The genus two G-function for the cubic elliptic singularity
In 2001 \textit{B. Dubrovin} and \textit{Y. Zhang} [``Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants'', Preprint, \url{arXiv:math/0108160}] gave a complicated explicit formula for the genus \(2\) generating function for semisimple Frobenius manifolds. In 2012, in collaboration with \textit{S.-Q. Liu} [Russ. J. Math. Phys. 19, No. 3, 273--298 (2012; Zbl 1325.53114)], they split it into contributions from genus \(2\) dual graphs and the so called genus \(2\) \(G\)-function \(G^{(2)}\). In 2015 \textit{X. Liu} and \textit{X. Wang} [Adv. Math. 274, 631--650 (2015; Zbl 1368.53058)] showed that \(G^{(2)}\) vanishes for simple singularities and \(\mathbb{P}^1\) orbifolds of Fano type. In this paper the author studies properties of \(G^{(2)}\) for the cubic elliptic singularity and derives closed form formulas for its derivatives along the mirror map. The derivation is based on the Saito-Givental theory for isolated polynomial singularities. First, Frobenius manifold structure is put on the deformation space of the singularity using Saito's relative differential form, which is semisimple at the generic point. Then Gromov-Witten invariants and the \(I\)-function are defined using Givental's formalism. From the \(I\)-function two series, \(X\) and \(L\), are defined, which are quasi-modular forms with the modular group \(\Gamma_0(3)\) up to analytic continuation and symplectic transformation. Finally, the derivative of \(G^{(2)}\) is expressed as an explicit polynomial in \(X\) and \(L\). The modular properties are naturally interpretable in terms of Gromov-Witten invariants of the corresponding \((3,3,3)\) type elliptic orbifold.
scientific article; zbMATH DE number 7451110
scientific article