42 research outputs found
A_{n-1} singularities and nKdV hierarchies
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
-singularity, the total descendent potential is a tau-function of the
KdV hierarchy. We derive this result from a more general construction for
solutions of the KdV hierarchy from solutions of the KdV hierarchy.Comment: 29 pages, to appear in Moscow Mathematical Journa
Explicit reconstruction in quantum cohomology and K-theory
Cohomological genus-0 Gromov-Witten invariants of a given target space can be
encoded by the "descendant potential," a generating function defined on the
space of power series in one variable with coefficients in the cohomology space
of the target. Replacing the coefficient space with the subspace
multiplicatively generated by degree-2 classes, we explicitly reconstruct the
graph of the differential of the restricted generating function from one point
on it. Using the Quantum Hirzebruch--Riemann--Roch Theorem from our joint work
with Valentin Tonita, we derive a similar reconstruction formula in genus-0
quantum K-theory. The results amplify the role of the divisor equations, and
the structures of -modules and -modules in quantum cohomology and
quantum K-theory with respect to Novikov's variables.Comment: 13 page