We associate a half-integer number, called { the quantum index}, to algebraic curves in the real plane satisfying to certain conditions. The area encompassed by the logarithmic image of such curves is equal to π² times the quantum index of the curve and thus has a discrete spectrum of values. We use the quantum index to refine real enumerative geometry in a way consistent with the Block-Göttsche invariants from tropical enumerative geometry.
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Grigory Mikhalkin (2017) studied this question.