2002/09/16 by Alexander Givental, Givental, Alexander
Mathematics · Physics and Astronomy · #14N35 #17B69 #32S30 #37K30 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #FOS: Physical sciences #Geometric and Algebraic Topology #Geometry and complex manifolds #Mathematical Physics (math-ph) #Representation Theory (math.RT) #math-ph #math.AG #math.MP #math.RT #msc:14N35 #msc:17B69 #msc:32S30 #msc:37K30
paper · pdf · doi:10.48550/arxiv.math/0209205
29 pages, to appear in Moscow Mathematical Journal
openalex publication_date 2002/09/16 · arxiv created 2003/05/28 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
According to a conjecture of E. Witten proved by M. Kontsevich, a certain generating function for intersection indices on the Deligne -- Mumford moduli spaces of Riemann surfaces coincides with a certain tau-function of the KdV hierarchy. The generating function is naturally generalized under the name the \em total descendent potential in the theory of Gromov -- Witten invariants of symplectic manifolds. The papers arXiv: math.AG/0108100 and arXive: math.DG/0108160 contain two equivalent constructions, motivated by some results in Gromov -- Witten theory, which associate a total descendent potential to any semisimple Frobenius structure. In this paper, we prove that in the case of K.Saito's Frobenius structure on the miniversal deformation of the An-1-singularity, the total descendent potential is a tau-function of the nKdV hierarchy. We derive this result from a more general construction for solutions of the nKdV hierarchy from n-1 solutions of the KdV hierarchy.