New strings for old Veneziano amplitudes II. Group-theoretic treatment

TL;DR

This paper employs group theory and polynomial invariants to reconstruct Veneziano amplitudes, providing an exact, mathematically rigorous approach that connects symmetries with scattering amplitude properties.

Contribution

It introduces a group-theoretic framework using polynomial invariants and supersymmetric quantum mechanics to exactly solve Veneziano amplitudes, extending previous methods.

Findings

Exact reconstruction of Veneziano amplitudes using group invariants

Demonstration of generating functions via supersymmetric quantum mechanics

Validation of the formalism's consistency with conformal field theory results

Abstract

In this part of our four parts work (e.g see Part I, hep-th/0410242) we use the theory of polynomial invariants of finite pseudo-reflection groups in order to reconstruct both the Veneziano and Veneziano-like (tachyon-free) amplitudes and the generating function reproducing these amplitudes. We demonstrate that such generating function can be recovered with help of the finite dimensional exactly solvable N=2 supersymmetric quantum mechanical model known earlier from works by Witten, Stone and others. Using the Lefschetz isomorphisms theorem we replace traditional supersymmetric calculations by the group-theoretic thus solving the Veneziano model exactly using standard methods of representation theory. Mathematical correctness of our arguments relies on important theorems by Shepard and Todd, Serre and Solomon proven respectively in early fifties and sixties and documented in the monograph by Bourbaki. Based on these theorems we explain why the developed formalism leaves all known results of conformal field theories unchanged. We also explain why these theorems impose stringent requirements connecting analytical properties of scattering amplitudes with symmetries of space-time in which such amplitudes act.