Heterotic Strings on ${\mathbb T^3}/{\mathbb Z_2}$, Nikulin involutions and M-theory

We first describe the low energy dynamics of ten dimensional heterotic supergravity compactified on the smooth, flat 3-manifold ${\mathbb T^3}/{\mathbb Z_2}$, without supersymmetry, and explain how it arises from flat heterotic gauge fields. The semi-classical theory has both Coulomb and Higgs branches of non-supersymmetric vacua. We then give an exact worldsheet description as asymmetric orbifolds of $\mathbb T^3$, where the orbifold generator involves a Nikulin non-symplectic involution $θ$ of the even self-dual lattice $Γ_{(19,3)}$. Along the way we briefly compare our findings with M-theory on $\text{K3}/θ$. Our construction gives a novel CFT description of the semi-classical field theory moduli space. In particular, the Wilson line parameters in the lattice $I\subset Γ_{(19,3)}$ of signature $(19-s,1)$ which is invariant under $θ$, and in its orthogonal complement $N$, correspond respectively to Coulomb and Higgs branch moduli. There is a rich pattern of transitions amongst Higgs and Coulomb branches which we describe using the worldsheet theory.

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