Homotopy quantization of real curve enumeration

The complexification of a real rational curve is composed
of two holomorphic disks glued along the real locus of the
curve. It turns out that each of these disks realizes an element
of a certain (non-commutative) Heisenberg-type group which
appears as the second homotopy group \pi_2(X,L,p) for some X
and L. Using this group we may recover the quantum index of
the real curve which in its turn is responsible for the refined
enumeration of complex rational curves in the plane.
Joint work with Sergey Galkin.