Description
A number of classical enumerative problems can be easily solved by methods of tropical
geometry. Tropical curves are planar metric graphs with certain requirements of balancing,
rationality of slopes and integrality.
We consider a generalization of tropical curves, removing requirements of rationality of
slopes and integrality. One of the classical enumerative problems is that of counting of (either
complex or real) rational curves through a collection of points in a toric variety. We explain this
counting procedure as a construction of some simple top-dimensional cycles on certain moduli
of (pseudo)tropical curves.
Cycles on these moduli turn out to be closely related to maps of spheres and associative and
Lie algebras. In particular, counting of both complex and real curves is related to the quantum
torus algebra. More complicated counting invariants (the so-called Gromov-Witten descendants)
are similarly related to the super-Lie structure on the quantum torus.
