Traces of Mirror Symmetry in Nature

Abstract

In this work we discuss the place of Veneziano amplitudes (the precursor of string models) and their generalizations in the Regge theory of high energy physics scattering processes. We emphasize that mathematically such amplitudes and their extensions can be interpreted in terms of the Laplace (respectively, miultiple Laplace) transform(s) of generating function for the Ehrhart polynomial associated with some integral polytope P (specific for each scattering process). Following works by Batyrev and Hibi to each polytope P it is possible to associate another (mirror) polytope P'. For this to happen, it is nesessary to impose some conditions on P and, hence, on the generating function for P. Since each of these polytopes is in fact encodes some projective toric variety, this information is used for development of new symplectic and supersymmetric models reproducing the Veneziano amplitudes. General ideas are illustrated on classical example of the pion-pion scattering for which the existing experimental data can be naturally explained with help of mirror symmetry arguments.