Higher Twisted Sector Couplings of $Z_N$ Orbifolds

TL;DR

The paper derives the full moduli-dependent four-twist correlation functions in symmetric $Z_N$ orbifold CFTs with a general antisymmetric background $B$-field, decomposing amplitudes into classical worldsheet instantons and quantum fluctuations. It performs explicit S- and U-channel factorizations to extract twist-anti-twist annihilation couplings $C^k_{f_b,f_a;p,w}$ and Yukawa couplings $Y^{k,l}_{f_a,f_b,f_c}^{k,l}$, including global monodromy considerations and the role of fixed tori. It demonstrates duality invariance of the couplings under a group generated by transformations of the background and lattice, aided by unitary twist-field redefinitions and Poisson resummation, thereby establishing that Yukawa couplings are automorphic functions for a large discrete group. The results underscore an automorphic structure of orbifold couplings and provide detailed, moduli-dependent formulas that are relevant for phenomenology (e.g., hierarchical Yukawas) and mathematical aspects of discrete symmetry in string compactifications. In prime $N$ cases, the expressions simplify, with Yukawa couplings depending only on fixed-point differences and admitting a transparent lattice-sum form.

Abstract

We derive the basic correlation functions of twist fields coming from arbitrary twisted sectors in symmetric $Z_N$ orbifold conformal field theories, keeping all the admissible marginal perturbations, in particular those corresponding to the antisymmetric tensor background field. This allows a thorough investigation of modular symmetries in this type of string compactification. Such a study is explicitly carried out for the group generated by duality transformations. Thus, apart from being of phenomenological use, our couplings are also interesting from the mathematical point of view as they represent automorphic functions for a large class of discrete groups.

Figures (1)

• Figure 1: The loops $C_1, C_2$ and $C_3$ for the case $p=2, q=1$.