## 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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