Mirror Counts of Spectral Curves
Abstract
Given a convex lattice polygon Δ⊂ ℝ2, let NΔ be the count of rational, nodal curves in the linear system of the ample line bundle LΔ on the toric variety ℙΔ defined by Δ, having fixed intersection with the toric boundary. We show that the relative Jacobian of the linear system defines an integrable system that has a mirror dual in the language of constructible sheaves on a two-torus microsupported on a Legendrian link. In this setting, we define the dual counting problem using an analogue of rulings of Legendrian links in three-space, then prove equivalence with NΔ. We perform calculations of NΔ in several examples using constructible methods, tropical curve counting, and localization in logarithmic Gromov-Witten theory to demonstrate the equality of the different approaches. Moreover, the rulings give rise to a stratification of the moduli of constructible sheaves. We conjecture that this ruling decomposition recovers the refined tropical invariants of Block-Göttsche, and hence encodes higher-genus logarithmic Gromov--Witten invariants, by a theorem of Bousseau.
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BibTeX
@misc{graber2026mirror,
title = {{Mirror Counts of Spectral Curves}},
author = {Tom Graber and Mingyuan Hu and Eric Zaslow},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06821},
eprint = {2609.06821},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06821},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Mirror Counts of Spectral Curves
AU - Tom Graber
AU - Mingyuan Hu
AU - Eric Zaslow
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06821
UR - https://arxiv.org/abs/2609.06821
AB - Given a convex lattice polygon Δ⊂ ℝ^2, let N_Δ be the count of rational, nodal curves in the linear system of the ample line bundle L_Δ on the toric variety ℙ_Δ defined by Δ, having fixed intersection with the toric boundary. We show that the relative Jacobian of the linear system defines an integrable system that has a mirror dual in the language of constructible sheaves on a two-torus microsupported on a Legendrian link. In this setting, we define the dual counting problem using an analogue of rulings of Legendrian links in three-space, then prove equivalence with N_Δ. We perform calculations of N_Δ in several examples using constructible methods, tropical curve counting, and localization in logarithmic Gromov-Witten theory to demonstrate the equality of the different approaches. Moreover, the rulings give rise to a stratification of the moduli of constructible sheaves. We conjecture that this ruling decomposition recovers the refined tropical invariants of Block-Göttsche, and hence encodes higher-genus logarithmic Gromov--Witten invariants, by a theorem of Bousseau.
ER -