Classification of Modular Symmetries in Non-Supersymmetric Heterotic String theories

We study modular symmetries in non-supersymmetric heterotic string theories on toroidal backgrounds with Wilson line modulus, constructed by stringy Scherk-Schwartz compactification. In particular, we focus on a subgroup of the T-duality group $O(D+16,D,\mathbb{Z})$ with $D=2$ given by an outer automorphism of the Narain lattice, which can be mapped to the Siegel modular group $\mathrm{Sp}(4,\mathbb{Z})$. We classify the modular symmetries and a $CP$-like symmetry on $T^2$ and its orbifolds with symmetric and asymmetric orbifold twists. It turns out that the non-supersymmetric heterotic string theories only enjoy a part of modular symmetries in contrast to supersymmetric ones. Furthermore, the gauge symmetry is maximally enhanced at fixed points of modular symmetries on $T^2$ on which we analyze the vacuum structure of eight-dimensional tachyon-free vacua.