Symmetries and Solutions of Getzler’s Equation for Coxeter and Extended Affine Weyl Frobenius Manifolds
نویسنده
چکیده
The G-function associated to the semisimple Frobenius manifold C/W (where W is a Coxeter group or an extended affine Weyl group) is studied. The general form of the Gfunction 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/W̃ (An−1) , where W̃ (An−1) is a certain extended affine Weyl group (or, equivalently, corresponding to the Hurwitz space M̂0;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.
منابع مشابه
Extended affine Weyl groups and Frobenius manifolds
We define certain extensions of affine Weyl groups (distinct from these considered by K. Saito [S1] in the theory of extended affine root systems), prove an analogue of Chevalley theorem for their invariants, and construct a Frobenius structure on their orbit spaces. This produces solutions F (t1, . . . , tn) of WDVV equations of associativity polynomial in t1, . . . , tn−1, exp tn.
متن کاملPolynomial and non-polynomial solutions set for wave equation with using Lie point symmetries
This paper obtains the exact solutions of the wave equation as a second-order partial differential equation (PDE). We are going to calculate polynomial and non-polynomial exact solutions by using Lie point symmetry. We demonstrate the generation of such polynomial through the medium of the group theoretical properties of the equation. A generalized procedure for polynomial solution is pr...
متن کاملIsomonodromic Tau-Function of Hurwitz Frobenius Manifolds and Its Applications
In this work we find the isomonodromic (Jimbo-Miwa) tau-function corresponding to Frobenius manifold structures on Hurwitz spaces. We discuss several applications of this result. First, we get an explicit expression for the G-function (solution of Getzler’s equation) of the Hurwitz Frobenius manifolds. Second, in terms of this tau-function we compute the genus one correction to the free energy ...
متن کاملConformal mappings preserving the Einstein tensor of Weyl manifolds
In this paper, we obtain a necessary and sufficient condition for a conformal mapping between two Weyl manifolds to preserve Einstein tensor. Then we prove that some basic curvature tensors of $W_n$ are preserved by such a conformal mapping if and only if the covector field of the mapping is locally a gradient. Also, we obtained the relation between the scalar curvatures of the Weyl manifolds r...
متن کاملNew Expression of Discrete Painlevé Equations
It is known that discrete Painlevé equations have symmetries of the affine Weyl groups. In this paper we propose a new representation of discrete Painlevé equations in which the symmetries become clearly visible. We know how to obtain discrete Painlevé equations from certain rational surfaces in connection with the extended affine Weyl groups. By means of this representation, we clarify the rel...
متن کامل