[Paper Review] Classification of the Chiral Z_2 x Z_2 Heterotic String Models

S.E.M. Nooij|ArXiv.org|Mar 6, 2006

Black Holes and Theoretical Physics42 references3 citations

TL;DR

This paper classifies the chiral spectrum of heterotic $ℤ_2 \times \mathbb{Z}_2$ orbifold string models using a free fermionic formulation, demonstrating that perturbative three-generation models require asymmetric shifts rather than symmetric ones. It establishes a direct mapping between orbifold and free fermionic constructions, showing that chiral content is governed by vacuum expectation values of background fields in an underlying $N=4$ theory, with the full gauge group $\mathbb{E}_6 \times U(1)^2 \times SO(8)^2$ and three generations arising from twisted sectors.

ABSTRACT

This thesis provides a classification of the chiral content of the heterotic $\mathbbm{Z}_2 imes \mathbbm{Z}_2$ orbifold models. We show that the chiral content of the heterotic $\mathbbm{Z}_2 imes \mathbbm{Z}_2$ orbifold models at any point in the moduli space can be described by a free fermionic model. We present a direct translation between the orbifold formulation and the free fermionic construction. We use the free fermionic description for the classification wherein we consider orbifolds with symmetric shifts. We show that perturbative three generation models are not obtained in the case of $\mathbbm{Z}_2 imes \mathbbm{Z}_2$ orbifolds with symmetric shifts on complex tori, and that the perturbative three generation models in this class necessarily employ an asymmetric shift. We show that the freedom in the modular invariant phases in the $N = 1$ vacua that control the chiral content, can be interpreted as vacuum expectation values of background fields of the underlying $N = 4$ theory, whose dynamical components are projected out by the $\mathbbm{Z}_2$ fermionic projections. In this class of vacua the chiral content of the models is determined by the underlying $N = 4$ mother theory.

Motivation & Objective

• To classify the chiral content of heterotic $\mathbb{Z}_2 \times \mathbb{Z}_2$ orbifold string models across the full moduli space.
• To establish a direct translation between the orbifold construction and the free fermionic formulation for these models.
• To determine whether perturbative three-generation models can be realized with symmetric shifts on complex tori.
• To investigate the role of modular invariant phases in $N=1$ vacua as vacuum expectation values of background fields in the $N=4$ mother theory.
• To assess the phenomenological viability of $\mathbb{E}_6$-unified models with three generations, particularly regarding $\mathbb{E}_6$ breaking mechanisms.

Proposed method

• The chiral content of $\mathbb{Z}_2 \times \mathbb{Z}_2$ orbifold models is mapped to a free fermionic model, enabling a systematic classification of the spectrum.
• A direct translation is derived between the orbifold twist sectors and the free fermionic basis vectors, preserving the gauge group and chiral spectrum.
• The analysis focuses on models with symmetric shifts on complex tori, identifying constraints on the GSO projection coefficients.
• The modular invariant phases in $N=1$ vacua are interpreted as vacuum expectation values of background fields in the $N=4$ parent theory.
• The chiral content is shown to be fully determined by the underlying $N=4$ theory, with dynamical components projected out by $\mathbb{Z}_2$ fermionic projections.
• Explicit GSO phase matrices are constructed for models with 32, 8, and 6 generations to illustrate the classification.

Experimental results

Research questions

• RQ1 Can perturbative three-generation models be realized in $\mathbb{Z}_2 \times \mathbb{Z}_2$ orbifold compactifications with symmetric shifts on complex tori?
• RQ2 How are the chiral states in $N=1$ vacua of these models related to vacuum expectation values of background fields in the $N=4$ mother theory?
• RQ3 What is the precise mapping between the orbifold construction and the free fermionic formulation for $\mathbb{Z}_2 \times \mathbb{Z}_2$ heterotic models?
• RQ4 Why do $\mathbb{E}_6$-unified models with three generations fail when broken via Wilson-line-like vectors in this class of models?
• RQ5 What is the role of the $N=4$ parent theory in determining the chiral spectrum of the resulting $N=1$ vacua?

Key findings

• Perturbative three-generation models in the $\mathbb{Z}_2 \times \mathbb{Z}_2$ orbifold class with symmetric shifts on complex tori do not exist; they require asymmetric shifts.
• The chiral content of $N=1$ vacua is fully determined by the underlying $N=4$ mother theory, with the modular invariant phases interpreted as vacuum expectation values of background fields.
• A direct and explicit translation is established between the orbifold formulation and the free fermionic construction, enabling a unified classification framework.
• The model with 32 generations arises from 16 from each of the first two twisted planes, with the full gauge group $\mathbb{E}_6 \times U(1)^2 \times SO(8)^2$.
• An eight-generation $SO(10)$ model is constructed with four generations from each of the first two planes, using a specific set of GSO projection coefficients.
• An $\mathbb{E}_6$-unified six-generation model is realized, but $\mathbb{E}_6$ breaking via Wilson-line-like vectors leads to truncation of fermion families, rendering it incompatible with three generations.