A combinatorial calculation of the Landau-Ginzburg model $M= \mathbb C^3, W=z_1 z_2 z_3$

TL;DR

This paper computes the Landau-Ginzburg A-model for a specific singularity in mirror symmetry, revealing its equivalence to the B-model of the pair-of-pants, thus illustrating a concrete example of mirror symmetry.

Contribution

It applies microlocal sheaf theory to explicitly calculate the Landau-Ginzburg A-model for a singularity, demonstrating its mirror equivalence to a well-known B-model.

Findings

Explicit calculation of the Landau-Ginzburg A-model for $M=\mathbb{C}^3, W=z_1 z_2 z_3$

Demonstration of the equivalence to the B-model of the pair-of-pants

Confirmation of mirror symmetry predictions in a specific example

Abstract

The aim of this paper is to apply ideas from the study of Legendrian singularities to a specific example of interest within mirror symmetry. We calculate the Landau-Ginzburg $A$-model with $M= \mathbb C^3, W=z_1 z_2 z_3$ in its guise as microlocal sheaves along the natural singular Lagrangian thimble $L = {\mathit Cone}(T^2)\subset M$. The description we obtain is immediately equivalent to the $B$-model of the pair-of-pants $\mathbb P^1 \setminus \{0, 1, \infty\}$ as predicted by mirror symmetry.