# Simple Elliptic Singularities: a note on their G-function

I.A.B. Strachan

Arxiv ID: 1004.2140•Last updated: 12/15/2020

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.

#### PaperStudio AI Chat

I'm your research assistant! Ask me anything about this paper.