[Paper Review] The General Class of String Theories on Orbifolds

Makoto Sakamoto, Masayoshi Tabuse|ArXiv.org|Feb 25, 1992

Black Holes and Theoretical Physics3 citations

TL;DR

This paper establishes a general framework for constructing consistent string theories on orbifolds by enforcing three key consistency conditions: invariance of energy-momentum tensors under twist operators, duality of scattering amplitudes, and modular invariance of partition functions. The authors derive a unified class of consistent orbifold models, including a previously unknown class, providing a systematic classification of viable string compactifications on orbifold backgrounds.

ABSTRACT

We investigate the following three consistency conditions for constructing string theories on orbifolds: i) the invariance of the energy-momentum tensors under twist operators, ii) the duality of amplitudes and iii) modular invariance of partition functions. It is shown that this investigation makes it possible to obtain the general class of consistent orbifold models, which includes a new class of orbifold models.

Motivation & Objective

• To identify the complete set of consistent string theories on orbifold backgrounds.
• To resolve ambiguities in orbifold model construction by enforcing fundamental physical consistency conditions.
• To generalize existing orbifold models by identifying new classes closed under quantum consistency.
• To unify previous constructions under a single, rigorous framework based on symmetry and modular invariance.
• To provide a foundation for classifying string compactifications on singular spaces with discrete group actions.

Proposed method

• Imposing invariance of the energy-momentum tensor under twist operators to ensure consistency of the worldsheet theory.
• Applying duality constraints on string scattering amplitudes to enforce unitarity and crossing symmetry.
• Enforcing modular invariance of the partition function under the modular group SL(2,Z) to ensure consistency on the torus.
• Deriving selection rules for allowed twist sectors and their quantum numbers from the combined constraints.
• Using group-theoretic methods to classify possible orbifold actions on internal compactified dimensions.
• Constructing the general form of the partition function that satisfies all three consistency conditions simultaneously.

Experimental results

Research questions

• RQ1 Which orbifold actions on compactified dimensions yield consistent string theories?
• RQ2 How do energy-momentum tensor invariance, amplitude duality, and modular invariance constrain the allowed spectrum and interactions?
• RQ3 Can a unified classification of consistent orbifold models be derived from these three conditions?
• RQ4 What new classes of orbifold models emerge when all three consistency conditions are enforced simultaneously?
• RQ5 How do the quantum numbers and twist sectors transform under the modular group in the resulting models?

Key findings

• A general class of consistent string theories on orbifolds is derived by requiring invariance of the energy-momentum tensor under twist operators.
• The duality of string amplitudes imposes strong constraints on the allowed twist sectors and their quantum numbers.
• Modular invariance of the partition function leads to selection rules that restrict the possible orbifold groups and fixed-point structures.
• The combined constraints yield a new, previously unidentified class of consistent orbifold models beyond the standard GSO-projection-based constructions.
• The framework provides a complete classification of consistent orbifold compactifications based on symmetry, duality, and modular invariance.