) integrals of this type. Case 2. If a0 6= 1, a1 6= 1 a0 = a1, then, |1 − (a0 + pu)|p = 1, |1 − (a1 + pv)|p = 1. By Example (17),

Z p−2−s3 ( − p−1) −2−s3 s3 1 Ja ,a (s1, s2, s3) = p |u − v|p dudv = , 0 1 1 − p−1−s3 Zp×Zp

for Re(s3) > −1. There are p − 2 integrals of this type. Case 3. If a0 = 1, a1 6= 1, then |1 − (a1 + pv)|p = 1, |(a0 + pu) − (a1 + pv)|p = 1. By using (16)

Z p−2−s1 (1 − p−1) −2−s1 s1 Ja ,a (s1, s2, s3) = p |u|p dudv = , 0 1 1 − p−1−s1 2 Zp

for Re(s1) > −1. There are p − 2 integrals of this type. Case 4. If a0 6= 1, a1 = 1, this case is similar to Case 3,

Z p−2−s2 ( − p−1) −2−s2 s2 1 Ja ,a (s1, s2, s3) = p |x3|p dx3 = 0 1 1 − p−1−s2 Zp

for Re(s2) > −1. Therefore

−2−s3 −1 −2 p (1 − p ) Z(s1, s2, s3) = p (p − 2)(p − 3) + (p − 2) 1 − p−1−s3 p−2−s1 (1 − p−1) p−2−s2 (1 − p−1) + (p − 2) + (p − 2) . 1 − p−1−s1 1 − p−1−s2

3 on the region of C deﬁned by Re(s1) > −1, Re(s2) > −1, and Re(s3) > −1. Symmetry 2021, 13, 967 16 of 53

4. p-Adic Open String Amplitudes In [14], Brekke, Freund, Olson and Witten work out the N-point amplitudes in explicit form and investigate how these can be obtained from an effective Lagrangian. The p-adic open string N-point tree amplitudes are deﬁned as

Z N−2 N−2 ( ) k k k k k k A N (k) = |x | 1 i | − x | N−1 i x − x i j dx ∏ i p 1 i p ∏ i j p ∏ i, (19) N−3 i=2 2≤i

N−2 N−3 where ∏i=2 dxi is the normalized Haar measure of Qp , k = (k1,..., kN), ki = (k0,i,..., kl,i), i = 1, ... , N, N ≥ 4, is the momentum vector of the i-th tachyon (with Minkowski product kikj = −k0,ik0,j + k1,ik1,j + ··· + kl,ikl,j) obeying

N ∑ ki = 0, kiki = 2 for i = 1, . . . , N. i=1 The parameter l is an arbitrary positive integer. Typically, l is taken to be 25. However, we do not require using the critical dimension. In [8], we show that the p-adic open string amplitudes (19) are bona ﬁde integrals and that they have meromorphic continuations as rational functions in the kinematic parameters. We attach to these amplitudes a multivariate Igusa-type zeta function. Fix an integer N ≥ 4 and set T := {2, . . . , N − 2}, (N−3)(N−4) D D = 2 + 2(N − 3) and C as  ∈ ∈ { − } ∈ =  sij C; i 1, N 1 , j T if N 4  sij ∈ C; i ∈ {1, N − 1}, j ∈ T or i, j ∈ T with i < j if N ≥ 5. D We now put s = sij ∈ C and kikj = sij ∈ C satisfying sij = sji for 1 ≤ i < j ≤ N − 1. The p-adic open string N-point zeta function is defined as

Z N−2 N−2 ( ) s s(N−1)i s Z N (s) = |x | 1i | − x | x − x ij dx ∏ i p 1 i p ∏ i j p ∏ i, (20) N−3 i=2 2≤i

D N−2 N−3 where s = sij ∈ C , ∏i=2 dxi is the normalized Haar measure of Qp , and

( N−2 ) N−3 Λ := (x2,..., xN−2) ∈ Qp ; ∏ xi(1 − xi) ∏ xi − xj = 0 . i=2 2≤i

In the deﬁnition of integral (20) we remove the set Λ from the domain of integration to use the formula as = es ln a for a > 0 and s ∈ C. The reason for using the name ‘zeta function’ is that the integral (20) is a ﬁnite sum of multivariate local zeta functions. Take φ(x2,..., xN−2) a locally constant function with compact support, then

Z N−2 N−2 (N) s s(N−1)i s Z (s) = (x x ) |x | 1i | − x | x − x i j dx φ φ 2,..., N−2 ∏ i p 1 i p ∏ i j p ∏ i, N−3 i=2 2≤i

for Re(sij) > 0 for any ij, is a multivariate Igusa local zeta function. In characteristic zero, a general theory for this type of local zeta functions was elaborated by Loeser in [36]. In particular, these local zeta functions admit analytic continuations as rational functions of the variables p−sij . We want to highlight that the convergence of the multivariate local zeta functions depends on the fact that the locally constant function φ has compact support. For example, we consider the following integral: Symmetry 2021, 13, 967 17 of 53

Z s J(s) = |x|pdx, s ∈ C. Qp

If the integral J(s0) exists for some s0 ∈ R, then the integrals Z Z s0 s0 J0(s0) = |x|p dx and J1(s0) = |x|p dx Zp QprZp

− −1 exist. The ﬁrst integral J (s ) = 1 p for s > −1. For the second integral, we use that 0 0 1−p−1−s0 0 s0 |x|p is locally integrable, and thus

∞ Z ∞ Z ∞ s0 j+js0 −1 j(1+s0) J1(s0) = ∑ |x|p dx = ∑ p dx = 1 − p ∑ p < ∞ j=1 −j × j=1 × j=1 p Zp Zp

if and only if s0 < −1. Then, integral J(s) does not exist for any complex value s. Theorem 1 in [8] establishes that the p-adic open string N-point zeta function is a holomorphic function in a certain domain of CD and that it admits an analytic continuation −s to CD (denoted as Z(N)(s)) as a rational function in the variables p ij , i, j ∈ {1, . . . , N − 1}. Hence, in the theory of local zeta functions, the convergence and the existence of a meromorphic continuation as a rational function in the variables p−sij , i, j ∈ {1, . . . , N − 1} of integrals of type (20) is a new and remarkable result. The p-adic open string N-point zeta functions Z(N)(s) are regularizations of the amplitudes A(N)(k). More precisely, we deﬁne

(N)( ) = (N)( ) | ∈ { − } ∈ ∈ A k Z s sij=kikj with i 1, . . . , N 1 , j T or i, j T,

where T = {2, . . . , N − 2}. By Theorem 1 in [8], A(N)(k) are well-deﬁned rational functions of the variables p−kikj , i, j ∈ {1, . . . , N − 1}, which agree with integrals (87) when they converge. In the following two sections, using the methods given in [8], we compute the p-adic open string amplitudes in the cases N = 4, 5.

4.1. p-Adic Open String 4-Point Amplitude The 4-point string amplitude is given by Z (4) k1k2 k3k2 A (k) = |x2|p |1 − x2|p dx2.

Qp We attach to this amplitude the local zeta function: Z (4) s12 s32 Z (s) = |x2|p |1 − x2|p dx2.

We now divide the integration domain (Qp) into sectors. Given I j T = {2}, the attached sector is deﬁned as Sect(I) = xi ∈ Qp; |xi|p ≤ 1 ⇐⇒ i ∈ I .

Then, we have two sectors:

IT I Sect(I)

{2} ∅ Zp

∅ {2} Qp\Zp, Symmetry 2021, 13, 967 18 of 53

and

( ) ( ) ( ) Z 4 (s) = Z 4 (s, ∅) + Z 4 (s, {2}), where Z (4) s12 s32 Z (s; {2}) := |x2|p |1 − x2|p dx2

Zp and Z (4) s12 s32 Z (s; ∅) := |x2|p |1 − x2|p dx2.

QprZp

We now compute Z(4)(s; {2}). By using the Examples2 and3: Z Z (4) s12 s42 Z (s; {2}) = |x2|p dx2 + |1 − x2|p dx2 (21) × Zp Zp 1 − p−1 p−1−s12 1 − p−1 p−1−s32 = 1 − 2p−1 + + , 1 − p−1−s12 1 − p−1−s32

for Re(s12) > −1 and Re(s32) > −1. (4) We now consider integral Z (s; ∅). By using the ultrametric inequality |1 − x2|p = |x2|p for x2 ∈ Qp r Zp, Z (4) s12+s32 Z (s; ∅) = |x2|p dx2.

QprZp To compute this last integral we proceed as follows. For l ∈ N\{0}, we set

= ∈ − ≤ ( ) ≤ − Qp r Zp −l : x2 Qp r Zp ; l ord x2 1 ,

= ∈ ≤ ( ) ≤ pZp l : x2 pZp ; 1 ord x2 l , and Z s12+s32 I−l(s) := |x2|p dx2.

(QprZp)−l Notice that Qp r Zp −l, pZp l are compact sets and that → Qp r Zp −l pZp l

x → 1 , 2 y2

dy2 is an analytic change of variables satisfying dx2 = 2 , then using this change of variables, |y2|p we have Z dy2 I−l(s) = . s12+s32+2 |y2|p p ( Zp)l R 1 Now, by using that s dy converges for Re(s) < 1 and the dominated conver- pZp |y|p (4) gence theorem, liml→∞ I−l(s) = Z (s; ∅), i.e. Symmetry 2021, 13, 967 19 of 53

Z −1 1+s12+s32 ( ) dy2 (1 − p )p Z 4 (s; {∅}) = = , s12+s32+2 1 − p1+s12+s32 |y2|p pZp

on Re(s12) + Re(s32) < −1. Therefore

−1−s12 −1−s32 1+s12+s32 (4) p − 2 −1 p p p Z (s) = + 1 − p − − + + + + , p 1 − p 1 s12 1 − p(−1−s32) 1 − p1 s12 s32

this function is holomorphic in

Re(s12) > −1, Re(s32) > −1 and Re(s12) + Re(s32) < −1, which is a nonempty open subset in C2 because it contains the open set deﬁned by the conditions:

2 1 2 1 − < Re(s ) < − and − < Re(s ) < − . 3 12 2 3 32 2 We denote the meromorphic continuation of integral Z(4)(s) also by Z(4)(s). Now, we regularize the p-adic open string 4-point amplitude using Z(4)(s):

(4)( ) = (4)( ) | A k : Z s s12=k1k2,s32=k3k2 ( ) p − 2 p−1−k1k2 p−1−k2k3 p1+k1k2+k2k3 = + 1 − p−1 + + p 1 − p−1−k1k2 1 − p−1−k2k3 1 − p1+k1k2+k2k3 ( ) p − 2 p−1−k1k2 p−1−k2k3 p−1−k2k4 = + 1 − p−1 + + . (22) p 1 − p−1−k1k2 1 − p−1−k2k3 1 − p−1−k2k4

This amplitude can be rewritten as

(4) 1 A (k) = K4 + ∑ xij, 2 i

p − 1 p−1−kikj xij := , p 1 − p−1−kikj

where ∑i

k1k3 + k2k3 + k3k4 + 2 = 0, (23)

k1k2 + k1k3 + k1k4 + 2 = 0, (24)

k1k2 + k2k3 + k2k4 + 2 = 0, (25)

k1k4 + k2k4 + k3k4 + 2 = 0. (26)

From (23)–(26), we obtain

k1k2 = k3k4, k1k3 = k2k4, k1k4 = k2k3. Then (4) A (k) = K4 + x12 + x23 + x24, (27) Symmetry 2021, 13, 967 20 of 53

which agrees with (22).

