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Gromov-Witten invariants of flag manifolds, via \(D\)-modules. (English) Zbl 1093.14076
The authors present a method for computing the three-point genus zero Gromov–Witten invariants of the flag manifold \(G/B\). For this one should define the quantum multiplication, i.e. choose the isomorphism between vector spaces \(QH^*(G/B)\) and \(H^*(G/B)\otimes \mathbb C[q_1,\ldots,q_r]\). This isomorphism is called an (abstract) quantum evaluation map and should satisfy some conditions, in particular, the integrability condition (vanishing of some particular differential 2-form). The existence of a basis that gives this map is equivalent to the existence of a particular \(D\)-module, which is called a quantization of \(QH^*(G/B)\). In general, such quantization gives an evaluation map that does not satisfy the integrability condition. To correct this, one should choose the particular operator. After a natural choice of a \(D\)-module that provides the quantization one gets the usual quantum product operation (Theorem \(1.5\)).
The authors prove that in fact the correcting operator is polynomial (Proposition 2.2). This means that the system of partial differential equations on it given by the integrability condition can be solved “by quadrature”. Finally, the authors consider the case of \(G= \text{GL}_n \mathbb C\) for \(n=2,3,4\) as an example and get the particular formulas for quantum evaluation maps.

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14M15 Grassmannians, Schubert varieties, flag manifolds
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