Tropical Refined Curve Counting with Descendants

Area of Science:

• Algebraic Geometry
• Tropical Geometry
• Mathematical Physics

Background:

• Logarithmic Gromov-Witten invariants are crucial in string theory and algebraic geometry.
• Tropical geometry offers combinatorial approaches to geometric problems.
• Understanding refined invariants and their geometric interpretations is an active research area.

Purpose of the Study:

• To prove a q-refined tropical correspondence theorem for higher genus descendant logarithmic Gromov-Witten invariants.
• To establish an agreement between these invariants and a q-refined count of tropical curves.
• To provide a geometric interpretation for previously defined tropical counts.

Main Methods:

• Utilizing the logarithmic degeneration formula.
• Developing a key technique to compute integrals against double ramification cycles.
• Connecting these integrals to the non-commutative KdV hierarchy.

Main Results:

• A q-refined tropical correspondence theorem is proven for higher genus descendant logarithmic Gromov-Witten invariants on toric surfaces.
• A generating series of these invariants is shown to equal a q-refined count of tropical curves with higher valency conditions.
• A geometric proof for the deformation invariance of the tropical count is obtained.

Conclusions:

• The study provides an algebro-geometric meaning to tropical counts.
• The results bridge logarithmic Gromov-Witten theory and tropical geometry.
• The techniques developed offer new tools for studying related invariants.