Global properties of topological string amplitudes and orbifold invariants

Abstract

We derive topological string amplitudes on local Calabi-Yau manifolds in terms of polynomials in finitely many generators of special functions. These objects are defined globally in the moduli space and lead to a description of mirror symmetry at any point in the moduli space. Holomorphic ambiguities of the anomaly equations are fixed by global information obtained from boundary conditions at few special divisors in the moduli space. As an illustration we compute higher genus orbifold Gromov-Witten invariants for \( {{{\mathbb{C}^3}} \mathord{\left/{\vphantom {{{\mathbb{C}^3}} {{\mathbb{Z}_3}}}} \right.} {{\mathbb{Z}_3}}} \) and \( {{{\mathbb{C}^3}} \mathord{\left/{\vphantom {{{\mathbb{C}^3}} {{\mathbb{Z}_4}}}} \right.} {{\mathbb{Z}_4}}} \).