Tropical refined curve counting from higher genera and lambda classes

Area of Science:

• Algebraic Geometry
• String Theory
• Mathematical Physics

Background:

• Tropical curves provide a combinatorial approach to enumerative geometry.
• Gromov-Witten invariants count curves in complex manifolds, fundamental in string theory.
• Logarithmic Gromov-Witten invariants extend these counts to curves with specific intersection properties.

Purpose of the Study:

• To provide a geometric interpretation of the Block-Göttsche invariants.
• To establish a connection between tropical curve counts and higher genus log Gromov-Witten invariants.
• To demonstrate the deformation invariance of these invariants.

Main Methods:

• Utilizing a change of variables transformation.
• Analyzing the resulting generating series.
• Connecting combinatorial counts to topological invariants.

Main Results:

• The Block-Göttsche q-number refinement is shown to be a generating series for higher genus log Gromov-Witten invariants.
• A lambda class insertion is identified within this generating series.
• The deformation invariance of the Block-Göttsche invariants is made explicit.

Conclusions:

• The study offers a new geometric perspective on the Block-Göttsche invariants.
• This work bridges combinatorial and topological approaches in enumerative geometry.
• The findings have implications for understanding string theory and related mathematical structures.