Аннотация:
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.