Yukawa Couplings for Bosonic $Z_N$ Orbifolds: Their Moduli and Twisted Sector Dependence

TL;DR

The paper provides a comprehensive calculation of all three-point Yukawa couplings for bosonic $Z_N$ orbifolds in the most general background compatible with the twist, including both metric $G$ and antisymmetric $B$ fields. Using a path-integral approach, it derives the four-point twist-field correlator by separating into quantum fluctuations and classical instanton contributions, and identifies a crucial global-monodromy constraint that introduces a gcd-related sector index $\phi$. The resulting Yukawa couplings are expressed in terms of a product of Gamma factors and a restricted-instanton sum over a moduli-dependent lattice, with selection rules ensuring nonvanishing couplings. The work establishes explicit duality invariance of the couplings under a family of target-space duality maps, clarifying how twisted-sector ground states mix and how phases must be chosen to preserve invariance. Collectively, these results validate twisted-sector duality in a broad class of string compactifications and refine prior formulations by incorporating complete monodromy considerations.

Abstract

The three point correlation functions with twist fields are determined for bosonic $Z_N$ orbifolds. Both the choice of the modular background (compatible with the twist) and of the (higher) twisted sectors involved are fully general. We point out a necessary restriction on the set of instantons contributing to twist field correlation functions not obtained in previous calculations. Our results show that the theory is target space duality invariant.