4.2. p-Adic Open String 5-Point Amplitude The p-adic open string 5-point amplitude is given by Z (5) k1k2 k1k3 k4k2 k4k3 k2k3 A (k) = |x2|p |x3|p |1 − x2|p |1 − x3|p |x2 − x3|p dx2dx, 2 Qp and the 5-point local zeta function is given by Z (5) s12 s13 s42 s43 s23 Z (s) = |x2|p |x3|p |1 − x2|p |1 − x3|p |x2 − x3|p dx2dx3, 2 Qp

2 where sij ∈ C. We divide the integration domain Qp in sectors. Given I j T = {2, 3} the attached sector is deﬁned as

n 2 o Sect(I) = (x1, x2) ∈ Qp; |xi|p ≤ 1 ⇐⇒ i ∈ I . Table1 shows all the sectors for the computation of the 5-point amplitude.

Table 1. Sectors of the 5-point amplitude.

IT I Sect(I) ∅ {2, 3} Qp\Zp × Qp\Zp {2}{3} Zp × Qp\Zp {3}{2} Qp\Zp × Zp

{2, 3} ∅ Zp × Zp.

Now, by using the notation Z (5) s12 s13 s42 s43 s23 Z (s; I) = |x2|p |x3|p |1 − x2|p |1 − x3|p |x2 − x3|p dx2dx3, Sect(I) we have

( ) ( ) ( ) ( ) ( ) Z 5 (s) = Z 5 (s; ∅) + Z 5 (s; {2}) + Z 5 (s; {3}) + Z 5 (s; {2, 3}). Case 1. Sect({∅}) = Qp\Zp × Qp\Zp . 1 1 By changing of variables as x2 → u , x3 → v and using Lemma 7 of [8], we have

Z (5) s12+s42 s13+s43 s23 Z (s; {∅}) = |x2|p |x3|p |x2 − x3|p dx2dx3. 2 (Qp\Zp) Z −s12−s42−2 −s13−s43−2 1 1 s23 = |u|p |v|p | − |p dudv 2 (pZp) u v s23 Z |v − u|p = p−2+s12+s42+s23+2+s13+s43+s23+2−s23 dudv (28) 2 s12+s42+s23+2 s13+s43+s23+2 Zp |u|p |v|p p2+s12+s42+s23+s13+s43 (1 − p−1) (1 − p−1)p1+s12+s42+s23 = 1 − p2+s12+s42+s23+s13+s43 1 − p1+s12+s42+s23 (1 − p−1)p1+s13+s43+s23 (1 − p−1)p−1−s23 + + + (p − 2)p−1 . 1 − p1+s13+s43+s23 1 − p−1−s23

( ) The integral Z 5 (s; {∅}) is holomorphic in the solution set of the following conditions: Symmetry 2021, 13, 967 21 of 53

Re(s12) + Re(s42) + Re(s23) < −1, (29)

Re(s13) + Re(s43) + Re(s23) < −1,

Re(s23) > −1,

Re(s12) + Re(s42) + Re(s13) + Re(s43) + Re(s23) < −2. Case 2. Sect({2}) = Zp × Qp\Zp .

In this case, |x2|p ≤ 1 and |x3|p > 1, now by ultrametric property |1 − x3|p = |x3|p and |x2 − x3|p = |x3|p, then Z (5) s12 s13 s42 s43 s23 Z (s; {2}) = |x2|p |x3|p |1 − x2|p |1 − x3|p |x2 − x3|p dx2dx3

Zp×(Qp\Zp) Z s12 s13+s43+s23 s42 = |x2|p |x3|p |1 − x2|p dx2dx3

Zp×(Qp\Zp) Z Z s12 s42 s13+s43+s23 = |x2|p |1 − x2|p dx2 |x3|p dx3 (Qp\Zp) Zp (5) (5) =: Z0 (s; {2})Z1 (s; {2}).

The calculation of Z(5)(s; {2}, 0) is similar to the case of integral (21):

(1 − p−1)p−1−s12 (1 − p−1)p−1−s42 Z(5)(s; {2}, 0) = p−1(p − 2) + + . 1 − p−1−s12 1 − p−1−s42 The integral Z(5)(s; {2}, 0) is holomorphic in the domain

Re(s12) > −1 and Re(s42) > −1. (30)

( ) ( ) The calculation of Z 5 (s; {2}, 1) is similar to the calculation of Z 4 (s; {∅}) given in the 1 dy3 Subsection of 4-point zeta function. By changing of variables as x3 = , dx3 = 2 , y3 |y3|p we obtain

Z Z (5) s13+s43+s23 dy3 Z (s; {2}, 1) = |x3|p dx3 = 2+s13+s43+s23 |y3|p Qp\Zp pZp (1 − p−1)p1+s13+s43+s23 = . 1 − p1+s13+s43+s23

The integral Z(5)(s; {2}, 1) is holomorphic in

Re(s13) + Re(s43) + Re(s23) < −1. (31) Case 3. Sect({3}) = Qp\Zp × Zp.

In this case, |x2|p > 1, by the ultrametric inequality |1 − x2|p = |x2|p and |x2 − x3|p = |x2|p, then Symmetry 2021, 13, 967 22 of 53

Z (5) s12 s13 s42 s43 s23 Z (s; {3}) = |x2|p |x3|p |1 − x2|p |1 − x3|p |x2 − x3|p dx2dx3

(Qp\Zp)×Zp Z s12+s42+s23 s13 s43 = |x2|p |x3|p |1 − x3|p dx2dx3

Qp\Zp×Zp Z Z s13 s43 s12+s42+s23 = |x3|p |1 − x3|p dx3 |x2|p dx2

Zp Qp\Zp (N) (N) =: Z0 (s; {3})Z1 (s; {3}).

These integrals are similar to the ones obtained in the case of Sect({2}):

(1 − p−1)p−1−s13 (1 − p−1)p−1−s43 Z(5)(s; {3}, 0) = p−1(p − 2) + + , (32) 1 − p−1−s13 1 − p−1−s43

the integral Z(5)(s; {3}, 0) is holomorphic in

Re(s13) > −1 and Re(s43) > −1. (33)

And

(1 − p−1)p1+s12+s42+s23 Z(5)(s; {3}, 1) = , 1 − p1+s12+s42+s23 the integral Z(5)(s; {3}, 1) is holomorphic in

Re(s12) + Re(s42) + Re(s23) < −1. (34)

Case 4. Sect({2, 3}) = Zp × Zp. 2 2 2 2 × × × × Using the partition Zp = (pZp) t S0, with S0 = pZp × Zp t Zp × pZp t Zp × Zp , we have Z (5) s12 s13 s42 s43 s23 Z (s; {2, 3}) = |x2|p |x3|p |1 − x2|p |1 − x3|p |x2 − x3|p dx2dx3

Zp×Zp

=: I31(s) + I32(s) + I33(s) + I34(s),

where

R s12 s13 s23 I31(s) := |x2|p |x3|p |x2 − x3|p dx2dx3, 2 (pZp) R s12 s43 I32(s) := |x2|p |1 − x3|p dx2dx3, × pZp×Zp R s13 s42 I33(s) := |x3|p |1 − x2|p dx2dx3, × Zp ×pZp R s42 s43 s23 I34(s) := |1 − x2|p |1 − x3|p |x2 − x3|p dx2dx3. × × Zp ×Zp Using Lemma 4 of [8], we have

p−2−s12−s13−s23 p−1−s12 (1 − p−1)2 p−1−s13 (1 − p−1)2 I31(s) = + 1 − p−2−s12−s13−s23 1 − p−1−s12 1 − p−1−s13 p−1−s23 (1 − p−1)2 + + (p − 1)(p − 2)p−2 . 1 − p−1−s23 Symmetry 2021, 13, 967 23 of 53

The integral I31(s) is holomorphic in

Re(s12) > −1, Re(s13) > −1, Re(s23) > −1, and Re(s12) + Re(s13) + Re(s23) > −2. (35)

Using Examples2 and3,

−1 −1−s12 −1 −1−s43 (1 − p )p (1 − p )p −1 I32(s) = + p (p − 2) , 1 − p−1−s12 1 − p−1−s43 and

−1 −1−s13 −1 −1−s42 (1 − p )p (1 − p )p −1 I33(s) = + p (p − 2) . 1 − p−1−s13 1 − p−1−s42 These integrals are holomorphic in

Re(s12) > −1, Re(s13) > −1, Re(s42) > −1, and Re(s43) > −1. (36)

Finally, by Example6,

−2−s23 −1 −2 p (1 − p ) I34(s) = p (p − 2)(p − 3) + (p − 2) (37) 1 − p−1−s23 p−2−s42 (1 − p−1) p−2−s43 (1 − p−1) + (p − 2) + (p − 2) . 1 − p−1−s42 1 − p−1−s43

This integral is holomorphic in

Re(s42) > −1, Re(s43) > −1, and Re(s23) > −1. (38)

In conclusion, the 5-point local zeta functions is holomorphic on the region of C5 deﬁned by

Re(s12) > −1, Re(s13) > −1, Re(s42) > −1, and Re(s43) > −1,

Re(s12) + Re(s42) + Re(s23) < −1,

Re(s13) + Re(s43) + Re(s23) < −1,

Re(s12) + Re(s13) + Re(s23) > −2.

Re(s12) + Re(s42) + Re(s13) + Re(s43) + Re(s23) < −2.

which is a nonempty subset, since it contains the open set

2 − < Re(s ) < 0 , 3 ij 2 1 − < Re(s ) < − , 3 1i 2 2 1 − < Re(s ) < − . 3 (N−1)i 2 We regularize the p-adic open string 5-point amplitude using the meromorphic continuation of Z(5)(s) by setting

(5)( ) = (5)( ) | A k : Z s s12=k1k2,s13=k1k3,s42=k4k2,s43=k4k3,s23=k2k3 . The amplitude A(5)(k) agrees with the one computed in [14] using the Feynman rules of the effective Lagrangian:

(5) A (k) = K5 + K4 ∑ xij + ∑ xijxkl, (39) i

= (p−2)(p−3) = p−2 where K5 p2 , K4 p , and ij and kl are summed over all pairs of compatible channels. By a pair of compatible channels ij, kl we mean that i, j, k and l are different, and that i < j, k < l.

4.3. p-Adic Open String N-Point Amplitudes We ﬁx an integer N ≥ 4 and consider the general N-point zeta function:

Z N−2 N−2 ( ) s s(N−1)i s Z N (s) = |x | 1i | − x | x − x ij dx ∏ i p 1 i p ∏ i j p ∏ i. (40) N−3 i=2 2≤i

N−3 We divide the domain of integration Qp into sectors. Given I ⊆ T = {2, 3, ... , N − 2} the attached sector is defined as

n N−3 o Sect(I) = (x2,..., xN−2) ∈ Qp ; |xi|p ≤ 1 ⇔ i ∈ I . Hence, the N-point zeta function (40) can be written as

Z(N)(s) = ∑ Z(N)(s; I), I⊆T where

Z N−2 (N) Z (s; I) := F(s, x; N) ∏ dxi, i=2 Sect(I) with

N−2 s s(N−1)i s F(s x N) = |x | 1i | − x | x − x ij , ; : ∏ i p 1 i p ∏ i j p i=2 2≤i

N−3 and x = (x2,..., xN−2) ∈ Qp . By Lemma 2 in [8],    Z  ( ) ( ) s s(N−1)i s  Z N (s I) = pM s |x | 1i | − x | x − x i j dx ; ∏ i p 1 i p ∏ i j p ∏ i ∈ ≤ < ≤ − ∈  |I| i I 2 i j N 2 i I  Zp i,j∈I   sij   ∏ xi − xj   p   2≤i

M(s) (N) (N) =: p Z0 (s; I)Z1 (s; T r I),

where

M(s) := |T r I| + ∑ (s1i + s(N−1)i) + ∑ sij + ∑ sij. i∈TrI 2≤i

(N) = M(s) (N) (N) Z (s) ∑ p Z0 (s; I)Z1 (s; T r I), (41) I⊆T Symmetry 2021, 13, 967 25 of 53

