Generalized Mariño-Vafa formula and local Gromov-Witten theory of orbi-curves

Abstract

We prove a generalized Mariño-Vafa formula for Hodge integrals over $\overline{\mathcal{M}}_{g, \gamma - \mu} (\mathcal{B}G)$ with $G$ an arbitrary finite abelian group. This formula can be viewed as a formula for the one-leg orbifold Gromov-Witten vertex where the leg is effective. We will prove the orbifold Gromov-Witten/Donaldson-Thomas correspondence between our formula and the formula for the orbifold DT vertex in J. Bryan, C. Cadman & B. Young, The orbifold topological vertex. We will also use this formula to study the local Gromov-Witten theory of an orbi-curve with cyclic stack points in a Calabi-Yau three-orbifold.