A tropical view on Landau-Ginzburg models

Abstract

This paper, largely written in 2009/2010, fits Landau-Ginzburg models into the mirror symmetry program pursued by the last author jointly with Mark Gross since 2001. This point of view transparently brings in tropical disks of Maslov index 2 via the notion of broken lines, previously introduced in two dimensions by Mark Gross in his study of mirror symmetry for ${\mathbb{P}}^2$. A major insight is the equivalence of properness of the Landau-Ginzburg potential with smoothness of the anticanonical divisor on the mirror side. We obtain proper superpotentials which agree on an open part with those classically known for toric varieties. Examples include mirror LG models for non-singular and singular del Pezzo surfaces, Hirzebruch surfaces and some Fano threefolds.

arXiv comment53 pages, 17 figures, 2010 version submitted as ancillary file. v2: Prop.5.6 corrected

References in corpus (7)

• Phases of $N=2$ Theories In Two Dimensions
• Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves
• Lagrangian Floer theory on compact toric manifolds I
• The tropical vertex
• Smoothing toroidal crossing spaces
• A homology theory for tropical cycles on integral affine manifolds and a perfect pairing
• Open Gromov-Witten invariants and superpotentials for semi-Fano toric surfaces

Cited by in corpus (6)

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• Strong positivity for the skein algebras of the $4$-punctured sphere and of the $1$-punctured torus
• Cluster structures and subfans in scattering diagrams
• Scattering diagrams, theta functions, and refined tropical curve counts
• Scattering diagrams for generalized cluster algebras