Quantum indices and refined enumeration of real plane curves

TL;DR

This paper introduces the quantum index, a new invariant for real algebraic curves, linking geometric area in logarithmic space to discrete quantum values, and refines real enumerative geometry with tropical invariants.

Contribution

It defines the quantum index for certain real algebraic curves and connects it to tropical enumerative invariants, providing a novel refinement of real enumerative geometry.

Findings

The area of the logarithmic image is proportional to the quantum index.

Quantum indices take discrete values, leading to a refined counting of curves.

The approach aligns with Block-G"ottsche invariants in tropical geometry.

Abstract

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-G\"ottsche invariants from tropical enumerative geometry.