Free Fermionic Constructions of Heterotic Strings

TL;DR

The chapter surveys the Free Fermionic Formulation (FFF) of heterotic strings, detailing how internal coordinates are fermionized, how worldsheet supersymmetry is realized among fermions, and how modular invariance constrains the construction to a finite set of basis vectors and GGSO phases. It explains the equivalence between free-fermion models and orbifold descriptions, and presents concrete N=1 vacua (e.g., flipped SU(5), Pati-Salam, Standard-like models) as well as symmetric-basis scans, non-supersymmetric constructions, and the fermionic-to-orbifold map. This framework yields a practical toolkit for computing spectra, superpotential couplings, and large-scale landscape scans, while enabling moduli deformations and SUSY-breaking analyses via the orbifold correspondence. Overall, the work provides a coherent, constructive approach to exploring phenomenologically interesting heterotic vacua within a solvable, modular-invariant setting and connects fermionic points to geometric orbifolds for moduli-space studies.

Abstract

This chapter is an introduction to the Free Fermionic Formulation of String Theory, with emphasis on heterotic model building. After a brief review of bosonization in two dimensional conformal field theories, we discuss how internal bosonic string coordinates can be consistently replaced by free fermionic degrees of freedom. In this framework, worldsheet supersymmetry may be realized entirely among free fermions. Embedding this construction into string theory leads to a number of constraints arising from modular invariance at one and higher genera. The solution of these constraints takes the form of a small number of model building rules from which the string spectrum and interactions may be analyzed. We review some of the most well-studied models in the literature and their classification, with emphasis on the symmetric basis. The explicit map of free fermionic models to the orbifold construction is presented in some detail.