Exploring the landscape of CHL strings on T^d

TL;DR

This paper systematically explores the moduli space of heterotic string compactifications on T^d/Z_2 orbifolds, identifying points of gauge group enhancement and analyzing the lattice structures involved.

Contribution

It provides a comprehensive computational analysis of the moduli space for d=1 and 2, revealing the structure of enhanced gauge groups and their fundamental groups.

Findings

Identified maximally enhanced gauge groups for d=1 and 2.

Mapped the lattice embeddings related to gauge group enhancements.

Extended results to higher dimensions d > 2.

Abstract

Compactifications of the heterotic string on special T^d/Z_2 orbifolds realize a landscape of string models with 16 supercharges and a gauge group on the left-moving sector of reduced rank d+8. The momenta of untwisted and twisted states span a lattice known as the Mikhailov lattice II_{(d)}, which is not self-dual for d > 1. By using computer algorithms which exploit the properties of lattice embeddings, we perform a systematic exploration of the moduli space for d=1 and 2, and give a list of maximally enhanced points where the U(1)^{d+8} enhances to a rank d+8 non-Abelian gauge group. For d = 1, these groups are simply-laced and simply-connected, and in fact can be obtained from the Dynkin diagram of E_{10}. For d = 2 there are also symplectic and doubly-connected groups. For the latter we find the precise form of their fundamental groups from embeddings ofof lattices into the dual of II_{(2)}. Our results easily generalize to d > 2.