NEW MODELS FOR VENEZIANO AMPLITUDES: COMBINATORIAL, SYMPLECTIC AND SUPERSYMMETRIC ASPECTS

Abstract

The bosonic string theory evolved as an attempt to find a physical/quantum mechanical model capable of reproducing Euler's beta function (Veneziano amplitude) and its multidimensional analogue. The multidimensional analogue of beta function was studied mathematically for some time from different angles by mathematicians such as Selberg, Weil and Deligne among many others. The results of their studies apparently were not taken into account in physics literature on string theory. In several recent publications, attempts were made to restore the missing links. As discussed in these publications, the existing mathematical interpretation of the multidimensional analogue of Euler's beta function as one of the periods associated with the corresponding differential form "living" on the Fermat-type (hyper) surface, happens to be crucial for restoration of the quantum/statistical mechanical models reproducing such generalized beta function. There is a number of nontraditional models — all interrelated — capable of reproducing the Veneziano amplitudes. In this work we would like to discuss two of such new models: symplectic and supersymmetric. The symplectic model is based on observation that the Veneziano amplitude is just the Laplace transform of the generating function for the Ehrhart polynomial. Such a polynomial counts the number of lattice points inside the rational polytope (i.e. polytope whose vertices are located at the nodes of a regular lattice) and at its boundaries. In the present case, the polytope is a regular simplex. It is a deformation retract for the Fermat-type (hyper) surface (perhaps inflated, as explained in the text). Using known connections between polytopes and dynamical systems, the quantum mechanical system associated with such a dynamical system is found. The ground state of this system is degenerate with degeneracy factor given by the Ehrhart polynomial. Using some ideas by Atiyah, Bott and Witten we argue that the supersymmetric model related to the symplectic can be recovered. While recovering this model, we demonstrate that the ground state of such a model is degenerate with the same degeneracy factor as for earlier obtained symplectic model. Since the wave functions of this model are in one to one correspondence with the Veneziano amplitudes, this exactly solvable supersymmetric (and, hence, also symplectic) model is sufficient for recovery of the partition function reproducing the Veneziano amplitudes thus providing the exact solution of the Veneziano model.