Heterotic string on the CHL orbifold of K3

We study $$ mathcal{n}=2 $$ compactifications of heterotic string theory on the chl orbifold $$ left(k3times {t}^2right)/{mathrm{mathbb{z}}}_n $$ with n = 2, 3, 5, 7. $$ {mathrm{mathbb{z}}}_n $$ acts as an automorphism on k3 together with a shift of 1/n along one of the circles of t 2. these compactifications generalize the example of the heterotic string on k3 × t 2 studied in the context of dualities in $$ mathcal{n}=2 $$ string theories. we evaluate the new supersymmetric index for these theories and show that their expansion can be written in terms of the mckay-thompson series associated with the $$ {mathrm{mathbb{z}}}_n $$ automorphism embedded in the mathieu group m 24. we then evaluate the difference in one-loop threshold corrections to the non-abelian gauge couplings with wilson lines and show that their moduli dependence is captured by siegel modular forms related to dyon partition functions of $$ mathcal{n}=4 $$ string theories.