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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.