Spectral Curves for Semisimple Conformal Frobenius Manifolds

We construct connected spectral curves for semisimple conformal Frobenius manifolds with flat unit, using the periods of the second structure equation. Eynard-Orantin topological recursion on these curves reproduces the stable ancestor correlators of Givental-Teleman reconstruction. The spectral curves are defined on simply connected domains in the complex plane. We apply the construction to the quantum cohomology of ${\mathbb{P}}^n$ for $n\ge 2$ at a point, obtaining explicit spectral curve data in terms of Hecke triangle groups and hypergeometric functions. Finally, for ${\mathbb{P}}^2$ and ${\mathbb{P}}^3$, we express the complete spectral curve data, including the bidifferentials, in classical modular and quasimodular forms.