Instantons and the Point Particle Field Theory Derived From Strings - Davis, Simon (Communications in Mathematics and Applications, 2023)
String theories with supersymmetry have perturbation series that are finite at each order with exponential bounds and do not reflect the presence of nonperturbative effects. Worldsheet instantons in superstring theory are surfaces which support fields with a finite Euclidean action. Dirichlet boundaries can be added to compact surfaces to represent the coupling of open and closed strings and yield an exponential term with a dependence on the coupling characteristic of strings rather than point-particle field theory. An additional set of worldsheet occurs in \(N=2\) string theory after the quantization of a \(U(1)\) symmetry. The \(N=2\) open string amplitudes with \(U(1)\) instantons may be derived from a cubic Yang-Mills theory. Nevertheless, the summation over the genus and \(U(1)\) instanton number includes other amplitudes without an exponential nonperturbative term. The expansion of \(N=2\) closed string amplitudes similarly consists of many vanishing terms, and couplings with the open string are required initially for the introduction of nonperturbative string effects. It is necessary to evaluate the action of a nontrivial solution to the effective field equations to find an exponential term with the dependence on the coupling of point-particle field theories. The theory then can be developed into a description of a model for elementary particles at larger distance scales. The theory of Eisenstein series and cusp forms is developed for the delineation between strings and point particles in the effective action.