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, 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 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 quantum Schubert polynomials are the specialization of quantum double Schubert polynomials with second set of variables equals to zero, and as a corollary obtain a simple formula for the quantum Schubert polynomials. 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. 
Anatol N. Kirillov and Toshiaki Maeno