Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2005-02-26
J.Geom.Phys. 56 (2006) 1433-1472
Physics
High Energy Physics
High Energy Physics - Theory
48 pages J.Geom.Phys.(in press, available on line)
Scientific paper
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].
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