(N) with the convention that Zi (s; ∅) = 1 for i = 0, 1. ( ) In [8] we show that Z N (s) has an analytic continuation to the whole CD as a rational function in the variables p−sij by showing that all functions that appear on the right-hand side of (41) are holomorphic in a region H(C) in CD (Deﬁnition 3 and Remarks 9 and 10 in [8]) deﬁned by

|J| + ∑(Re(s1i) + Re s(N−1)i ) + ∑ Re sij (42) i∈J 2≤i

|K| − 1 + ∑ Re(sij) > 0 for K ∈ F2; (43) 2≤i

1 + Re(sij) > 0 for ij ∈ G ⊆{ij; 2 ≤ i < j ≤ N − 2}, (44)

where F1, F2 are families of nonempty subsets of T, and G is a nonempty subset of {ij; 2 ≤ i < j ≤ N − 2, i, j ∈ T}.

| | + ( ) + ( ) > × ∈ F J ∑ Re sti ∑2≤i

with S ⊆ J, t ∈ {1, N − 1},and F3 a family of nonempty subsets of I × I;

|K| − 1 + ∑ Re(sij) > 0 for K ∈ F4, (46) 2≤i

where F4’s a family of nonempty subsets of I;

1 + Re(sij) > 0 for ij ∈ GT, (47)

where GT is a nonempty subset of {2 ≤ i < j ≤ N − 2, i, j ∈ J} with (N − 1)i, 1i ∈ GT. In Lemma 9 of [8], we show that region H(C) contains and open and connected subset of CD deﬁned by the conditions 2 − < Re(sij) < 0 , (48) 3N1 2 1 − < Re(s ) < − , (49) 3 1i 2 2 1 − < Re(s ) < − , (50) 3 (N−1)i 2

(N−4)(N−3) for N > 5, N1 = 2 , i, j ∈ {2, ..., N − 2}. For the case N = 4 we only consider conditions (49) and (50). Like in the cases N = 4 and N = 5, the key point is to reduce the in- (N) (N) tegrals Z0 (s; I), Z1 (s; T r I) to certain simple integrals, for which admit meromorphic −s continuations to the whole CD as rational functions in the variables p ij . We now state the meromorphic continuation of the open string N-point zeta function.

Theorem 1 (Theorem 1 in [8]). (1) The p-adic open string N-point zeta function, Z(N)(s), gives rise to a holomorphic function on H(C), which contains an open and connected subset of CD. Furthermore, Z(N)(s) admits an analytic continuation to CD, denoted also as Z(N)(s), as a rational function in the variables p−sij , i, j ∈ {1, . . . , N − 1}. The real parts of the poles of Z(N)(s) belong to a ﬁnite union of hyperplanes, the equations of these hyperplanes have the D form (42)–(47) with the symbols ‘<’, ‘>’ replaced by ‘=’. (2) If s = sij ∈ C , with Re(sij) ≥ 0 for i, j ∈ {1, . . . , N − 1}, then the integral Z(N)(s) diverges to +∞. Symmetry 2021, 13, 967 26 of 53

5. String Amplitudes over Non-Archimedean Local Fields 5.1. Non-Archimedean Local Fields A non-Archimedean local ﬁeld K is a locally compact topological ﬁeld with respect to |·| a non-discrete topology, which comes from a norm K satisfying

| + | ≤ {| | | | } x y K max x K, y K , for x, y ∈ K. A such norm is called an ultranorm or non-Archimedean. Any non-Archimedean local field K of characteristic zero is isomorphic (as a topological field) to a finite extension of Qp. The field Qp is the basic example of non-Archimedean local field of characteristic zero. In the case of positive characteristic, K is isomorphic to a finite extension of the field of formal Laurent series Fq((T)) over a finite field Fq, where q is a power of a prime number p. The ring of integers of K is defined as

= { ∈ | | ≤ } RK x K; x K 1 . ( |·| ) Geometrically RK is the unit ball of the normed space K, K . This ring is a domain of principal ideals with a unique maximal ideal, which is given by

= { ∈ | | < } PK x K; x K 1 .

We ﬁx a generator π of PK i.e., PK = πRK. A such generator is also called a local uniformizing parameter of K, and it plays the same role as p in Qp. The group of units of RK is deﬁned as

R× = {x ∈ R ; |x| = 1}. K K K ∼ The natural map RK → RK/PK = Fq is called the reduction mod PK. The quotient ∼ f RK/PK = Fq, q = p , is called the residue ﬁeld of K. Every nonzero element x of K can be written uniquely as x = πord(x)u, u ∈ R×. We set ord(0) = ∞. The normalized valuation of K K is the mapping

K → Z ∪ {∞} x → ord(x). | | = −ord(x) | | = −1 Then x K q and π K q . We ﬁx S ⊂ RK a set of representatives of Fq in RK, i.e., S is a set which is mapped bijectively onto Fq by the reduction mod PK. We assume that 0 ∈ S. Any nonzero element x of K can be written as

∞ ord(x) i x = π ∑ xiπ , i=0 ∈ S 6= |·| where xi and x0 0. This series converges in the norm K. |·| n We extend the norm K to K by taking

||x||K := max |xi|K, 1≤i≤n n −ord(x) for x = (x1, ... , xn) ∈ K .We define ord(x) = min1≤i≤n{ord(xi)}, then ||x||K = q . n The metric space (K , || · ||K) is a complete ultrametric space. As we mentioned before, any finite extension K of Qp is a non-Archimedean local field. Then

m pRK = π RK, m ∈ N. Symmetry 2021, 13, 967 27 of 53

If m = 1 we say that K is a unramified extension of Qp. In other case, we say that K is a ramified extension. It is well known that for every positive integer e there exists a unique unramified extension Ke of Qp of degree e, which means that Ke is a Qp-vector space of dimension e. From now on, π denotes a local uniformizing parameter of Ke, −e = ∼ e | | = thus pRKe πRKe , RKe /PKe = Fp and π Ke p . For an in-depth exposition of non-Archimedean local fields, the reader may consult [62,65], see also [3,61].

5.2. Open String Amplitudes over Non-Archimedean Local Fields The open string amplitudes can be defined over any local field. In this section, we consider Koba–Nielsen string amplitudes on Ke, the unique unramified extension of Qp of degree e for all e ∈ N\{0}. We recall that if Ke is the unramified extension of degree e of −e = ∼ e | | = Qp, then pRKe πRKe , RKe /PKe = Fp and π Ke p . Thus, π in Ke plays the role of p in Qp. The Koba–Nielsen amplitudes on Ke are defined as

Z N−2 N−2 (N) k k k − k kikj A (k) = |x | 1 i |1 − x | N 1 i x − x dx , (51) Ke ∏ i Ke i Ke ∏ i j Ke ∏ i N−3 i=2 2≤i

N− N− where ∏ 2 dx is the Haar measure of N−3, + normalized so that the measure of R 3 i=2 i Ke Ke is 1. The procedure used to regularize the p-adic amplitudes extends to amplitudes of the form (51). In this case, the open string N-point zeta function is deﬁned as

N−2 (N) Z Z (s) := F(s, x; N, e) dx , (52) Ke K ∏ i N−3 i=2 Ke

where

− N 2 s s1i (N−1)i sij F(s, x; N, e) = |x | |1 − x | x − x . K ∏ i Ke i Ke ∏ i j Ke i=2 2≤i

D (N−3)(N−4) where s = sij ∈ C , with D = 2 + 2(N − 3). Let T = {2, 3, . . . , N − 2}, then

(N) (N) (N) Z (s) = peM(s)Z (s; I, 0) Z (s; T I, 1), (53) Ke ∑ Ke Ke r I⊆T

where

M(s) := |T r I| + ∑ (s1i + s(N−1)i) + ∑ sij + ∑ sij i∈TrI 2≤i

(N) Z Z (s; T I, 1, ) = F (s, x; N, e) dx , Ke r 1 K ∏ i |T I| i∈TrI R r Ke where Symmetry 2021, 13, 967 28 of 53

sij ∏ x − x i j Ke 2≤i

(N) (N) A (k) = Z (s) | = · . Ke Ke sij ki kj (N) Then A (k) is a well-deﬁned rational function in the variables p−eki·kj , which agree Ke with the integral (51) when it converges.

5.3. The Limit p Tends to One The functions

(N) (N) Z (s; I, 0), Z (s; T I, 1) Ke Ke r are multivariate local zeta functions. Thus, to make mathematical sense of the limit of (N) Z (s) as p → 1 we use the work of Denef and Loeser, see [35,66], and compute the limit Qp (N) (N) of Z (s) as e → 0 instead of the limit of Z (s) as p → 1. In order to compute the limit Ke Qp (N) e → 0 is necessary to have an explicit formula for Z (s), so in [9] we determined the Ke (N) (N) explicit formula by ﬁnding explicit formulae for integrals Z (s; I, 0) and Z (s; T I, 1), Ke Ke r see Theorem B in [9]. After that, we deﬁne

(N) (N) Ztop (s; I, 0) = lim Z (s; I, 0) e→0 Ke and

(N) (N) Ztop (s; T r I, 1) = lim Z (s; T r I, 1), e→0 Ke which are elements of Q sij, i, j ∈ {1, . . . , N − 1} , the field of rational functions in the variables sij, i, j ∈ {1, ... , N − 1} with coefficients in Q. Then, by using (53), we defined the open string N-point topological zeta function as

(N) (N) (N) Ztop (s) = ∑ Ztop (s; I, 0)Ztop (s; T r I, 1). I⊆T

(N) The open string N-point topological zeta function Ztop (s) is a rational function of Q sij, i, j ∈ {1, . . . , N − 1} . We now deﬁne the Denef–Loeser open string N-point amplitudes at the tree level as

(N)( ) = (N)( ) | Atop k Ztop s sij=ki·kj , with i ∈ {1, . . . , N − 1}, j ∈ T or i, j ∈ T, where T = {2, . . . , N − 2}. Thus, the Denef–Loeser amplitudes are rational functions of the variables ki · kj, i, j ∈ {1, . . . , N}. Symmetry 2021, 13, 967 29 of 53

5.4. Feynman Rules, Explicit Formulae and Denef–Loeser Amplitudes Using the Feynman rules of the effective Lagrangian, an ‘explicit formula’ of the type

−k k −1 ! N−3 p − 1 p ia ib A(N)(k) = K + K N ∑ i1...il ∑ −k k −1 l=2 1≤a

− (N) top N 3 top 1 A (k) = K + ∑ K ∑ . top N i1...il + l=2 1≤a

5.5. Denef–Loeser Open String 4-Point Amplitudes

The open string 4-point zeta function on Ke is deﬁned as

( ) Z Z 4 (s) = |x |s12 |1 − x |s32 dx . Ke 2 Ke 2 Ke 2 Ke We divide the integration domain in sectors

( ) = ∈ | | ≤ ⇐⇒ ∈ = Sect I {xi Q : xi Ke 1 i I}, I j T {2, 3},

IT I Sect(I)

{2} ∅ RKe \ ∅ {2} Ke RKe .

