Gravitational couplings in N $$ \mathcal{N} $$ = 2 heterotic compactifications with Wilson lines

In this paper we compute the gravitational couplings of the heterotic string compactified on (K3 × T2)/ℤN and E8 × E8 and predict the Gopakumar Vafa invariants of the dual Calabi Yau manifold in presence of Wilson lines. Here ℤN acts as an automorphism on K3 associated with the conjugacy classes of M23 and a shift of 1/N on one of the S1 of T2. We study in detail the cases N = 2, 3 for standard and several non-standard embeddings where K3 is realized as toroidal orbifolds T4/ℤ4 and T4/ℤ3. From these computations we extract the polynomial term in perturbative pre-potential for these orbifold models in presence of a single Wilson line. We also show for standard embeddings the integrality of the Gopakumar Vafa invariants depend on the integrality of Fourier coefficients of Fourier transform of the twisted elliptic genus of K3 in presence of n < 8 Wilson lines.