Stationary Phase Integrals, Quantum Toda Lattices, Flag Manifolds and the Mirror Conjecture

This operator is a quantization of the Hamiltonian of the Toda lattice on n+1 identical particles with configuration coordinates t0, ..., tn and with the exponential interaction potential exp(ti−ti−1) of neighbors. The Toda lattice is known to be integrable on both classical and quantum levels: there exist commuting differential polynomials Dm(~∂/∂t, exp t, ~), m = 0, ..., n, which play the role of quantum conservation laws (i.e. [H,D0] = ... = [H,Dn] = 0) and whose symbols Dm(p, exp t, 0) form a complete set of Poisson-commuting first integrals of the classical Toda lattice. In this paper we study solutions S(t) of the differential system D0S = ... = DnS = 0 whose characteristic Lagrangian variety L is the most degenerate invariant Lagrangian variety of the Toda lattice. According to [12, 13] this Lagrangian variety is the spectrum of the quantum cohomology algebra of the manifold of complete flags 0 ⊂ C ⊂ ... ⊂ C. We represent solutions S by stationary phase integrals in n(n+1)/2 complex variables and point out the role these solutions play in the quantum cohomology theory. As we explain in the last section, our results prove the mirror conjecture [9] in the case of the flag manifolds.