On the Construction of Asymmetric Orbifold Models

TL;DR

<3-5 sentence high-level summary>: The paper investigates asymmetric orbifold constructions built from chiral shifts and chiral reflections, focusing on maintaining both modular invariance and a consistent Hilbert-space interpretation across twisted sectors. It develops and applies a chiral-splitting framework in which the full partition function is assembled from paired left and right chiral blocks via a modulus-insensitive pairing matrix $K$, with $K$ determined by one-loop degenerations and higher-genus consistency. It demonstrates that the order of chiral reflections must be 4 (not 2) to preserve modular covariance, and it constructs explicit one-loop partition functions for several models, including the $s_R$, $s_R^2$, and the Kachru-Kumar-Silverstein–type $f$ and $g$ orbifolds, including the inclusion of worldsheet fermions. The work clarifies how to perform higher-loop analyses in asymmetric orbifolds by leveraging chiral splitting and symmetry considerations, offering a concrete algorithmic path for building viable models in this challenging regime and connecting to known symmetric-block constructions.</p>

Abstract

Various asymmetric orbifold models based on chiral shifts and chiral reflections are investigated. Special attention is devoted to the consistency of the models with two fundamental principles for asymmetric orbifolds : modular invariance and the existence of a proper Hilbert space formulation for states and operators. The interplay between these two principles is non-trivial. It is shown, for example, that their simultaneous requirement forces the order of a chiral reflection to be 4, instead of the naive 2. A careful explicit construction is given of the associated one-loop partition functions. At higher loops, the partition functions of asymmetric orbifolds are built from the chiral blocks of associated symmetric orbifolds, whose pairings are determined by degenerations to one-loop.