Towards higher Kontsevich-Manin numbers – three approaches

Kontsevich-Manin numbers are numbers of holomorphic maps from the Riemann surface $X$ to complex manifold $Y$ passing through given cycles in $Y$. They satisfy quite an interesting quadratic relation called WDVV equation. We propose to generalize it to the case where $X$ and $Y$ are complex manifolds and dimension of $X$ is higher than $1$ as follows: we propose to take a set of pairs $(C_{X,a}, C_{Y,a} )$ and count the number of maps such that image of $C_{X,a}$ intersects $C_{Y,a}$.
We argue that such definition is well-defined at least when $X$ and $Y$ are toric manifolds. In such case one may try to consider compactification of the space of maps by quasimaps that turns out to be also toric. First approach implies replacement of cycles $C_{Y,a}$ in the target by smooth differential forms that are dual to these cycles. Then one may compute the integral of the pull-backs of corresponding differentials forms ignoring the fact that they are ill-defined on the compactifying strata. We conjecture that it gives the right counting and it the case when $\dim X=1$ it was checked by numerical experiments.
Second approach implies the replacement of the original problem by easier problem, known in physics as counting GLSM numbers. Their computation is easy, however, it differs from the original problem at compactifying locus where map is a proper quasimap. We call such contribution freckled contribution, to get correct number it should be subtracted from the GLSM numbers, we will give examples how it work. Here one has to construct a technology of subtraction of freckles, that is possibly doable.
The third approach is based on understanding of holomorphic maps of toric manifold as higher Mors-Bott-Novikov theory. Namely, for $\dim X=1$ holomorphic maps are known to be trajectories of a vector field on a loop space in manifold $Y$. For $\dim X=d$ they are common trajectories of $d$ commuting vector fields on the space of $d$-loops in $Y$. The $d$-version of Morse theory for $d=2$ was studied by Soukhanov who showed that counting such trajectories leads to the $L$-infinity algebra of the Algebra of the Infrared. We propose that such $L$-infinity algebra would underline the $d$ dimensional generalization of WDVV equations.