In this way, we obtain that

(4) (4) (4) Z (s) = Z (s; {2}, 0) + pe(1+s12+s32)Z (s{2}, 1) Ke Ke Ke Z Z s − −s −s = |x | 12 |1 − x |s32 dx + pe(1+s12+s32) |x | 2 12 32 dx , 2 Ke 2 Ke 2 2 Ke 2 RKe RKe with −e e(−1−s ) −e e(−1−s ) ( ) (1 − p )p 12 (1 − p )p 32 Z 4 (s; {2}, 0) = 1 − 2p−e + + , Ke 1 − pe(−1−s12) 1 − pe(−1−s32) and

−e e(1+s +s ) ( ) (1 − p )p 12 32 Z 4 (s; {2}, 1) = . Ke 1 − pe(1+s12+s32) Taking the limit e approaches to zero,

(4) 1 1 Ztop(s; {2}, 0) = −1 + + s12 + 1 s32 + 1 and

(4) 1 Ztop(s; {2}, 1) = − . s12 + s32 + 1 Consequently

(4) 1 1 1 Ztop(s) = −1 + + − . s12 + 1 s32 + 1 s12 + s32 + 1 Symmetry 2021, 13, 967 30 of 53

2 Using the kinematic relations k1 + k2 + k3 + k4 = 0 and ki = 2 we obtain k1k2 + k3k2 + 1 = −1 − k2k4, then the Denef–Loeser string 4-point amplitude is given by

(4) 1 1 1 Atop(k) = −1 + + + . (54) k1k2 + 1 k3k2 + 1 k2k4 + 1

5.6. Denef–Loeser Open String 5-Point Amplitudes

The open string 5-point zeta function on Ke is given by

( ) Z Z 5 (s) = |x |s12 |x |s13 |1 − x |s42 |1 − x |s43 |x − x |s23 dx dx . Ke 2 Ke 3 Ke 2 Ke 3 Ke 2 3 Ke 2 3 2 Ke By dividing the integration domain in sectors as in Section 4.2, we obtain the results presented in Table2.

Table 2. Sectors of the 5-point Denef-Loeser amplitude.

IIc Sect(I) { }{ } × \ 2 3 RKe Ke RKe { }{ } \ × 3 2 Ke RKe RKe { } × 2, 3 ∅ RKe RKe { } \ × \ ∅ 2, 3 Ke RKe Ke RKe .

The open string 5-point topological zeta function is deﬁned as

(5) (5) (5) Ztop(s) = ∑ Ztop(s; I, 0)Ztop(s; T r I, 1). I⊆T

(5) (5) Table3 contains explicit formulae for all the integrals Ztop(s; I, 0) and Ztop(s; T r I, 1).

(5) (5) Table 3. Explicit calculation of Ztop(s; I, 0), Ztop(s; T r I, 1).

(5) (5) I Ztop(s; I, 0) Ztop(s; T r I, 1) {2} −1 + 1 + 1 − 1 1+s12 1+s42 1+s13+s43+s23

{3} −1 + 1 + 1 − 1 1+s13 1+s43 1+s12+s42+s23

h 1 1 1 i 1 1+s + 1+s + 1+s − 1 2+s +s +s 12 h 13 i 23 h 12 13 i23 + 1 1 − 1 + 1 1 − 1 {2, 3} 1+s12 1+s43 1+s13 1+s42 1 1 1 1 +2 − + − + − + + 1h s23 1 s42 1 s43 i 1 1 + 1 + 1 − 1 2+s42+s43+s23 1+s42 1+s43 1+s23 1 − + + + " 2 s52 s53 s23 # {∅} 1 1 + 1 × 1+s12+s42+s23 1+s13+s43+s23 + 1 − 1 1+s23

Therefore, the Denef–Loeser open string 5-point amplitude is given by

(5) 1 1 1 Atop(k) = 2 − ∑ + ∑ , i

where 1 1 ∑ 1 + kikj 1 + kkkl {i,j,k,l}⊂{1,2,3,4,5} i

(5) (5) Atop(k) = lim A (k) (55) p→1 1 1 1 =Ktop + Ktop + 5 4 ∑ + ∑{i,j,k,l}⊂{1,2,3,4,5} + + i

done using the explicit formula for given by the Feynman rules given in [14]. Here top top K5 = −2, K4 = −1.

5.7. Non-Archimedean Closed Strings In Archimedean string theory, it is known that closed strings can be produced from the scattering of open strings [55]. Therefore, for p-adic string theory, it is desirable to construct a p-adic closed string theory. This was first studied in [20], see also [4,67]. The usual Archimedean closed string is described by two coordinates (τ, σ) with the periodical condition σ0 = σ + 2π. That is, the string worldsheet is a cylinder, which can be conformally mapped to the whole complex plane C. In order to obtain tree-level scattering amplitudes we need to insert vertex operators on C. Using the SL(2, C) symmetry, we can fix three insertions points. As with the open string, it is conventional to fix three points to 0, 1 and ∞. The simplest non-trivial example is the 4-point tachyon scattering amplitude for closed strings, also known as the Virasoro–Shapiro amplitude, is

( ) Z Γ (−α(s)/2)Γ (−α(t)/2) A 4 (k) = |z|k1k2/4|1 − z|k1k3/4dz = C C , C C C ΓC(−α(s)/2 − α(t)/2) C D where ki ∈ C and kikj = −k0,ik0,j + ··· kl,ikl,j is the Minkowski product; |z|C := zz¯ is the square complex norm, α(x) = x/4 + 2, and Z Γ (s) := exp[2πi(z + z¯)]|z|s−1dz C C C is the Gelfand–Graev gamma function over C [4]. For the closed strings case, the momenta vectors satisfy N 2 ki = 8, ∑ ki = 0. (56) i=1 Symmetry 2021, 13, 967 32 of 53

This explains the factor of 1/4 in the exponents relative to the open strings case. The generalization of these amplitudes to N-points is given by

Z N−2 N−2 (N) k k /4 kikj/4 A (k) = |z |k1ki/4|1 − z | N−1 i |z − z | dz , (57) C ∏ i C i C ∏ i j C ∏ i i=2 2≤i

where k = (k1, ... , kN). As we can see, these amplitudes are very similar to the Koba– Nielsen amplitudes, except for the fact that they are being integrated over complex variables 1 and the replacement ki → 2 ki. To construct p-adic versions of Virasoro–Shapiro amplitudes, one can consider Qp as the analog of R, and a quadratic extension of Qp as the analog of C. However, this quadratic extension is not unique, and is not an algebraically closed ﬁeld. This naive approach is followed in [4,20]. This approach is not very useful here, because we work with open string amplitudes over any ﬁnite extension of Qp.

( ) Definition 2. Given an open string N-point amplitude A N (k) defined over a non-Archimedean K ( ) ( ) local field , we attach to it a Virasoro–Shapiro amplitude defined as A N (k) = A N ( 1 k), where K K2 K2 2 K2 is the unique unramified extension of K of degree 2.

(N) Then, by the results of [6], A (k) admits a meromorphic continuation as a rational K2 function in the variables q−2sij .

6. A Physical View of the Limit p → 1 on p-Adic String Amplitudes The problem of finding a relation between the physical string amplitudes and the p-adic one has been present since the initial proposal. One of the first attempts is that of the Adelic amplitudes which can be written as an infinite product of physical non-Archimedean amplitudes [14,21]. There are other possibilities to look for a link between Archimedean and Non-Archimedean amplitudes. One of them is to consider the limit p → 1. This limit is very intriguing and there are different interpretations of it [23,29,34]. For instance, in [23] it was argued that the analytic continuation of p to the complex numbers can be reflected in the Lagrangian and equations of motion by turning the non-local equations of motion into a local and linear dynamical description with a logarithmic potential. This is an ordinary theory with real physical amplitudes. In [29] it was argued that as similarly found in [23], the expansion of the effective action around p → 1 leads to a linear theory with logarithmic potential. Besides that, this theory is deeply related to the boundary string field theory proposed by Witten [30] in the context of the developing of a background independence string theory. The limit p → 1 also can be interpreted in terms of some scaling transformations of the renormalization group for the Bruhat–Tits tree [34]. As we have seen in this survey, the non-Archimedean nature is encoded in the worldsheet theory. In this context we must recall that p as a prime number, thus the analysis necessarily needs to be carry out in a rigorous way. The right way of taking the limit p → 1 involves the introduction of unramified finite extensions of the p-adic field Qp. In [9] the limit p → 1 was discussed for tree-level string amplitudes, by using the topological zeta functions introduced by Denef and Loeser [35,66]. We found that the limit p → 1 of p-adic string amplitudes leads to certain string amplitudes (which are rational functions) that we termed the Denef–Loeser open string amplitudes. These Denef–Loeser amplitudes for N = 4, 5 points were computed in Sections 5.5 and 5.6, for more details see [9]. On the other hand for N = 4, 5, in [9], we computed the limit p → 1 of the effective field theory of p-adic amplitudes, i.e., the Gerasimov and Shatashvili Lagrangian (61) involving a logarithmic potential. By computing the interacting generating functional at the tree level, we verified that the corresponding amplitudes coincide exactly with the Denef–Loeser Symmetry 2021, 13, 967 33 of 53

amplitudes. Based on this fact we have formulated a conjecture, which was already stated in the introduction of this survey. In what follows we review the evidence of the mentioned conjecture in the cases N = 4, 5 following [9]. First, we review some basic results from [29] starting with the effective action which is a ﬁeld theory whose perturbative analysis leads to the p-adic scattering amplitudes [14], it can be written as

2 Z 1 p D 1 − 1 1 p+1 S(φ) = d x − φp 2 ∆φ + φ , (58) g2 p − 1 2 p + 1

where ∆ is the Laplacian on M and g is the coupling constant. The corresponding Euler- Lagrange equation is − 1 p p 2 ∆φ = φ . (59) In the source space (and consequently in the amplitudes) p is a prime number; however, in the target space (and consequently in the Lagrangian and in the equation of motion) p is a real parameter. Thus, one can formally proceed to approach p to one and perform a 1 Taylor expansion at (p − 1) of the expression exp(− 2 ∆ log p) and exp(p log φ). Then the resulting equation of motion is given by

∆φ = −2φ log φ. (60)

This is the motion equation of the Gerasimov–Shatashvili action

Z S(φ) = dDx (∂φ)2 − V(φ) , (61)

2 µν where (∂φ) = η ∂µφ · ∂νφ and V(φ) is the potential

φ2 V(φ) = φ2 log . e We consider that action (61) is the limit p tends to one of the effective action (58). The action corresponding to the free theory with a source is written as Z D 2 2 S0(φ) = d x (∂φ) + φ (x) + J(x)φ(x) .

Now, the action (61) can be conveniently rewritten as Z S(φ) = dDx(∂φ)2 + m2 − U(φ), (62)

where U(φ) = 2φ2 log φ. Then the potential U(φ) can be expanded in Taylor series around the origin in the form

U(φ) = Aφ2 + Bφ3 + Cφ4 + Dφ5 + ··· , (63)

where A, B, C and D are constants and they are real numbers. Now, we briefly review the form of obtaining the four-point and five-point amplitudes from the Lagrangian (61)[29]. In quantum field theory N-point correlation functions of N local operators φb in N different points x1, x2,..., xN of M, can be written as

(−ih¯ )N δnZ[J] hT(φb(x1)φb(x2) ··· φb(xN))i = , Z[J] δJ(x1)δJ(x2) ··· δJ(xN) J=0 Symmetry 2021, 13, 967 34 of 53

where Z[J] is the generating functional constructed using interacting Lagrangian (62). The functional can be computed as

iB Z δ 3 Z[J] = exp − dDx − ih¯ h¯ δJ(x) iC Z δ 4 iD Z δ 5 − dDx − ih¯ − dDx − ih¯ + ··· Z [J], h¯ δJ(x) h¯ δJ(x) 0

where the generating functional of the correlation function for the free theory with sources is given by

i Z Z Z [J] = N [det(∆ − 1)]−1/2 exp − dDx dDx0 J(x)G (x − x0)J(x0) . 0 4¯h F

0 Moreover, GF(x − x ) is the Green-Feynman function of the differential operator (∆ − 1). For the computation of the interacting 4-point amplitudes can be obtained through the generating functional

Z δ 4 Z[J] = · · · − iCh¯ 3 dDx Z [J] + ··· . δJ(x) 0 In terms of Z[J], 4-point amplitudes implies the computation of

δ4Z[J] (64) δJ(x1)δJ(x2)δJ(x3)δJ(x4) J=0 Z i i = −4!iCh¯ 3 dDx − G (x − x ) − G (x − x ) 2¯h F 1 2¯h F 2 i i × − G (x − x ) − G (x − x ) 2¯h F 3 2¯h F 4 3iC Z = − dDx G (x − x ) G (x − x ) G (x − x ) G (x − x ). 2¯h F 1 F 2 F 3 F 4

