New Strings for Old Veneziano Amplitudes III. Symplectic Treatment

TL;DR

This paper develops a symplectic approach to understanding Veneziano amplitudes by linking Ehrhart polynomials of inflated polytopes to generating functions, providing new physical models and insights into scattering data.

Contribution

It introduces a symplectic treatment connecting Ehrhart polynomials with Veneziano amplitudes, expanding the mathematical framework for string scattering models.

Findings

Veneziano amplitude as Laplace transform of Ehrhart polynomial generating function.

Symplectic models reproduce Veneziano-like amplitudes.

Application of mirror symmetry explains experimental pion scattering data.

Abstract

A d-dimensional rational polytope P is a polytope whose vertices are located at the nodes of d-dimensional Z-lattice. Consider a number of points inside the inflated polytope (with coefficient of inflation k, k=1,2, 3...). The Ehrhart polynomial of P counts the number of such lattice points (nodes) inside the inflated P and (may be) at its faces (including vertices). In Part I (hep-th/0410242) of our four parts work we noticed that the Veneziano amplitude is just the Laplace transform of the generating function (considered as a partition function in the sence of statistical mechanics) for the Ehrhart polynomial for the regular inflated simplex obtained as a deformation retract of the Fermat (hyper) surface living in complex projective space. This observation is sufficient for development of new symplectic (this work) and supersymmetric (hep-th/0411241)physical models reproducing the Veneziano (and Veneziano-like) amplitudes. General ideas (e.g.those related to the properties of Ehrhart polynomials) are illustrated by simple practical examples (e.g. use of mirror symmetry for explanation of available experimental data on pion-pion scattering) worked out in some detail. Obtained final results are in formal accord with those earlier obtained by Vergne [PNAS 93 (1996) 14238].