Tropical refined curve counting from higher genera and lambda classes

Abstract

Block and Göttsche have defined a $q$-number refinement of counts of tropical curves in ${\mathbb{R}}^2$. Under the change of variables $q=e^{iu}$, we show that the result is a generating series of higher genus log Gromov-Witten invariants with insertion of a lambda class. This gives a geometric interpretation of the Block-Göttsche invariants and makes their deformation invariance manifest.

arXiv comment70 pages, revised version, identical to the one published in Inven. Math

References in corpus (8)

• The intrinsic normal cone
• Virasoro constraints and the Chern classes of the Hodge bundle
• Hodge integrals and degenerate contributions
• Curves on K3 surfaces and modular forms
• The tropical vertex
• Refined curve counting with tropical geometry
• Refined curve counting on complex surfaces
• Block-Göttsche invariants from wall-crossing

Cited by in corpus (18)

• The quantum tropical vertex
• The canonical wall structure and intrinsic mirror symmetry
• Holomorphic anomaly equation for $({\mathbb{P}}^2,E)$ and the Nekrasov-Shatashvili limit of local ${\mathbb{P}}^2$
• Stable maps to Looijenga pairs
• Scattering diagrams, stability conditions, and coherent sheaves on ${\mathbb{P}}^2$
• Quantum mirrors of log Calabi-Yau surfaces and higher genus curve counting
• Strong positivity for the skein algebras of the $4$-punctured sphere and of the $1$-punctured torus
• Scattering diagrams, theta functions, and refined tropical curve counts
• A proof of N.Takahashi's conjecture for $({\mathbb{P}}^2,E)$ and a refined sheaves/Gromov-Witten correspondence
• Donaldson-Thomas invariants from tropical disks
• Refined floor diagrams from higher genera and lambda classes
• Stable maps to Looijenga pairs: orbifold examples
• Tropical curves in abelian surfaces I: enumeration of curves passing through points
• Computation of refined toric invariants II
• The local/logarithmic correspondence and the degeneration formula for quasimaps
• Enumerative geometry of surfaces and topological strings
• Tropical vertex and real enumerative geometry
• The log-open correspondence for two-component Looijenga pairs