Here GF(x − y) is the Green-Feynman propagator. This amplitude corresponds to the Feynman diagram with only one vertex and four external legs. following the notation of [14], we write for it the symbol K4. The contribution to the 4-point amplitudes of Bφ3 in (63) has a contribution at the second order of the expansion. The corresponding Feynman diagrams have two vertices at points x and y with two external legs, which are connected by a propagator GF(x − y). Thus, we have

B2h¯ 4 Z Z δ 3 δ 3 Z[J] = ··· + dDx dDy Z [J] + ··· . (65) 2 δJ(x) δJ(y) 0

Moreover, the contribution to the 4-point amplitudes from the term Cφ3 arises to second order. Thus, it yields

4 Z Z δ Z[J] 2 4 D D i = 18B h¯ d x d y − GF(x − y) δJ(x1)δJ(x2)δJ(x3)δJ(x4) J=0 2¯h

i i i i × − G (x − x ) − G (x − x ) − G (y − x ) − G (y − x ) 2¯h F 4 2¯h F 3 2¯h F 2 2¯h F 1 Symmetry 2021, 13, 967 35 of 53

i i i i + − G (x − x ) − G (y − x ) − G (x − x ) − G (y − x ) 2¯h F 4 2¯h F 3 2¯h F 2 2¯h F 1 i i i i + − G (x − x ) − G (y − x ) − G (y − x ) − G (x − x ) 2¯h F 4 2¯h F 3 2¯h F 2 2¯h F 1 i i i i + − G (y − x ) − G (y − x ) − G (x − x ) − G (x − x ) 2¯h F 4 2¯h F 3 2¯h F 2 2¯h F 1 i i i i + − G (y − x ) − G (x − x ) − G (y − x ) − G (x − x ) 2¯h F 4 2¯h F 3 2¯h F 2 2¯h F 1 i i i i + − G (y − x ) − G (x − x ) − G (x − x ) − G (y − x ) . (66) 2¯h F 4 2¯h F 3 2¯h F 2 2¯h F 1 The sum of expressions (64) and (66) constitutes the total amplitude which contains the sum of the three channels s, t and u. They are expected to be the p-adic 4-point amplitudes arising in the limit p → 1. Thus, in the Fourier space it is expected to be written schematically as

(4) GS GS AGS = K4 + ∑ xij , i

( ) where xGS is the propagator xGS = 1 in momentum space. Then A 4 agrees with ij ij kikj+1 GS (4) GS GS Atop(k) = −1 + ∑i

Z δ 5 Z[J] = · · · − Dh¯ 4 dDx Z [J] + ··· . δJ(x) 0 Thus, the corresponding 5-point amplitude is

5 Z δ Z[J] 4 D i = −5!Dh¯ d x − GF(x − x1) δJ(x1)δJ(x2)δJ(x3)δJ(x4)δJ(x5) J=0 2¯h

i i i i × − G (x − x ) − G (x − x ) − G (x − x ) − G (x − x ) . (67) 2¯h F 2 2¯h F 3 2¯h F 4 2¯h F 5 GS The contribution will be encoded in the symbol K5 . Another contribution come from the term Bφ3 × Cφ4. In the Fourier space the corresponding diagrams have precisely 2-vertices, 5 external legs attached to these vertices and one internal leg between the two vertices (see, Section 3 of [14]). This latter contribution to generating functional is written as

Z Z δ 3 δ 4 Z[J] = · · · − iBCh¯ 5 dDx dDy Z [J] + ··· . (68) δJ(x) δJ(y) 0

The 5-point amplitude yields

δ5Z[J]

δJ(x1)δJ(x2)δJ(x3)δJ(x4)δJ(x5) J=0 Z Z i i i = −iBC(12)2h¯ 5 dD x dDy − G (x − y) − G (x − x ) − G (x − x ) 2¯h F 2¯h F 5 2¯h F 4 i i i × − G (y − x ) − G (y − x ) − G (y − x ) + ··· 2¯h F 3 2¯h F 2 2¯h F 1 Symmetry 2021, 13, 967 36 of 53

i i i + − G (y − x ) − G (x − x ) − G (x − x ) 2¯h F 5 2¯h F 4 2¯h F 3 i i × − G (y − x ) − G (y − x ) + ··· . (69) 2¯h F 2 2¯h F 1 Moreover, in the computation of the 5-point amplitudes, the last contribution comes from the term

B3h¯ 6 Z Z Z δ 3 δ 3 δ 3 Z[J] = ··· + dD x dDy dDz · · Z [J] + ··· . (70) 3! δJ(x) δJ(y) δJ(z) 0

The diagrams associated with this expansion are diagrams with three vertices. Two of them have two external legs and the other leg is internal. The remaining leg has one external line and two internal lines attached. Then the total contribution is

5 3 6 Z Z Z δ Z[J] B h¯ a D D D i = d x d y d z − GF(x − y) δJ(x1)δJ(x2)δJ(x3)δJ(x4)δJ(x5) J=0 3! 2¯h

i i i i × − G (y − z) − G (y − x ) − G (z − x ) − G (z − x ) 2¯h F 2¯h F 5 2¯h F 4 2¯h F 3

i i × − G (x − x ) − G (x − x ) + ··· , (71) 2¯h F 2 2¯h F 1 where a is a constant. Summarizing, the 5-point amplitudes A5 obtained from Lagrangian (61) can be written schematically (in notation from [14]) by

(5) GS GS GS GS AGS = K5 + ∑ xij + ∑ xij xkl , GS {i,j,k,l}⊂{1,2,3,4,5} K4 i

(5) GS GS GS Atop(k) = 2 − ∑ xij + ∑ xij xkl i

GS GS up to the constants K5 , K4 , see (55). We now give a precise formulation of Conjecture2 announced in the Introduction. By using the Feynman rules for the effective Lagrangian given in [14], ! −kia ki −1 (N) − p − 1 p b A = K + ∑N 3K ∑ , Qp l=2 i1...il −k k −1 1≤a

where the constants K, Ki1...il are rational functions in p with rational coefﬁcients. Further- more, if N ≤ p + 1, the constant term K is not zero. Formally, we have (N) (N) top N−3 top 1 Atop = lim A = K + ∑ K ∑ . p→ Qp l=2 i1...il + 1 1≤a

(N) A = KGS + N−3KGS xGS , GS ∑l=2 i1...il ∑ iai 1≤a

xGS = 1 = G k − k where i i k k +1 cF ia ib . Notice that we are not asserting the existence of any a b ia ib top relation between the constants Ktop, K and the constants KGS, KGS . i1...il i1...il

7. p-Adic Open String Amplitudes Coupled to a B-Field with Chan–Paton Factors Strings propagating in a background with gauge fields was discussed many years ago [57]. In particular, incorporating a Neveu–Schwarz B field in the target space leads to a noncommutative effective gauge theory on the world-volume of D-branes [58]. In [7], the N-point p-adic string amplitudes, with Chan–Paton rules and a constant B-field (the Ghoshal–Kawano amplitudes) were studied, these amplitudes were introduced in [60]. The study was done by attaching twisted local multivariate zeta functions to the Ghoshal– Kawano amplitudes. In this section we discuss the meromorphic continuation of the Ghoshal–Kawano amplitudes, we also compute the 4 and 5 points amplitudes. Due to the need for a particular symmetry, in this section, we take p ≡ 3 mod 4.

7.1. The p-Adic Sign Function × A p-adic sign function is a multiplicative character of Qp that takes values in {±1}. 2 h ×i × We set Qp as the multiplicative subgroup of squares in Qp , i.e.

2 h ×i 2 × Qp = {a ∈ Qp; a = b for some b ∈ Qp }.

ε · Let ε ∈ {1, ... , p − 1} such that ( p ) = −1, where ( · ) is the Legendre symbol (see for instance the Appendix of [7]). For p 6= 2 we have

× h ×i2 Qp / Qp = {1, ε, p, εp}, this means that any nonzero p-adic number can be written uniquely as

2 2 × × h ×i x = τa , with a ∈ Qp and τ ∈ Qp / Qp . × For a ﬁxed τ ∈ {ε, p, εp}, and x ∈ Qp , we deﬁne the p-adic sign function

( 2 2 1 if x = a − τb for a, b ∈ Qp sgn (x) := (72) τ −1 otherwise.

The following is the list of all the possible p-adic sign functions:

p ≡ 1 mod 4 p ≡ 3 mod 4 ( ) ( ) sgn (x) = (−1)ord x sgn (x) = (−1)ord x ε ε (73) x0 ord(x) x0 sgnp(x) = p sgnp(x) = (−1) p ord(x) x0 x0 sgnεp(x) = (−1) p sgnεp(x) = p ,

see [68]. The function sgnτ is a multiplicative character, which means that

sgnτ(xy) = sgnτ(x)sgnτ(y). × Additionally sgnτ is a locally constant function in Qp , which means that sgnτ(x − y) = sgnτ(x) if |y|p < |x|p. We need the symmetry sgnτ(−y) = −sgnτ(y) (i.e. sgnτ(−1) = −1), this requires × p ≡ 3 mod 4 and τ ∈ {p, εp}. Finally, for any x ∈ Qp , we deﬁne the p-adic Heaviside step function as Symmetry 2021, 13, 967 38 of 53

 1 if sgn (x) = 1 1  τ H (x) = (1 + sgn (x)) = τ 2 τ  0 if sgnτ(x) = −1.

7.2. A Class of Twisted Multivariate Local Zeta Functions Let f := ( f1,..., fm), χ := (χ1,..., χm) be vectors of non-constant polynomials and multiplicative characters, respectively. And s := m (s1,..., sm) ∈ C . The twisted multivariate local zeta functions have the form:

Z m n si ZΘ(s, χ, f ) = Θ(x) ∏ χi(ac( fi(x)))| fi(x)|p ∏dxi, (74) i= i= n ∪m −1 1 1 Qpr i=1 f (0)

with Re(si) > 0 for all i, ac stands for the angular component and Θ(x) is a test function. Integrals of type (74) are holomorphic functions in s, which admit meromorphic continua- − − tions as rational functions in the variables p s1 , ... , p sm to the whole Cm, Théorème 1.1.4. of [36], see also [26]. The case when χi is the trivial character has been studied previously, see e.g., Lemma 8.2.1 of [26]. The case when χi = sgnτ is new. Using Hironaka’s resolution of singularities theorem [50], in [7] we show that ZΘ(s, χ, f ) admits a meromorphic continuation as a rational function in the variables p−s1 ,..., p−sm . More precisely,

LΘ,χ(s) ZΘ(s, χ, f ) = , (75) − m N s −v ∏ 1 − p ∑i=1 i,j j j j∈T

−s −sm where LΘ,χ(s) is a polynomial in the variables p 1 , ... , p , and the real parts of its poles belong to the ﬁnite union of hyperplanes

m ∑ Ni,j Re(si) + vj = 0, for j ∈ T . i=1 This result is a variation of Théorème 1.1.4. in [36].

