Pre-string theory

In this note, I recollect a two-week period in September 1968 when I factorized the Veneziano model using string variables in Chicago. Professor Yoichiro Nambu went on to calculate the N-particle dual resonance model and then to factorize it on an exponential degeneracy of states. That was in 1968 and the following year 1969 he discovered the string action. I also include some other reminiscences of Nambu who passed away on July 5, 2015.

EXPLICIT MODEL REALIZING PARTON–HADRON DUALITY

We present a model that realizes both resonance-Regge (Veneziano) and parton–hadron (Bloom–Gilman) duality. We first review the features of the Veneziano model and we discuss how parton–hadron duality appears in the Bloom–Gilman model. Then we review limitations of the Veneziano model, namely that the zero-width resonances in the Veneziano model violate unitarity and Mandelstam analyticity. We discuss how such problems are alleviated in models that construct dual amplitudes with Mandelstam analyticity (so-called DAMA models). We then introduce a modified DAMA model, and we discuss its properties. We present a pedagogical model for dual amplitudes and we construct the nucleon structure function F2(x, Q2). We explicitly show that the resulting structure function realizes both Veneziano and Bloom–Gilman duality.

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

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.