Quantum index, Arnold-Rokhlin surfaces, and real enumerative geometry

The main goal of this note is to relate two different Welschinger-type rules of signs that were used for definition of real enumerative invariants of toric surfaces (the invariants considered being relative to the toric boundary).

The relation between them intertwines Mikhalkin's quantum index and geometry of real and complex point sets of counted real curves.

As a by-product, we suggest a new definition of quantum index, which can be used for refined invariant enumeration of real algebraic curves on surfaces in a non-toric setup.