Exploring the landscape of heterotic strings on Td
Anamaría Font, Bernardo Fraiman, Mariana Graña, Carmen A. Núñez, Héctor Parra De Freitas
Abstract
A bstract 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 × E 8 and Spin(32) / ℤ 2 heterotic theories.