Zeta functions and oscillatory integrals for meromorphic functions

Veys, Willem; Zúñiga–Galindo, W. A.

doi:10.1016/j.aim.2017.02.022

Cited by 19 publications

(16 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Such as it was mentioned in the introduction of [28], see also [37], there are deep connections between local zeta functions with string amplitudes and quantum field theory amplitudes that still are not fully understood.…”

Section: Final Remarksmentioning

confidence: 99%

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

Bocardo-Gaspar

García‐Compeán

Zúñiga–Galindo

2018

J. High Energ. Phys.

View full text Add to dashboard Cite

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

show abstract

Section: Final Remarksmentioning

confidence: 99%

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

Bocardo-Gaspar

García‐Compeán

Zúñiga–Galindo

2018

J. High Energ. Phys.

View full text Add to dashboard Cite

show abstract

“…Combining Theorem 1.4 with the results in [21] and [23], one may formulate a monodromy conjecture for rational functions, like the original one in [7]. For previous works in this direction, see for example [9] and [28]. Note that if in a coordinate system F and G depend on separated variables we can easily see that our b mero f,m (s) coincides with the b-function b F (s) of the holomorphic function F .…”

Section: Introductionmentioning

confidence: 72%

On a Berstein-Sato polynomial of a meromorphic function

Takeuchi¹

2021

Preprint

View full text Add to dashboard Cite

We define Berstein-Sato polynomials for meromorphic functions and study their basic properties. In particular, we prove a Kashiwara-Malgrange type theorem on their geometric monodromies, which would be useful also in relation with the monodromy conjecture. A new feature in the meromorphic setting is that we have several b-functions whose roots yield the same set of the eigenvalues of the Milnor monodromies.

show abstract

“…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 p l and with exponential sums mod p l [28].…”

Section: Introductionmentioning

confidence: 99%

Local Zeta Functions and Koba–Nielsen String Amplitudes

et al. 2021

View full text Add to dashboard Cite

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 rise to a boundary string field theory, which was previously proposed by Witten in the context of background independent string theory. Explicit computations in the cases of 4 and 5 points show that the Feynman amplitudes at the tree level of the Gerasimov–Shatashvili Lagrangian are related to the limit p→1 of the p-adic Koba–Nielsen amplitudes. At a mathematical level, this phenomenon is deeply connected with the topological zeta functions introduced by Denef and Loeser. A Koba–Nielsen amplitude is just a new type of local zeta function, which can be studied using embedded resolution of singularities. In this way, one shows the existence of a meromorphic continuations for the Koba–Nielsen amplitudes as functions of the kinematic parameters. The Koba–Nielsen local zeta functions are algebraic-geometric integrals that can be defined over arbitrary local fields (for instance R, C, Qp, Fp((T))), and it is completely natural to expect connections between these objects. The limit p tends to one of the Koba–Nielsen amplitudes give rise to new amplitudes which we have called Denef–Loeser amplitudes. Throughout the article, we have emphasized the explicit calculations in the cases of 4 and 5 points.

show abstract

Zeta functions and oscillatory integrals for meromorphic functions

Cited by 19 publications

References 36 publications

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

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

On a Berstein-Sato polynomial of a meromorphic function

Local Zeta Functions and Koba–Nielsen String Amplitudes

Contact Info

Product

Resources

About