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

We study the algebraic aspects of equivariant quantum cohomology algebra of the flag manifold. We introduce and study the quantum double Schubert polynomials S̃w(x, y), which are the Lascoux– Schützenberger type representatives of the equivariant quantum cohomology classes. Our approach is based on the quantum Cauchy identity. We define also quantum Schubert polynomials S̃w(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 S̃w(x) = S̃w(x, y)|y=0 and as corollary obtain a simple formula for the quantum Schubert polynomials S̃w(x) = ∂ (y) ww0S̃w0(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 extended Ehresman–Bruhat order on the symmetric group and formulate the equivariant quantum Pieri rule. On leave from Steklov Mathematical Institute, Fontanka 27, St.Petersburg, 191011, Russia Supported by JSPS Research Fellowships for Young Scientists