FJRW rings and Landau-Ginzburg Mirror Symmetry

TL;DR

This paper explores the Landau-Ginzburg mirror symmetry by establishing isomorphisms between A-models and B-models for invertible potentials, and interprets Arnold's strange duality as a form of mirror symmetry.

Contribution

It introduces the dual group construction and proves a mirror symmetry theorem at the Frobenius algebra level for invertible potentials.

Findings

Established state space isomorphism between FJRW A-model and orbifold Milnor ring B-model.

Proved mirror symmetry theorem for G^max at Frobenius algebra level.

Interpreted Arnold's strange duality as mirror symmetry between W/J and W^SD.

Abstract

In this article, we study the Berglund--H\"ubsch transpose construction W^T for invertible quasihomogeneous potential W. We introduce the dual group G^T and establish the state space isomorphism between the Fan-Jarvis-Ruan-Witten A-model of W/G and the orbifold Milnor ring B-model of W^T/G^T. Furthermore, we prove a mirror symmetry theorem at the level of Frobenius algebra structure for G^max. Then, we interpret Arnol'd strange duality of exceptional singularities W as mirror symmetry between W/J and its strange dual W^SD.