Ternary Codes and $Z_3$-Orbifold Constructions of Conformal Field Theories

TL;DR

This paper explores the construction of conformal field theories from ternary codes and lattices, revealing connections to the Monster module and Niemeier lattices, which are crucial for classifying self-dual conformal field theories at central charge 24.

Contribution

It introduces new methods to construct conformal field theories from ternary codes and lattices, extending previous binary code results and providing insights into the classification of self-dual theories.

Findings

Complete construction of Niemeier lattices from ternary codes

Connections between lattice constructions and the Monster module

Progress towards classifying self-dual c=24 conformal field theories

Abstract

We describe a pair of constructions of Eisenstein lattices from ternary codes, and a corresponding pair of constructions of conformal field theories from lattices which turn out to have a string theoretic interpretation. These are found to interconnect in a similar way to results for binary codes, which led to a generalisation of the triality structure relevant in the construction of the Monster module. We therefore make some comments regarding a series of constructions of $V^\natural$. In addition, we present a complete construction of the Niemeier lattices from ternary codes, which in view of the above analogies should prove to be of great importance in the problem of the classification of self-dual $c=24$ conformal field theories. Other progress towards this problem is summarised, and some comments arise from this discussion regarding the uniqueness of the Monster conformal field theory. (Talk presented at the "Monster Bash", Ohio State University, May 1993.)