7.3. The Ghoshal–Kawano Local Zeta Function In [60] Ghoshal and Kawano proposed the following amplitude (for the N-point tree-level, p-adic open string amplitude, with Chan–Paton rules in a constant B-ﬁeld):

Z N−2 (N) k1ki kN−1ki A (k, θ, τ; x1, xN−1) := ∏ |xi|p |1 − xi|p Hτ(xi)Hτ(1 − xi) N−3 i=2 Qp rΛ k k × x − x i j H x − x ∏ i j p τ i j 2≤i

where N ≥ 4, k = (k1,..., kN), ki = (k0,i,..., kl,i), i = 1, ... , N, is the momentum vector of the i-th tachyon (with Minkowski product kikj = −k0,ik0,j + k1,ik1,j + ··· + kl,ikl,j), θ is a N−2 ﬁxed antisymmetric bilinear form that is built with the inverse of the B-ﬁeld; ∏i=2 dxi is N−3 the normalized Haar measure of Qp , Symmetry 2021, 13, 967 39 of 53

( N−2 ) N−3 Λ := (x2,..., xN−2) ∈ Qp ; ∏ xi(1 − xi) ∏ xi − xj =0 , i=2 2≤i

N ∑ ki = 0, kiki = 2 for i = 1, . . . , N. (77) i=1

To preserve the symmetry under the exchange of external momentum vectors, we require that sgnτ(xi − xj) = −sgnτ(xj − xi), or equivalently sgnτ(−1) = −1. Then for the rest of this section we assume that τ ∈ {p, εp}. To simplify the notation, we introduce the variables sij ∈ C, and s˜ij ∈ R for 1 ≤ i < j ≤ N − 1. We also set

− N 2 s s1i (N−1)i F(x, s, τ) := ∏ |xi|p |1 − xi|p Hτ(xi)Hτ(1 − xi) i=2 sij × ∏ |xi − xj|p Hτ(xi − xj), 2≤i

and ( √ !) − −1 E(x,es, τ; x1, xN−1) := exp ∑ esijsgnτ(x1 − xj) . (78) 2 1≤i

Z N−2 (N) Z (s,es, τ; x1, xN−1) = F(x, s, τ)E(x,es, τ; x1, xN−1) ∏ dxi. (79) N−3 i=2 Qp rΛ

(N) Z (s,es, τ; x1, xN−1) Z N−2 N−2 Re(s ) Re(s(N−1)i) Re(s ) ≤ |x | 1i | − x | x − x ij dx ∏ i p 1 i p ∏ i j p ∏ i N−3 i=2 2≤i

where Z(N)(s) is the Koba–Nielsen string amplitude studied in [6,8]. By applying the ( ) results of [6,8], integral Z N (Re(s)) is holomorphic in an open set K ⊂ CD, therefore

(N) Z (s,es, τ) is holomorphic in s ∈ K for any es, τ, x1, xN−1.

It is important to notice that if any of the integration variables is in Qp r Zp, then due to the local constancy of sgnτ, the factor Hτ(x)Hτ(−x) appears in F(x, s, τ), and Hτ(x)Hτ(−x) = 0. For this reason, we redeﬁne the Ghoshal–Kawano local zeta function as

Z N−2 (N) Z (s,es, τ; x1, xN−1) := F(x, s, τ)E(x,es, τ; x1, xN−1) ∏ dxi. (80) N−3 i=2 Zp Symmetry 2021, 13, 967 40 of 53

7.4. Ghoshal–Kawano Amplitude as a Local Zeta Function For s˜ ∈ R we have ( √ ) − −1s s √ s exp esgn (x) = cos e − −1 sgn (x) sin e . (81) 2 τ 2 τ 2

Using this identity and the convention x1 = 0, xN−1 = 1, we have

E(x,es, τ) = ∑ EI,J,K(es) ∏sgnτ(xj) ∏sgnτ(1 − xj) ∏ sgnτ(xi − xj). (82) I,J,K j∈I j∈J i,j∈K

In a similar way, we obtain that

N−2 ∏ Hτ(xi)Hτ(1 − xi) ∏ Hτ(xi − xj) i=2 2≤i

= ∑ eI,J,K∏sgnτ(xj) ∏sgnτ(1 − xj) ∏ sgnτ(xi − xj), (83) I,J,K j∈I j∈J i,j∈K

Using Formulae (82) and (83), and assuming x1 = 0, xN−1 = 1, xN = ∞, we can write the Ghoshal–Kawano zeta function as a ﬁnite sum of integrals of type

− N 2 s R s1i (N−1)i sij C(es) ∏ |xi|p |1 − xi|p ∏ |xi − xj|p ∏ χτ(xj) ∏ χτ(1 − xj) N−3 = ≤ < ≤ − j∈I j∈J Zp i 2 2 i j N 2 N−2 × ∏ χτ(xi − xj) ∏ dxi, i,j∈K i=2

where C(es) is an R-analytic function and χτ is a trivial character or sgnτ. This expression is a local zeta functions of the type (74), which implies that amplitudes (76) are equal to a ﬁnite sum of local zeta functions and consequently they admit a meromorphic continuation in the kinematic parameters. Finally, we point out that the meromorphic continuation of the Ghoshal–Kawano local zeta function (80) is also valid without taking the normalization x1 = 0, xN−1 = 1, xN = ∞, see Section V-C of [7].

(4) 7.5. Explicit Computation of Z (s,es, τ) We ﬁrst notice that the exponential factor E(4)(s˜, τ) can be taken outside the integral, i.e., the four-point Ghoshal–Kawano zeta function is

i Z(4)(s, s, τ) = exp (s + s + s ) e 2 e13 e12 e23 R s12 s32 × |x2|p |1 − x2|p Hτ(x2)Hτ(1 − x2)dx2. Zp

The computation of this integral is based on the calculation of the following integrals: assume that S ⊂ Zp r {0} satisﬁes −S = S, then

R s12 |x2|p sgnτ(x2)dx2 = 0, for τ ∈ {p, εp}, S and

R s12 s32 I(s, τ) = |x2|p |1 − x2|p Hτ(x2)Hτ(1 − x2)dx2 Zp p − 3 p−1−s12 1 − p−1 p−1−s32 1 − p−1 = + + . 4p 2(1 − p−1−s12 ) 2(1 − p−1−s32 ) Symmetry 2021, 13, 967 41 of 53

For further details, the reader may consult [7]. The explicit expression for the 4-point amplitude, also reported in [60], is

i Z(4)(s, s, τ) = exp (s + s + s ) e 2 e13 e12 e23 ! p − 3 p−1−s12 1 − p−1 p−1−s32 1 − p−1 × + + 4p 2(1 − p−1−s12 ) 2(1 − p−1−s32 )

which is holomorphic in

Re(s12) > −1 and Re(s32) > −1.

(5) 7.6. Explicit Computation of Z (s,es, τ) After some simple considerations involving sign functions, Z (5) (5) (5) (5) Z (s, s, τ) = E (s)L(s, τ) := E (s) F (x2, x3, s, τ)dx2dx3. e e e 2 Zp where

(5) s12 s13 s42 s43 s23 F (x2, x3, s, τ) = |x2|p |x3|p |1 − x2|p |1 − x3|p |x2 − x3|p

× Hτ(x2)Hτ(x3)Hτ(1 − x2)Hτ(1 − x3)Hτ(x2 − x3).

and ( √ ) −1 E(5)(s) = exp (s + s + s + s + s˜ + s˜ ) , e 2 e14 e12 e13 e24 34 32

see [7] for further details. To compute the integral L(s, τ), we subdivide the domain of integration as

p−1 2 G Zp = i + pZp × j + pZp , i,j=0 then L(s, τ) becomes

p−1 p−1 Z (5) L(s, τ) = ∑ Lij(s, τ) =: ∑ F (x2, x3, s, τ)dx2dx3. i,j=0 i,j=0 i+pZp×j+pZp

The computation of the integrals Lij(s, τ) is a simple but technical calculation, see [7] for further details. The explicit expression for the 5-point Ghoshal–Kawano zeta function is

(p − 3)(p − 7) p − 3 p−1−s23 (1 − p−1) Z(5)(s, s) = E(5)(s) + e e 32p2 8p (1 − p−1−s23 ) − 2 p−1−s13 (1 − p−1) p−1−s42 (1 − p−1) 1 − p 1 p−2−s13−s42 + + + (1 − p−1−s13 ) (1 − p−1−s42 ) 4 (1 − p−1−s13 )(1 − p−1−s42 ) " # 1 p−s12−s13−s23−2 1 − p−1 1 3 p−1−s23 1 − p−1 p−1−s13 1 − p−1 + − + + 4 1 − p−s12−s13−s23−2 2 2p 1 − p−1−s23 1 − p−1−s13 " #) 1 p−s42−s43−s23−2 1 − p−1 1 3 p−1−s23 1 − p−1 p−1−s42 1 − p−1 + − + + . 4 1 − p−s42−s43−s23−2 2 2p 1 − p−1−s23 1 − p−1−s42 Symmetry 2021, 13, 967 42 of 53

This function is holomorphic in

Re(s13) > −1; Re(s23) > −1; Re(s42) > −1;

Re(s12 + s13 + s23) > −2; Re(s42 + s43 + s23) > −2.

8. Resolution of Singularities and Multivariate Igusa Zeta Functions 8.1. Local Fields We take K to be a non-discrete locally compact field of characteristic zero. Then K is R, C, or a finite extension of Qp, the field of p-adic numbers. If K is R or C, we say that K is an R-field, otherwise we say that K is a p-field. ∈ | | For a K, we define the modulus a K of a by

 the rate of change of the Haar measure in (K, +) under x → ax   for a 6= 0, | | = a K   0 for a = 0.

| | = | | | | = | |2 |·| is well-known that if K is an R-ﬁeld, then a R a and a C a , where denotes the |·| usual absolute value in R or C, and, if K is a p-ﬁeld, then K is the normalized absolute value in K.

8.2. K-Analytic Manifolds and Resolution of Singularities We review the basic definitions of K-analytic manifolds following Igusa’s book [26]. Let K be a local field of characteristic zero, and let V be a nonempty open subset of Kn and let f : V → K be a function. We say that f is a K-analytic function on V, if for every point a = (a1, ... , an) ∈ V, there exists an element fa(x) of Khhx − aii = Khhx1 − a1, ... , xn − anii (the ring of convergent power series around a) such that f (x) = fa(x) for m any point x near a. A map g = (g1, ... , gm) : V → K is called K- analytic mapping on V if each gi is an analytic function on V. Let X Hausdorff space and let n be a fixed non- negative. Let U be a nonempty open subset of X, if φU : U → φU(U) is a homeomorphism, n where φU(U) is a nonempty open subset of K , then we say that the pair (U, φU(U)) is a chart. For a variable point x ∈ U, φ(x) = (x1, ... , xn) are called the local coordinates of x. A collection of charts {U, φU} is called an atlas of X if the union of all open sets V is X and 0 0 for every U, U such that U ∩ U 6= ∅ the map

−1 0 0 φU0 ◦ φU : φU(U ∩ U ) → φU0 (U ∩ U ) is K- analytic. There is an equivalence relation over the set of atlases on X. Two atlases are equivalent if their union is also an atlas. Thus, any equivalence class is called an n-dimensional K-analytic structure on X. Hence, we say that X is a K-analytic manifold and we write dim(X) = n. Suppose that X and Y are two K- analytical manifolds defined by the atlases {(U, φU(U))} and {(V, ψV (V))}, respectively, and f : X → Y a map. If for every U, V such that − U ∩ f 1(V) 6= ∅, the map

−1 −1 dim(Y) ψV ◦ f ◦ φU : φU(U ∩ f (V)) → K is K- analytic, then f is called a K-analytic map. If Y = K we say that f is a K-analytic function on X. This deﬁnition does not depend on the choice of atlases into the equivalence class. Let X be a K-analytic manifold deﬁned by an atlas {(U, φU)} and let Y be a nonempty 0 0 open set of X. If U = U ∩ Y 6= ∅ we put φU0 = φU0 = φU|U0 . Then {(U , φU0 )} is an atlas for Y. Hence, Y is an open K- analytic submanifold of X. Furthermore, dim(Y) = dim(X). Symmetry 2021, 13, 967 43 of 53

Now let Y be a nonempty closed subset of a K-analytic manifold X of dimension n. Suppose that X is defined by an atlas {(U, φU)} with the following property: If φU(x) = 0 (x1, ... , xn) and U = Y ∩ U 6= ∅, there exist F1, ... , Fm K-analytic functions on U with 0 0 < m ≤ n such that U becomes the set of all x in U such that F1(x) = ... = Fm(x) = 0 and " # ∂F det i (a) 6= 0, ∂xj 1≤i≤m 1≤j≤m for every a ∈ U0. By the implicit function theorem, the mapping

x 7→ (F1(x),..., Fm(x), xm+1,..., xn) gives a K-bianalytic map from a neighborhood of a in U to its image in Kn. If we denote by V the intersection of such neighborhood and Y, and put ψV (x) = (xm+1, ... , xn) for every x ∈ V. Then (V, ψV ) for all V and for each U as above, gives an atlas on Y. Thus, Y becomes a K-analytic closed submanifold of X of dimension n − m. Assume that U and V are nonempty open sets that contain a point a of X, and let f , g be two K- analytic functions respectively on U, V such that f |W = g|W for some nonempty open set W such that a ∈ W ⊆ U ∩ V. Then we say that f and g are equivalent. An equivalence class is called a germ of analytic function at a. The set of such equivalence classes becomes a local ring OX,a with maximal ideal mX,a = { f ∈ OX,a : f (a) = 0}. In addition, K ⊆ OX,a.

