Tropical geometry is a powerful instrument in algebraic geometry, allowing for a
simple combinatorial treatment of various enumerative problems. Tropical curves are
planar metric graphs with certain requirements of balancing, rationality of slopes and
integrality.
An addition of a ribbon structure (and a removal of rationality/integrality
requirements) lead to a particularly simple combinatorial construction of moduli of
ribbon (pseudo)tropical curves. Refined Block-Goettsche counting of rational tropical
curves turns into a construction of some simple top-dimensional cycles on these moduli
and maps of spheres.
These cycles turn out to be closely related to associative algebras; curves with
"flat" vertices necessitate a passage from associative to Lie algebras. In particular,
counting of (both complex and real) curves in toric varieties is related to the
quantum torus algebra.
More complicated counting invariants (the so-called Gromov-Witten descendants, or
relative Welschinger invariants) are treated similarly and are related to the
super-Lie structure on the quantum torus. As a by-product we obtain a new
one-parameter family of weights for a refined counting of the descendants.
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