Simple Elliptic Singularities: a note on their G-function

TL;DR

This paper explicitly constructs the G-function for elliptic hypersurface singularities, revealing its dependence on a single variable and exploring its properties and applications in integrable systems.

Contribution

It provides the first explicit construction of the G-function for elliptic hypersurface singularities using Noumi and Yamada's flat structure results.

Findings

G-function depends on only one variable, the marginal deformation.

Explicit formulas for G-function for 6, 7, 8 singularities.

Analysis of G-function's behavior under modular group actions.

Abstract

The link between Frobenius manifolds and singularity theory is well known, with the simplest examples coming from the simple hypersurface singularities. Associated with any such manifold is a function known as the $G$-function. This plays a role in the construction of higher-genus terms in various theories. For the simple singularities the G-function is known explicitly: G=0. The next class of singularities, the unimodal hypersurface or elliptic hypersurface singularities consists of three examples, \widetilde{E}_6,\widetilde{E}_7,\widetilde{E}_8 (or equivalently P_8, X_9,J_10). Using a result of Noumi and Yamada on the flat structure on the space of versal deformations of these singularities the $G$-function is explicitly constructed for these three examples. The main property is that the function depends on only one variable, the marginal (dimensionless) deformation variable. Other examples are given based on the foldings of known Frobenius manifolds. Properties of the $G$-function under the action of the modular group is studied, and applications within the theory of integrable systems are discussed.