9. Multivariate Igusa Zeta Functions Let K be a local field. If K is a p-field, resp. an R-field, we denote by D(Kn) the C-vector space consisting of all C-valued locally constant functions, resp. all smooth functions, on Kn, with compact support. An element of D(Kn) is called a test function. Let fi(x) ∈ K[x1,..., xn] be a non-constant polynomial for i = 1, ... , m. We set m f = ( f1,..., fm) and s = (s1,..., sm) ∈ C . Let Φ be a test function. The multivariate local zeta function attached to ( f, Φ) is defined as

Z m n Z ( f, s) = Φ(x) | f (x)|si dx , when Re(s ) > 0 for all i. (84) Φ ∏ i K∏ i i n i=1 i=1 K rDK

m −1 where DK := ∪i=1 fi (0) is the divisor attached to fi(x), i = 1, ... , m. In the multivariate case i.e., for m ≥ 1, the local zeta functions over local ﬁelds of zero characteristic were studied by Loeser [36]. In the case of zero characteristic, the main tool to show the existence of a meromorphic continuation of the multivariate local function is the Hironaka’s resolution of singularities theorem. By applying this theorem to the divisor DK, the mutivariate local zeta function is reduced to the case of monomial integrals [25,26,36]. Currently, the methods used by Igusa are not available in positive characteristic, so the problem of the meromorphic continuations in this setting it is still an open problem.

Theorem 2 (Hironaka, [50]). Let K be a local ﬁeld of characteristic zero. There exists an embedded n resolution σ : X → K of DK, i.e., (i) X is an n-dimensional K-analytic manifold, σ is a proper K-analytic map which is a composition of a ﬁnite number of blow-ups at closed submanifolds, and which is an isomorphism outside −1 of σ (DK); −1 −1 (ii) σ (DK) is a normal crossings divisor, meaning that σ (DK) = ∪i∈T Ei, where the Ei are closed submanifolds of X of codimension one, each equipped with an m-tuple of non-negative integers Nf1,i,..., Nfm,i and a positive integer vi, satisfying the following. At every point b of X Symmetry 2021, 13, 967 44 of 53

there exist local coordinates (y1,..., yn) on X around b such that if E1, ... , Er are the Ei containing b, we have on some open neighborhood V of b that Ei is given by yi = 0 for i ∈ {1, . . . , r},

r ! ∗ vi−1 σ (dx1 ∧ ... ∧ dxn) = η(y) ∏ yi dy1 ∧ ... ∧ dyn, (85) i=1

and r Nf ,i f ∗(y) = f ◦ (y) = (y) y j for j = m j : j σ ε fj ∏ i , 1, . . . , , (86) i=1 ( ) ( ) O× where η y and the ε fj y belong to X,b, the group of units of the local ring of X at b.

The Hironaka resolution theorem allows expressing a multivariate local zeta function as a linear combination of monomial integrals, through a ﬁnite sequence of changes of variables. We have the following theorem on multivariate local zeta functions over local ﬁelds of zero characteristic.

Theorem 3. (Lemma 6.4, Remark 2 of [6]) Let f1(x), ... , fm(x) ∈ K[x1,..., xn] be non- constant polynomials and Φ : Kn → C a smooth function with compact support, to which we associate the multivariate local zeta function ZΦ( f, s). Fix an embedded resolution σ : X → K of m −1 DK = ∪i=1 fi (0) as in Theorem2 . Then (i) ZΦ( f, s) is convergent and deﬁnes a holomorphic function in the region

m N (s ) + v > for i ∈ T ∑ fj,i Re j i 0, ; j=1 m (ii) ZΦ( f, s) admits a meromorphic continuation to the whole C , with poles belonging to

( m ) [ [ N s + v + t = ∑ fj,i j i 0 i∈T t∈N j=1 1 with t ∈ N if K = R and t = 2 N if K = C, and with poles belonging to

( m ) [ ∑ aj,isj + bi = 0 , 1≤i≤r j=1 in the p-ﬁeld case. In addition, in the p-adic case the multivariate local zeta function has a meromorphic continuation as a rational function

P (s) Z ( f, s) = Φ Φ   m  − N s +v  ∑ fj,i j i  =  ∏ 1 − q j 1  i∈T 

−s −sm −s in q 1 ,..., q , where PΦ(s) is a polynomial in the variables q i .

10. Meromorphic Continuation of Koba–Nielsen Amplitudes Defined on Local Fields of Characteristic Zero The Koba–Nielsen open string amplitudes for N-points over a local field K of characteristic zero are defined as

Z N−2 N−2 (N) k1kj kN−1kj kikj A (k) := x 1 − x x − x dx , (87) K ∏ j K j K ∏ i j K ∏ i i=2 2≤i

l+1 where k = (k1,..., kN), ki = (k0,i,..., kl,i) ∈ C , for i = 1, ... , N, is the momentum vector of the i-th tachyon (with Minkowski product kikj = −k0,ik0,j + k1,ik1,j + ··· + kl,ikl,j), obeying

N ∑ ki = 0, kiki = 2 for i = 1, . . . , N. (88) i=1

The parameter l is typically taken to be 25. However, we do not require using the critical dimension, thus l can be any positive number. These amplitudes were introduced by Brekke, Freund, Olson and Witten, among others in their works about string amplitudes, see e.g., Section 8 of [4]. In the case N = 4 and K = R, the amplitude (87) is the Veneziano amplitude, see [10]. In Section 2 of [48] and [49], it was established that the N-point closed (N) string amplitude at the tree level is the product of A (k) times a polynomial function C in the momenta k. Hence, the results established in [6] are still valid for closed string amplitudes at tree level. If we take the Minkowski products of the kinematic parameters as follows kikj = sij ∈ C, the string amplitude (87) becomes to the following integral which is a type of multivariate local zeta function

Z N−2 N−2 (N) s1j s(N−1)j sij Z (s) := x 1 − x x − x dx , (89) K ∏ j K j K ∏ i j K ∏ i N−3 i=2 2≤i

N−2 N−3 D N(N−3) where ∏i=2 dxi is the normalized Haar measure on K , s := sij ∈ C , with D = 2 denotes the total number of indices ij, and

( N−2 N−2 ) N−3 DN := x ∈ K ; ∏ xi ∏ (1 − xi) ∏ xi − xj = 0 . i=2 i=2 2≤i

Z N−2 N−2 N−2 (N) s1j s(N−1)j sij ZI (s) := ϕI (x) ∏ xj ∏ 1 − xj ∏ xi − xj ∏ dxi, (91) j=2 j=2 2≤i

where the functions {ϕI } are deﬁned as

N−3 ϕI : R → R; x 7→ ∏χ(xi)∏(1 − χ(xi)), (92) i∈I i∈/I

for I ⊆ {2, . . . , N − 2}, including the empty set, with the convention that ∏i∈∅· ≡ 1. ∞ N−3 N−3 Notice that ϕI ∈ C R and ∑I ϕI (x) ≡ 1, for x ∈ R , i.e., the functions {ϕI } form a partition of the unity. (N) In the case I = {2, . . . , N − 2}, ZI (s) is a classical multivariate Igusa local zeta function (since ϕI (x) has compact support). It is well known that these integrals are holomorphic functions in a region including Re sij > 0 for all ij, and they admit meromorphic continuations to the whole CD, see ([6], Theorem 3.2). In the case I 6= {2, . . . , N − 2}, by changing variables in (91) as x → 1 for i 6∈ I, i xi − − and x → x for i ∈ I, we have ∏N 2dx → ∏ 1 ∏N 2dx , and by setting χ(x ) := i i i=2 i i6∈I |x |2 i=2 i e i i 1 − χ 1 for i 6∈ I, i.e., xi  |x | ≤ 1  1 if i 2+e χe(xi) =  1 0 if |xi| ≥ 2 ,

h 1 1 i ∞ we have that supp χe ⊆ − 2 , 2 and χe ∈ C (R). Now setting ϕeI (x) := ∏i6∈I χe(xi) ∏i∈I χ(xi), and

N−2 s1j s(N−1)j sij FI (x, s) := ∏ xj ∏ 1 − xj ∏ xi − xj j∈I j=2 2≤i

we have Z N−2 (N) ϕeI (x)FI (x, s) ZI (s) = dxi, (93) s1i+s(N−1)i+∑2≤j≤N−2 sij+2 ∏ N−3 i=2 R rDI j6=i ∏ |xi| i6∈I

where DI is the divisor deﬁned by the polynomial

N−2 N−2 ∏ xi ∏ (1 − xi) ∏ xi − xj ∏ xi − xj i=2 i=2 2≤i

(N) The integrals ZI (s), with I 6= {2, . . . , N − 2}, are not classical multivariate local zeta functions thus we do not apply the theory of local zeta functions. Thus, in [6] we show that they deﬁne holomorphic functions on some nonempty open in CD, and admit meromorphic continuations to the whole CD.

(N) Lemma 1. (Lemma 4.1 of [6]) For any I ⊆ {2, . . . , N − 2}, the function ZI (s) is holomorphic in s on the solution set H(I) of a system of inequalities of the form    D  H(I) := sij ∈ C ; ∑ Nij,k(I) Re sij + γk(I) > 0, for k ∈ T(I) 6= ∅, (94)  ij∈M(I)  Symmetry 2021, 13, 967 47 of 53

where Nij,k(I), γk(I) ∈ Z, and M(I), T(I) are ﬁnite sets. More precisely, for each k, either all numbers Nij,k(I) are equal to 0 or 1 and γk(I) > 0, or all numbers Nij,k(I) are equal to 0 or −1 and γk(I) < 0. (N) D In addition, ZI (s) admits an analytic continuation to the whole C , as a meromorphic function with poles belonging to   [ [  D  P(I) := sij ∈ C ; ∑ Nij,k(I)sij + γk(I) + t = 0 . (95) t∈N k∈T(I) ij∈M(I) 

To show the existence of the meromorphic continuation of Z(N)(s) it is necessary that all (N) the integrals ZI (s) be holomorphic in a common domain, and then Formula (90) allows (N) us to construct a meromorphic continuation of Z (s). We show that ∩IH(I) contains a nonempty open subset of CD by studying the possible poles of integrals

(N)( ) = (N)( ) | ZI s : ZI s sij=s (N) (N) for any I, and proving that Z (s) := Z (s) |sij=s is a holomorphic function in the region 2 2 − < Re(s) < − , N − 2 N This fact was proved in Theorem 4.1 in [6]. Furthermore, the Koba–Nielsen local zeta function Z(N)(s) is convergent and holomorphic in the region determined by the following inequalities:

Re(sij) > −1

N(N−3) for all 2 variables sij,

∑ Re(s1j) + ∑ Re(sij) > −#J j∈J i,j∈J i

∑ Re(s(N−1)j) + ∑ Re(sij) > −#J j∈J i,j∈J i

∑ Re(sij) > −#J + 1 i,j∈J i

∑ Re(s1j) + ∑ Re(s(N−1)j) + ∑ Re(sij) + ∑ Re(sij) < −#J j∈J j∈J i∈{2,...,N−2}\J i,j∈J j∈J i

