The genus two G-function for cubic elliptic orbifold and modularity

In this paper, we study the genus two G-function which was introduced by Dubrovin, Liu and Zhang for the cubic elliptic orbifold. As results, we first prove the quasi-modularity for the descendant correlation functions of certain type in all genus. Then we prove any derivatives of the genus two G-function of certain type are quasi-modular forms after a mirror transformation. In particular, we compute the explicit closed formula for its certain first derivative. Our proof mainly relies on two techniques: Givental quantization formalism for semisimple Frobenius manifold and the tautological relations on the moduli space of stable curves.