Let \(X\) be a non-singular compact symplectic toric variety. Then \(X\) can be obtained by the symplectic reduction of \({\mathbb C}^N\) with its standard symplectic structure \(\text{Im}\sum d\bar{z}_i\wedge dz_i/2\) by the action of a subtorus \(T^k\) in the maximal torus \(T^N\) of diagonal unitary matrices on a generic level of the momentum map. In this situation \(k\) is called the Picard number of \(X\). Let \(u_1,u_2,\ldots,u_N\in H^2(X)\) denote the Poincaré duals of the fundamental cycles determined by the \(N\) compact toric hypersurfaces in \(X\) which are the \(T^k\)-reductions of the \(T^N\)-invariant coordinate hyperplanes in \({\mathbb C}^N\). Let \(Y\subset X\) be a non-singular complete intersection defined by global sections of \(\ell\) (\(\ell\geq 0\)) non-negative line bundles \(\mathcal L_a\), \(a=1,2,\ldots,\ell\), with first Chern classes \(v_1,v_2,\ldots,v_{\ell}\in H^2(X)\). Write \(\mathcal V=\bigoplus_a\mathcal L_a\). Then the bundle \(\mathcal V\) is convex. It can be endowed with the action of \(T^{\ell}\) by scalar multiplication on the fibers of each summand \(\mathcal L_a\). Then one may study the equivariant Gromov-Witten theory of the pair \((X,\mathcal V)\) with respect to the action of the torus \(T=T^N\times T^{\ell}\). Write \(H^*(\mathcal V)\subset H^*(Y)\) for the subring multiplicatively generated by the \(u_i\)’s. One has \(c_1(\mathcal T_X)=u_1+\cdots +u_N\), and \(c_1(Y)=u_1+\cdots +u_N-v_1-\cdots -v_{\ell}\), where \(v_a=c_1(\mathcal L_a)\).
Gromov-Witten theory associates to \(Y\) its quantum cohomology \(\mathcal D\)-module and its quantum cohomology algebra. Let \((t_1,\ldots,t_k)\) denote coordinates on \(H^2(Y)\) with respect to a basis \((p_1,\ldots,p_k)\) of integral symplectic classes. \(t_0\) will denote the coordinate on \(H^0(Y)\). The \(p_i\) define quantum operators \(p_i^*\) acting on vector functions \(s=s(t_0,t)\) with values in \(H^*(Y,{\mathbb C})\). They give rise to a system of linear PDE’s with associated \(\mathcal D\)-module generated by a single formal vector function \(J_Y=J_Y(t,\hbar^{-1})\) with coefficients in \(H^*(Y,{\mathbb Q})\). One can give a construction of \(J_Y\) in terms of intersection theory on moduli spaces of stable maps. Write \(J\) for the orthogonal projection of \(J_Y\) to the subalgebra \(H^*(\mathcal V,{\mathbb Q})\subset H^*(Y,{\mathbb Q})\). Let \(\Lambda\) be the semigroup of classes of compact holomorphic curves in \(Y\subset X\). For \(d\in\Lambda\) write \(D_j(d)=\int_{[d]}u_j\), \(L_a(d)=\int_{[d]}v_a\). Consider the formal vector function \(I=I(t,\hbar^{-1})\) with values in \(H^*(\mathcal V,{\mathbb Q})\) (‘hypergeometric series’): \[ \begin{split} I=\exp\left((t_0+\sum_{i=1}^kp_it_i)/\hbar\right)\sum_{d\in\Lambda} \exp\left(\sum_{j=1}^kt_jd_j\right)\times\\ \times{{\prod_{a=1}^{\ell} \prod_{m=-\infty}^{L_a(d)}(v_a+m\hbar)\prod_{j=1}^N\prod_{m=-\infty} ^0(u_j+m\hbar)}\over{\prod_{a=1}^{\ell}\prod_{m=-\infty}^0(v_a+m\hbar) \prod_{j=1}^N\prod_{m=-\infty}^{D_j(d)}(u_j+m\hbar)}} \end{split} \] One can now state the main result
\((*)\): Let \(Y\) be a non-singular toric complete intersection with non-negative first Chern class. Then \(I\) and \(J\) with coefficients in \(H^*(\mathcal V,{\mathbb Q})\) coincide up to a triangular weighted homogeneous change of variables \[ t_0\mapsto t_0+f_0(q)\hbar+h(q),\quad\log q_i\mapsto\log q_i+f_i(q), \] where \(q=e^t\), and \(h,f_0,\ldots,f_k\) are weighted homogeneous power series supported on \(\Lambda-\{0\}\), and \(\deg f_0=\deg f_i=0\).
As a matter of fact one may prove a more general result depending on additional variables \(\lambda\). The above result can be interpreted as the specialization for \(\lambda\rightarrow 0\). By a result of V. Batyrev one deduces that \((*)\) confirms the mirror conjecture for Calabi-Yau toric complete intersections.
For the entire collection see [Zbl 0905.00081].