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Quantum cohomology of flag manifolds \(G/B\) and quantum Toda lattices. (English) Zbl 1054.14533
From the introduction: Let \(G\) be a connected semi-simple complex Lie group, \(B\) its Borel subgroup, \(T\) a maximal complex torus contained in \(B\), and Lie \((T)\) its Lie algebra. This setup gives rise to two constructions; the generalized nonperiodic Toda lattice and the flag manifold \(G/B\). The Toda lattice for \((G,B,T)\) is the dynamical system on the cotangent bundle \(T^* \text{Lie}(T)\) endowed with the canonical holomorphic symplectic form and the holomorphic Hamiltonian function \[ H(p,q)=(p,p)-\sum_{\text{simple roots } \alpha_i}(\alpha_i,\alpha_i) \exp\bigl(\alpha_i (q)\bigr), \] where \((,)\) is any fixed nonzero multiplication of the Killing form on each simple component of Lie \((G)\) and the simple roots are given by the roots of \(B\) with respect to \(T\). This system is known to he completely integrable [B. Kostant, Sel. Math., New Ser. 2, 43–91 (1996; Zbl 0868.14024) and Adv. Math. 34, 195–338 (1979; Zbl 0433.22008)]. Therefore the variety defined by the ideal generated by the integrals of motions is the Lagrangian analytic submanifold of \(T^*\text{Lie} (T)\). On the other hand, for the flag manifold \(G/B\) we have the small quantum cohomology ring \(QH^*(G/B,\mathbb{C})\) which is generated by second cohomology classes and parameters. Denoting as \(q_i\) the coordinates of the parameter space \(H^2(G/B,\mathbb{C})\) defined by \(q_i(\sum a_jp_j)= \exp (a_i)\) (here \(p_j\) is the cohomology class corresponding to the fundamental weights), the author describes the ring structure of \(QH^* (G,B)\):
Theorem 1. The small quantum cohomology ring \(QH^*(G/B, \mathbb{C})\) is canonical isomorphic to \(\mathbb{C}[p_1,\dots, p_l,q_1, \dots, q_l]/I\), where \(I\) is the ideal generalized by the nonconstant complete integrals of motions of the Toda lattice for the Langlands-dual Lie group \((G^v,B^v,T^v)\) o \((G,B,T)\).
In fact the author proves more in this paper. Using the quantum hyperplane section principle it is possible to compute the virtual numbers of rational curves in Calabi-Yau 3-fold complete intersections in homogeneous spaces with the knowledge of the quantum \({\mathcal D}\)-module structure of the ambient spaces. The author shows that the \({\mathcal D}\)-module structure for \(G/B\) is governed by the conservation laws of quantum Toda lattices which are the quantizations of the Toda lattices and still integrable [B. Kostant, Invent. Math. 48, 101–184 (1978; Zbl 0405.12013) and London Math. Soc. Lect. Note Ser. 34, 287–316 (1979; Zbl 0474.58010)], A. Reyman and M. Semenov-Tiam-Shansky, Invent. Math. 54, 81–100 (1979; Zbl 0403.58004)]. The Hamiltonian operator he considers is \[ \widehat H=\Delta-\sum_{\text{simple roots }\alpha_i} (\alpha_i, \alpha_i) \exp \bigl(\alpha_i(q) \bigr), \] where \(\Delta\) is the Laplacian on Lie \(T\) associated with the invariant form \((,)\). Let \({\mathcal D}\) be the differential operator algebra over \(\mathbb{C}\), generated by \(\hbar\frac {\partial} {\partial t_i}\), multiplication by \(\hbar\) and \(\exp t_i\).
Theorem II. The quantum \({\mathcal D}\)-module of \(G/B\) is canonically isomorphic to \({\mathcal D}/{\mathcal I}\) where \({\mathcal I}\) is the left ideal generated by the nonconstant complete quantum integrals of motions of the quantum Toda lattice for the Langlands-dual Lie group \(G^\vee, B^\vee,T^\vee)\) of \((G,B,T)\).

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14M15 Grassmannians, Schubert varieties, flag manifolds
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
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