Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa-Intriligator formula

摘要

We study algebraic aspects of equivariant quantum cohomology algebra of the flag manifold. We introduce and study the quantum double Schubert polynomials G-tilder _w(x, y), which are the Lascoux-Schutzenberger type representatives of the equivariant quantum cohomology classes. Our approach is based on the quantum Cauchy identity. We define also quantum Schubert polynomials G-tilder _w(x) as the Gram-Schmidt orthogonalization of some set of monomials with respect to the scalar product, defined by the Grothendieck residue. Using quantum Cauchy identity, we prove that G-tilder _w(x) = G-tilder _w(x, y)|_(y = 0) and as a corollary obtain a simple formula for the quantum Schubert polynomials G-tilder _w(x) = partial deriv_(ww0)~((y)) G-tilder _(w0)(x, y)|_(y = 0). We also prove the higher genus analog of Vafa-Intriligator's formula for the flag manifolds and study the quantum residues generating function. We introduce the Ehresmann-Bruhat graph on the symmetric group and formulate the equivariant quantum Pieri rule.