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Explicit reconstruction in quantum cohomology and \(K\)-theory. (English. French summary) Zbl 1360.14130

The author proves that if \(\sum_{d}I_{d}Q^{d}\) where \(I_{d}(z,z^{-1})\) are cohomology valued Laurent \(z\)-series (representing a point on the graph of d\(\mathcal{F}\), the differential of genus-0 descendent potential, in symplectic loop space \(\mathcal{H}\)) and if \(\Phi_{\alpha}\) are polynomials in \(p_{1},\ldots , p_{r}\) then the family \[ I(\tau)=\sum_{d}T_{d}Q^{d}\text{exp}\Big\{ \frac{1}{z}\sum_{\alpha}\tau_{\alpha} \Phi_{\alpha}(p_{1}-zd_{1},\ldots , p_{r}-zd_{r}) \Big \} \] lies on the graph of d\(\mathcal{F}\). Here \(Q^{d}\) stands for the element corresponding to \(d\) in the semigroup ring of Mori cone \(\mathcal{M}\) of the compact Kähler manifold \(X\). Moreover, for arbitrary scalar power series \(C_{\alpha}(z)=\sum_{k\geq 0}\tau_{\alpha,k}z^{k}\), the linear combination \(\sum_{\alpha}c_{\alpha}(z)z\partial_{\tau_{\alpha}}I\) of the derivatives also lies on the graph. Furthermore, in case when \(p_{1}\ldots ,p_{r}\) generate \(H^{*}(X,\mathbb{Q})\), and \(\Phi_{\alpha}\) represents a linear basis, such linear combinations comprise the whole graph.

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
19L10 Riemann-Roch theorems, Chern characters
32Q15 Kähler manifolds
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References:

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