Curve counting on the Enriques surface and the Klemm–Mariño formula

We determine the Gromov–Witten invariants of the local Enriques surfaces for all genera and curve classes and prove the Klemm–Mariño formula. In particular, we show that the generating series of genus 1 invariants of the Enriques surface is the Fourier expansion of a certain power of Borcherds automorphic form on the moduli space of Enriques surfaces. We also determine all Vafa–Witten invariants of the Enriques surface. The proof uses the correspondence between Gromov–Witten theory and Pandharipande–Thomas theory. On the Gromov–Witten side, we prove the relative Gromov–Witten potentials of elliptic Enriques surfaces are quasi-Jacobi forms and satisfy a holomorphic anomaly equation. On the sheaf side, we relate the Pandharipande–Thomas invariants of the Enriques–Calabi–Yau threefold in fiber classes to the 2-dimensional Donaldson–Thomas invariants by a version of Toda’s formula for local K3 surfaces. Altogether, we obtain sufficient modular constraints to determine all invariants from basic geometric computations.