Symmetries and Solutions of Getzler's Equation for Coxeter and Extended Affine Weyl Frobenius Manifolds

Abstract

The G-function associated to the semi-simple Frobenius manifold C^n/W (where W is a Coxeter group or an extended affine Weyl group) is studied. The general form of the G function is given in terms of a logarithmic singularity over caustics in the manifold. The main result in this paper is a universal formula for the G-function corresponding to the Frobenius manifold C^n/W^(k)(A_{n-1}) where W^(k)(A_{n-1}) is a certain extended affine Weyl group (or, equivalently, corresponding to the Hurwitz space M_{0;k-1,n-k-1}), together with the general form of the G-function in terms of data on caustics. Symmetries of the G function are also studied.