We associate a half-integer number, called {\em 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 $\pi^2$ 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-Gottsche invariants from tropical enumerative geometry.

中文翻译：

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实平面曲线的量子指数和精细枚举

我们将一个称为 {\em 量子指数} 的半整数与满足某些条件的实平面中的代数曲线相关联。此类曲线的对数图像所包含的面积等于曲线量子指数的 $\pi^2$ 倍，因此具有离散的值谱。我们使用量子索引以与热带枚举几何中的 Block-Gottsche 不变量一致的方式细化实枚举几何。