New strings for old Veneziano amplitudes: III. Symplectic treatment

A d  -dimensional rational polytope P is a polytope whose vertices are located at the nodes of Zd lattice. Consider the number |kP∩Zd| of points inside the inflated P with coefficient of inflation k(k=1,2,3,…)k(k=1,2,3,…). The Ehrhart polynomial of P counts the number of such lattice points inside the inflated P and (may be) at its faces (including vertices). In Part I [A.L. Kholodenko, New string for old Veneziano amplitudes. I. Analytical treatment, J. Geom. Phys. 55 (2005) 50–74] of our four parts work we noticed that Veneziano amplitude is just the Laplace transform of the generating function (considered as a partition function in the sense of statistical mechanics) for the Ehrhart polynomial for the regular inflated simplex obtained as deformation retract of the Fermat (hyper) surface living in the complex projective space. This observation is sufficient for development of new symplectic (this work) and supersymmetric (Part II) 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 ππ scattering, etc.) worked out in some detail. Obtained final results are in formal accord with those earlier obtained by Vergne [M. Vergne, Convex polytopes and quanization of symplectic manifolds, Proc. Natl. Acad. Sci. 93 (1996) 14238–14242].