Grigory Mikhalkin: Logarithmic area of real plane curves and refined enumerative geometry

We note that under certain conditions, the logarithmic images of real plane curves bound regions whose areas are half-integer multiples of pi square. The half-integer number can be interpreted as a quantum state of the real curve and used to refine real enumerative geometry in a way consistent with the Block-Göttsche invariants from the tropical world.