Givental's conjecture identifying the Floer and quantum cohomology Heisenberg modules

Let $M$ be a symplectic manifold whose cohomology is generated by classes in $H^2(M,\mathbb{Z})$. Let $\mathcal{D}$ be the Heisenberg algebra of differential operators generated by $q_k=e^{t_k}$ and $\hbar\frac{\partial}{\partial t_k}$, and consider the $H^{2*}(M,\mathbb{C})$-valued function

$$F=\sum_{d\in\check{\mathcal{K}}}e^{\sum_{k=1}^{r}t_k(\frac{\omega_k}{\hbar}+\int_d\omega_k)}\frac{e_{S^1}(\mathcal{N}_d^+)}{e_{S^1}(\mathcal{N}_0^+)}.$$

Equivalently, with $q_k=e^{t_k}$ and $d_k=\int_d\omega_k$, this is

$$F=e^{(t_1\omega_1+\dots+t_r\omega_r)/\hbar}\sum_{d\in\check{\mathcal{K}}}q^d\frac{e_{S^1}(\mathcal{N}_d^+)}{e_{S^1}(\mathcal{N}_0^+)},$$

where $q^d=q_1^{d_1}\dots q_r^{d_r}$ and the ratio is set equal to $1$ for $d=0$. Let $G$ be the function defined using the space of stable maps, which generates the quantum cohomology $\mathcal{D}$-module.

Givental's conjecture. The Floer $\mathcal{D}$-module and the quantum cohomology $\mathcal{D}$-module are the same; equivalently, $F=G$, with the components of $F$ giving appropriate regularizations of $\mathcal{F}_{T_\beta}(\Delta)$ and generating the quantum cohomology $\mathcal{D}$-module as $\mathcal{D}/I_0$, where $I_0$ is the ideal of operators annihilating the components of $F$.

The conjecture proposes a regularization of the Fourier transform of the Floer fundamental cycle and identifies it with the stable-map construction of the quantum cohomology module. In the toric case the relevant ratio of equivariant Euler classes is asserted to be definable, while its definition for general symplectic manifolds is conjectural.