[Paper Review] Towards the classification of Z2XZ2 fermionic models

Alon E. Faraggi, Costas Kounnas+2 more|ArXiv.org|Nov 7, 2003

Black Holes and Theoretical Physics3 citations

TL;DR

This paper develops a systematic formalism to classify four-dimensional $\mathbb{Z}_2 \times \mathbb{Z}_2$ heterotic string models with $SO(10)$ grand unification, separating the orbifold twist from lattice shifts. It shows that perturbative three-generation models require asymmetric shifts, not just symmetric ones, and identifies a subclass where phenomenological features like generation count are predetermined by the $N=4$ lattice, enabling complete classification via algebraic formulas and GSO projection coefficients.

ABSTRACT

We develop a formalism that allows a complete classification of four-dimensional Z2XZ2 heterotic string models. Three generation models in a sub-class of these compactifications are related to the existence of three twisted sectors in Z2XZ2 orbifolds. In the work discussed here we classify the sub-class of these models that produce spinorial representations from all three twisted planes, and including symmetric shifts on the internal lattice. We show that perturbative three generation models are not obtained solely with symmetric shifts on complex tori, but necessitate the action of an asymmetric shift. In a subclass of these models we show that their chiral content is predetermined by the choice of the N=4 lattice. The implication of the results and possible geometrical interpretation are briefly discussed.

Motivation & Objective

• To systematically classify $\mathbb{Z}_2 \times \mathbb{Z}_2$ heterotic string models with $SO(10)$ grand unification using a fixed basis of fermionic vectors.
• To separate the orbifold twist action from lattice shifts and Wilson lines to clarify chirality generation and family reduction mechanisms.
• To identify conditions under which the chiral spectrum, particularly the number of generations, is determined at the $N=4$ level.
• To demonstrate that three-generation models cannot be obtained with symmetric shifts alone, necessitating asymmetric shifts.
• To lay the groundwork for a complete classification of this class of models using analytical formulas and projection coefficients.

Proposed method

• The model is defined by a fixed set of 12 basis vectors in the free fermionic formulation, including $v_1 = 1$, $v_2 = S$, $v_{2+i} = e_i$ for $i=1,\dots,6$, $v_9 = b_1$, $v_{10} = b_2$, $v_{11} = z_1$, and $v_{12} = z_2$, representing $N=4$ supersymmetry, orbifold twists, and shifts.
• The spectrum is generated via generalized GSO projections defined by $2^{N(N-1)/2}$ independent projection coefficients $c_{v_i}^{v_j}$ for $i>j$, ensuring modular invariance.
• The formalism decouples the $\mathbb{Z}_2 \times \mathbb{Z}_2$ orbifold twist (from $b_1$, $b_2$) from symmetric shifts ($e_i$) and Wilson lines ($z_1$, $z_2$), enabling independent analysis.
• The number of generations is computed algebraically using formulas derived from the GSO projection structure and fermion number operators.
• Models are classified by fixing the basis vectors and varying the projection coefficients, with a focus on subclasses where phenomenological features are inherited from the $N=4$ parent theory.
• A computer program is used to enumerate models in the subclass where $c_{b_i}^{z_m} = c_{b_i}^{e_i} = +1$, ensuring decoupling of twist and shift actions.

Experimental results

Research questions

• RQ1 Can a complete classification of $\mathbb{Z}_2 \times \mathbb{Z}_2$ heterotic string models with $SO(10)$ unification be achieved using a fixed basis of fermionic vectors and projection coefficients?
• RQ2 What is the role of symmetric versus asymmetric shifts in generating three-generation models?
• RQ3 Under what conditions are the phenomenological features of the model, such as the number of generations, predetermined by the $N=4$ lattice?
• RQ4 Why are three-generation models not obtainable with symmetric shifts alone on complex tori?
• RQ5 What is the geometric and dualistic significance of models where the $N=4$ structure determines the low-energy spectrum?

Key findings

• Perturbative three-generation models in the $\mathbb{Z}_2 \times \mathbb{Z}_2$ fermionic model class require an asymmetric shift; symmetric shifts on complex tori alone are insufficient.
• A subclass of models exists where the chiral spectrum, including the number of generations, is completely determined by the $N=4$ lattice and is independent of the orbifold twist and Wilson lines.
• The formalism allows a complete classification of this class of models, with a total of $2^{39}$ models in the decoupled subclass, reducible by symmetry to a manageable set.
• The mechanism of family reduction is identified as a consequence of the GSO projection structure, not solely from Wilson lines or shifts.
• The results imply that $Z_2 \times Z_2$ Calabi–Yau compactifications cannot yield exactly three generations without non-zero torsion, indicating a need for exotic compactifications.
• The necessity of asymmetric shifts for three-generation models suggests a special role for self-dual points under T-duality and potential implications for moduli stabilization and dualities.