Sv-map between type-I and heterotic sigma models

Abstract

The scattering amplitudes of gauge bosons in heterotic and open superstring theories are related by the single-valued map (sv-map): heterotic amplitudes are obtained by selecting a subset of multiple zeta value (MZV) coefficients in the $\alpha'$ (string tension parameter) expansion of open string amplitudes. In this dissertation, we argue that this relation holds also at the level of low-energy expansion (or individual Feynman diagrams) of the respective effective actions, by investigating the beta functions of two-dimensional sigma models describing world-sheets of open and heterotic strings coupled with gauge backgrounds. We will analyze the sigma model Feynman diagrams generating identical effective action terms in both theories and show that the heterotic coefficients are given by the single-valued projection of the open ones. The gauge backgrounds of the nonlinear sigma model of the heterotic string is usually studied under the fermionic representation. Here we will also propose a Wilson loop representation for it. When it is under the fermionic representation, the sv-map appears as a result of summing over all radial orderings of heterotic vertices on the complex plane. When it is under the proposed Wilson loop representation, the sv-map comes from a sum of path-ordered integrals along two opposite-directed contours, which has a simple geometric origin manifested in the nonabelian Stokes's theorem. (This dissertation is based on our published work [1,2].)