Exploring the landscape of heterotic strings on T^d

TL;DR

This paper analyzes heterotic string compactifications on T^d, providing criteria for gauge group realization, exploring moduli space algorithms, and confirming duality predictions with F-theory on K3 surfaces.

Contribution

It introduces lattice embedding criteria and algorithms to determine gauge groups and moduli in heterotic T^d compactifications, advancing understanding of string dualities.

Findings

Classified all maximally enhanced gauge groups for d=1,2.

Established criteria for gauge group realization via lattice embeddings.

Confirmed duality correspondence between heterotic T^2 compactifications and elliptic K3 surfaces.

Abstract

Compactifications of the heterotic string on T^d are the simplest, yet rich enough playgrounds to uncover swampland ideas: the U(1)^{d+16} left-moving gauge symmetry gets enhanced at special points in moduli space only to certain groups. We state criteria, based on lattice embedding techniques, to establish whether a gauge group is realized or not. For generic d, we further show how to obtain the moduli that lead to a given gauge group by modifying the method of deleting nodes in the extended Dynkin diagram of the Narain lattice II_{1,17}. More general algorithms to explore the moduli space are also developed. For d=1 and 2 we list all the maximally enhanced gauge groups, moduli, and other relevant information about the embedding in II_{d,d+16}. In agreement with the duality between heterotic on T^2 and F-theory on K3, all possible gauge groups on T^2 match all possible ADE types of singular fibers of elliptic K3 surfaces. We also present a simple method to transform the moduli under the duality group, and we build the map that relates the charge lattices and moduli of the compactification of the E_8 x E_8 and Spin(32)/Z_2 heterotic theories.