Computation of refined toric invariants II

TL;DR

This paper advances the computation of refined counts of real rational curves in toric surfaces by removing previous assumptions and relating classical invariants to tropical geometry, enhancing understanding of their structure and computation.

Contribution

It introduces a method to compute the quantum index of any oriented real rational curve without previous restrictions and connects classical invariants to tropical refined invariants.

Findings

Quantum index computation for any oriented real rational curve

Relation between classical and tropical refined invariants

Generalization of Mikhalkin's and author's previous results

Abstract

In 2015, G.~Mikhalkin introduced a refined count for real rational curves in toric surfaces. The counted curves have to pass through some real and complex points located on the toric boundary of the surface, and the count is refined according to the value of a so called quantum index. This count happens only to depend on the number of complex points on each toric divisors, leading to an invariant. First, we give a way to compute the quantum index of any oriented real rational curve, getting rid of the previously needed "purely imaginary" assumption on the complex points. Then, we use the tropical geometry approach to relate these classical refined invariants to tropical refined invariants, defined using Block-G\"ottsche multiplicity. This generalizes the result of Mikhalkin relating both invariants in the case where all the points are real, and the result of the author where complex points are located on a single toric divisor.