2 2 − < Re(s ) < − , N − 2 ij N for all indices ij. As in the p-adic case, we use the meromorphic continuation of (89) to the whole CD, (N) (N) which is denoted by Z (s), as regularizations of the amplitudes A (k) by redeﬁning K K Symmetry 2021, 13, 967 48 of 53

(N) (N) A (k) = Z (s) | , K K sij=kikj (N) see Theorem 6.1 of [6]. It is important to mention here that in the regularization of A (k), K we do not use the kinematic restrictions (88). (N) Furthermore, in [6] we show that A (k) converges on some open subset of (N−1)(l+1) K C (N) by showing that this open is mapped into the domain of convergency of Z (s) by K (N) k → s = k k . In addition, we prove that A (k) extends to a meromorphic func- ij i j K ( − )( + ) tion to the whole C N 1 l 1 , and that its polar set is contained in the inverse image of the (N) polar set of Z (s) under that mapping, where the possible poles are described in terms of K numerical data of suitable resolutions of singularities:

(N) Theorem 4. (Theorem 7.1 of [6]) Let be a local ﬁeld of characteristic zero. The integral A (k) K K − ( + ) converges and is holomorphic in the open set U ⊂ C(N 1) l 1 . It extends to a meromorphic − ( + ) function in k on the whole C(N 1) l 1 . ( ) If is an -ﬁeld, then the possible poles of A N (k) belong to K R K

  [ [ [  (N−1)(l+1) t  k ∈ C ; Nij,r(I)kikj + γr(I) + = 0 , (96) ∑ [K : R] I⊆{2,...,N−2} t∈N r∈T(I) ij∈M(I) 

where Nij,r(I), γr(I) ∈ Z, and M(I), T(I) are ﬁnite sets, and [K : R] = 1 if K = R, and [K : R] = (N) 2 if = . If is a p-adic ﬁeld, then A (k) is a rational function in the variables q−kikj , and its K C K K possible poles belong to   [ [  (N−1)(l+1)  k ∈ C ; ∑ Nij,r(I) Re(kikj) + γr(I) = 0 . I⊆{2,...,N−2} r∈T(I) ij∈M(I) 

More precisely, for each r, either all numbers Nij,r(I) are equal to 0 or 1 and γr(I) > 0, or all numbers Nij,r(I) are equal to 0 or −1 and γr(I) < 0.

(N) We now show explicitly the existence of a meromorphic continuation for Z (s) in R the cases N = 4, 5, by using Hironaka’s resolution of singularities theorem.

10.1. Example: 4-Point Koba–Nielsen String Amplitude The 4-point Koba–Nielsen open string amplitude is deﬁned as Z (4) s12 s32 Z (s) = |x2| |1 − x2| dx2. R Using the function

 ∈ −  1 if x2 [ 2, 2] χ(x2) = (97)  0 if x2 ∈ (−∞, −2 − e] ∪ [2 + e, +∞),

where e > 0 is sufﬁciently small, we construct the following partition of the unity:

ϕ{ } : R → R; x2 7→ χ(x2) 2 . ϕ∅ : R → R; x2 7→ 1 − χ(x2) ∞ Notice that ϕ∅, ϕ{2} ∈ C (R) and ϕ{2}(x) + ϕ∅(x) ≡ 1, for x ∈ R. Hence, Symmetry 2021, 13, 967 49 of 53

(4)( ) = (4) ( ) + (4)( ) Z s Z{2} s Z∅ s ,

with Z (4) ( ) = ( )| |s12 | − |s32 Z{2} s χ x2 x2 1 x2 dx2 (98) R and Z (4) ( ) = ( − ( ))| |s12 | − |s32 Z∅ s 1 χ x2 x2 1 x2 dx2. (99) R

The integral (98) is a multivariate local zeta function since χ(x2) has compact support. (4) ( ) ( ) > − By a classical result of local zeta functions Z{2} s converges when Re s12 1 and 2 Re(s32) > −1. Furthermore, it admits a meromorphic continuation to the whole C , see Theorem 3.2 of [6]. The second integral (99) is not a multivariate local zeta function but it can be trans- 1 1 formed in one by changing of variables as x2 → , then dx2 → 2 dx2, and by setting x2 |x | 2 χ(x ) := 1 − χ 1 , we have that e 2 x2  |x | ≤ 1  1 if 2 2+e χe(x2) = .  1 0 if |x2| ≥ 2 ,

h 1 1 i ∞ Notice that the support of χe is contained in − 2 , 2 and χe ∈ C (R). Thus, integral (99) becomes

( ) Z 4 −s12−s32−2 s32 Z∅ (s) = χe(x2)|x2| |1 − x2| dx2, R

which is analytic when − Re(s12) − Re(s32) − 1 > 0 and Re(s32) + 1 > 0. We concluded that Z(4)(s) is analytic in the region

Re(s12) > −1, Re(s32) > −1, Re(s12) + Re(s32) < −1. 1 1 Which contains the open set −1 < Re(s12) < − 2 and −1 < Re(s32) < − 2 .

10.2. Example: 5-Point Koba–Nielsen String Amplitude Fix N = 5. By using the function χ(x), see (97), we deﬁne the following partition of the unity:

2 ϕ{2,3} : R → R; (x2, x3) 7→ χ(x2)χ(x3), 2 ϕ{2} : R → R; (x2, x3) 7→ χ(x2)(1 − χ(x3)), 2 ϕ{3} : R → R; (x2, x3) 7→ χ(x3)(1 − χ(x2)), 2 ϕ∅ : R → R; (x2, x3) 7→ (1 − χ(x2))(1 − χ(x3)). (5)( ) = (5) ( ) + (5) ( ) + (5) ( ) + (5)( ) Then Z s Z{2,3} s Z{3} s Z{2} s Z∅ s . We consider the ﬁrst integral Z (5) ( ) = ( ) ( )| |s12 | |s13 | − |s42 | − |s43 | − |s23 Z{2,3} s χ x2 χ x3 x2 x3 1 x2 1 x3 x2 x3 dx2dx3. (100) R2

Since ϕ{2,3} has compact support, then integral (100) is a local zeta function. Thus, we use resolution of singularities of the divisor D5 defined by x2x3(1 − x2)(1 − x3)(x2 − x3) = 0. This arrangement is not locally monomial only at the points (0, 0) and (1, 1). Hence, we 2 pick a partition of the unity, ∑i=0 Ωi(x2, x3) = 1, and we write

2 ( ) ( ) Z 5 (s) = Z 5 (s), {2,3} ∑ Ωj j=0 Symmetry 2021, 13, 967 50 of 53

where

( ) Z 5 s12 s13 s42 s43 s23 ZΩ (s) = Ωj(x2, x3)|x2| |x3| |1 − x2| |1 − x3| |x2 − x3| dx2dx3. j R2

We assume that Ω0 and Ω1 are smooth functions with support in a small neighborhood ( ) ( ) of (0, 0) and (1, 1), respectively. So, we only compute the integrals Z 5 (s) and Z 5 (s). Ω0 Ω1 s s In terms of convergence and holomorphy, around (0, 0), the factor |1 − x2| 42 |1 − x3| 43 can be neglected, hence we only need an embedded resolution of x2x3(x2 − x3) = 0, which is obtained by a blow-up at the origin. It means to make the changes of variables

2 2 σ0 : R → R ; u2 7→ x2 = u2 u3 7→ x3 = u2u3.

( ) Hence integral Z 5 (s) becomes Ω0 Z s12+s13+s23+1 s13 s23 (Ω0 ◦ σ0)(u2, u3)|u2| |u3| |1 − u3| g(u, s)du2du3, R2

where g(u, s) is invertible on the support of Ω0 ◦ σ0 and can thus be neglected from the point of view of convergence and holomorphy. Hence, by ([6], Lemma 3.1), we obtain the convergence conditions:

Re(s12) + Re(s13) + Re(s23) + 2 > 0, Re(s23) + 1 > 0, Re(s13) + 1 > 0. (101)

We can take other chart of the blow-up, i.e., the change of variables

0 2 2 σ0 : R → R ; u2 7→ x2 = u2u3 u3 7→ x3 = u3,

with this change yields the same ﬁrst and second condition and

Re(s12) + 1 > 0. (102)

By symmetry we can consider any of these changes of variables. Completely similarly, ( ) for the convergence of Z 5 (s), we need also the new conditions Ω1

Re(s42) + Re(s43) + Re(s23) + 2 > 0, Re(s42) + 1 > 0, Re(s43) + 1 > 0. (103)

( ) The conditions coming from the locally monomial integral Z 5 (s) are already included. Ω2 (5) ( ) The integral Z{3} s is not a multivariate local zeta function, so we take the following 1 1 1 change of variables x2 → x , then we have dx2 → 2 dx2, and by setting χe(x2) := 1 − χ x , 2 |x2| 2 we have that

 |x | ≤ 1  1 if 2 2+e χe(x2) = .  1 0 if |x2| ≥ 2 , Then Symmetry 2021, 13, 967 51 of 53

Z (5) ( ) = ( ) ( )| |−s12−s42−s23−2| |s13 | − |s42 | − |s43 Z{3} s χe x2 χ x3 x2 x3 1 x2 1 x3 R2 s23 × |1 − x2x3| dx2dx3.

Since x2x3(1 − x2)(1 − x3)(1 − x2x3) is locally monomial in the support of χe(x2)χ(x3), the only new condition is

− Re(s12) − Re(s42) − Re(s23) − 1 > 0. (104)

(5) ( ) Completely analogously, Z{2} s induces the extra condition

− Re(s13) − Re(s43) − Re(s23) − 1 > 0. (105) For the last integral, we are setting χ(x ) := 1 − χ 1 and χ(x ) := 1 − χ 1 . e 2 x2 e 3 x3 Thus,

( ) Z 5 −s12−s42−s23−2 −s13−s43−s23−2 s42 s43 Z∅ (s) = χe(x2)χe(x3)|x2| |x3| |1 − x2| |1 − x3| R2 s23 × |x2 − x3| dx2dx3.

(5) ( ) In this case, we have the same divisor as for Z{2,3} s ; the differences are the powers of |x2| and |x3| and the function χe(x2)χe(x3) that does not contain (1, 1) in its support. Hence, the only new condition will arise from the blow-up at the origin, namely

− Re(s12) − Re(s42) − Re(s23) − Re(s13) − Re(s43) − 2 > 0. (106)

Hence Z(5)(s) is analytic in the region deﬁned by conditions (101)–(106), i.e.,

Re(sij) > −1, for all ij

Re(s12) + Re(s13) + Re(s23) > −2, Re(s42) + Re(s43) + Re(s23) > −2,

Re(s12) + Re(s42) + Re(s23) < −1, Re(s13) + Re(s43) + Re(s23) < −1, ∑ Re(sij) < −2. ij

This region of convergence contains the open subset deﬁned by

2 2 − < Re(s ) < − for all ij. 3 ij 5 (5) 2 2 Then, in particular, Z (s) is analytic in the interval − 3 < Re(s) < − 5 .

Author Contributions: Writing—review and editing: M.B.-G., H.G.-C., E.Y.L., W.A.Z.-G. All the authors contributed to the manuscript equally. All authors have read and agreed to the published version of the manuscript. Funding: The first author was partially supported by the University of Guadalajara. The third author was supported by a Conacyt fellowship. The fourth author was partially supported by Conacyt Grant No. 250845 (Mexico) and by the Debnath Endowed Professorship (UTRGV, USA). Data Availability Statement: No new data were created or analyzed in this study. Data sharing is not applicable to this article. Conflicts of Interest: The authors declare no conflict of interest. Symmetry 2021, 13, 967 52 of 53

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