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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 $$(G,B,T)$$.

##### MSC:
 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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