Notes on 8 Majorana Fermions

TL;DR

This work analyzes the $d=1+1$ dimensional system of eight Majorana fermions through the lens of SO(8) triality, showing self- and chiral-trialities that relate $8_v$, $8_s$, and $8_c$ representations via ${f Z}_2$ gauge fields and Arf invariants. It develops a robust framework based on bosonization, partition-function identities, and background spin structures to connect free fermions with gauge-theoretic duals, and extends these dualities to the worldsheet of the superstring, including Type II and Type 0 theories. The paper then leverages these dualities to address symmetric mass generation, demonstrating how Spin(8) and Spin(7) invariant interactions can gap the eight Majoranas while preserving discrete ${f Z}_2$ symmetries, with Arf terms driving the phase structure. Collectively, the results illuminate how spin structure, gauge sectors, and dualities orchestrate the spectrum and ground-state structure across 2D dual descriptions, with implications for SPT physics and string quantization.

Abstract

Eight Majorana fermions in $d=1+1$ dimensions enjoy a triality that permutes the representation of the $SO(8)$ global symmetry in which the fermions transform. This triality plays an important role in the quantization of the superstring, and in the analysis of interacting topological insulators and the associated phenomenon of symmetric mass generation. The purpose of these notes is to provide an introduction to the triality and its applications, with careful attention paid to various ${\bf Z}_2$ global and gauge symmetries and their coupling to background spin structures.