Gromov-Witten invariants of toric fibrations (Q2929554)

The author proves the following result: Decompose the \(J\)-function of the overruled Lagrangian cone \(\mathcal{L}_{B}\), corresponding to the base \(B\) of a toric fibration \(E \rightarrow B\), according to the degrees of curves; NEWLINE\[NEWLINEJ(z,\tau)=\sum_{D\in MC(B)}J_{D}(z,\tau)Q^{D}NEWLINE\]NEWLINE and introduce the hypergeometric modification NEWLINE\[NEWLINE I_{E}(z,t,\tau,q,Q):=e^{Pt/z}\sum_{d\in \mathbb{Z}^{K}, D\in MC(B)} \frac{J_{D}(z,\tau)Q^{D}q^{d}e^{dt}}{\prod_{j=1}^{N}\prod_{m=1}^{U_{j}(\mathcal{D})}(U_{j}+mz).}NEWLINE\]NEWLINE Then, for all \((t,\tau)\), the series \(I_{E}(-z)\) lies in the overruled Lagragian cone \(\mathcal{L}_{E}\) corresponding to the total space \(E\).NEWLINENEWLINE Based on this theorem, a conjecture of \textit{A. Elezi} [Int. Math. Res. Not. 2005, No. 55, 3445--3458 (2005; Zbl 1099.14046)] in a generalized form suggested by Givental is proved. Moreover, the result of the paper gives a new proof of mirror theorems by \textit{A. Givental} [Prog. Math. 160, 141--175 (1998; Zbl 0936.14031)] and \textit{H. Iritani} [Topology 47, No. 4, 225--276 (2008; Zbl 1170.53071)] for toric manifolds.