Simple singularities and integrable hierarchies

Kavli Affiliate: Todor E. Milanov

| First 5 Authors: Alexander B. Givental, Todor E. Milanov, , ,

| Summary:

The paper math.AG/0108100 gives a construction of the total descendent
potential corresponding to a semisimple Frobenius manifold. In math.AG/0209205,
it is proved that the total descendent potential corresponding to K. Saito’s
Frobenius structure on the parameter space of the miniversal deformation of the
A_{n-1}-singularity satisfies the modulo-n reduction of the KP-hierarchy. In
this paper, we identify the hierarchy satisfied by the total descendent
potential of a simple singularity of the A,D,E-type. Our description of the
hierarchy is parallel to the vertex operator construction of Kac — Wakimoto
except that we give both some general integral formulas and explicit numerical
values for certain coefficients which in the Kac — Wakimoto theory are studied
on a case-by-case basis and remain, generally speaking, unknown.

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