×

Extended affine Weyl groups and Frobenius manifolds. (English) Zbl 0964.32020

To a “Frobenius manifold” one may associate a “monodromy group” acting on a linear space [see B. Dubrovin, Lect. Notes Math. 1620, 120-348 (1996; Zbl 0841.58065)]. The authors construct Frobenius manifolds with prescribed monodromy group. More precisely they show that, given a certain extension of a Weyl group of a root system, there exists a Frobenius manifold with this group as monodromy. An explicit list of all Frobenius manifolds arising in this way and having dimension \(\leq 4\) is given.
For the construction procedure first “Fourier polynomials” are studied, then a ring of certain “Fourier polynomials” invariant under the given group is defined. The Frobenius manifold now arises as the spectrum of this ring. As a variety, this spectrum is just isomorphic to the affine space. The main point is that this variety can be equipped with a Frobenius structure such that one obtains the desired monodromy.
In an endnote, the authors point out that there is related work by K. Wirthmüller [Acta Math. 157, 159-241 (1986; Zbl 0635.14015)].

MSC:

32M10 Homogeneous complex manifolds
14B07 Deformations of singularities
20H15 Other geometric groups, including crystallographic groups
PDFBibTeX XMLCite
Full Text: DOI arXiv