The genus two G-function for cubic elliptic orbifold and modularity (Q6561403)

The aim of this paper is to study the genus two G-function which was introduced by \textit{B. Dubrovin} et al. [Russ. J. Math. Phys. 19, No. 3, 273--298 (2012; Zbl 1325.53114)] for the cubic elliptic orbifold. The author proves the quasi-modularity for the descendant correlation functions of certain type in all genus and proves that all derivatives of the genus two G-functions of certain type are quasi-modular forms after a mirror transformation. In particular, he computes an explicit closed formula for its certain first derivative. The proof mainly relies on two techniques: the Givental quantization formalism for semi-simple Frobenius manifold and the tautological relations on the moduli space of stable curves. \N\N\NThe paper is organized as follows. Section 1 is an introduction to the subject and summarizes the main results. Section 2 recalls some basic knowledge in orbifold Gromov-Witten theory of the elliptic orbifold. In Section 3, the author computes explicitly the Givental \(S\)- and \(R\)-matrix for the Gromov-Witten theory of cubic elliptic orbifolds. Section 4 is devoted to the quasi-modularity property. The author recalls some basic knowledge in modular forms and connects it to the Gromov-Witten invariants of the elliptic orbifold \(\mathbb{P}^1_{3,3,3}\) (the compact elliptic orbifold with three orbifold points of type \((3,3,3)\)). In Section 5, the author studies the descendant correlation function of the elliptic orbifold \(\mathbb{P}^1_{3,3,3}\) in all genera via the Givental quantization formalism and the tautological recursion relations on the moduli space of stable curves. He uses the topological recursion relations on the moduli space of curves to give a much more direct proof for quasi-modularity of the descendant invariants. Section 6 investigates the study the derivatives of the genus two G-function for the elliptic orbifold \(\mathbb{P}^1_{3,3,3}\). As an application of the result obtained in the previous section, the author proves a quasi-modularity property of the genus two G-function. In Section 7, the author computes a closed formula for the genus one generating function. He reduces the derivatives of the genus two G-function on the small phase space to lower genus correlation functions by the topological recursion relation on the moduli space of stable curves.