V.I.Arnol's seminar
September 21, 1999

S.M.Natanzon

Frobenius structures on the spaces of versal deformations of
$A_n$ and $B_n$ singularities.

In the beginning of 90-s E.Witten, R.Dijkgraaf, E.Verlinde,
H.Verlinde discovered a remarkable system of differential
equations (WDVV-equations). B.Dubrovin found that any solution
of this system describes a rich differential geometrical
structure, which he called Frobenius structure. This structure
connects a lot of different regions of mathematics: theory of
singularities, integrable systems, quantum cohomology, etc.
Frobenius structure appears in particular on the spaces of
versal deformations of singularities. Dubrovin described it for
simple singularities ($A_n$, $B_n$ etc.). In this case corresponding
solutions of WDVV-equations are polynomials. They were known only
for $n\leqslant 4$. In the talk they will be found for all $A_n$ and $B_n$.