Landau-Ginzburg Mirror Symmetry for Orbifolded Frobenius Algebras

TL;DR

This paper proves the Landau-Ginzburg Mirror Symmetry Conjecture for a broad class of invertible singularities, establishing an isomorphism between Frobenius algebras in FJRW theory and orbifolded Milnor rings.

Contribution

It demonstrates the mirror symmetry conjecture at the algebraic level for various invertible singularities, including sums of loops and Fermats with arbitrary symmetry groups.

Findings

FJRW Frobenius algebra is isomorphic to the orbifolded Milnor ring of the dual polynomial.

The result applies to a large class of invertible singularities.

The proof covers arbitrary sums of loops and Fermats with arbitrary symmetry groups.

Abstract

We prove the Landau-Ginzburg Mirror Symmetry Conjecture at the level of (orbifolded) Frobenius algebras for a large class of invertible singularities, including arbitrary sums of loops and Fermats with arbitrary symmetry groups. Specifically, we show that for a quasi-homogeneous polynomial W and an admissible group G within the class, the Frobenius algebra arising in the FJRW theory of [W/G] is isomorphic (as a Frobenius algebra) to the orbifolded Milnor ring of [W^T/G^T], associated to the dual polynomial W^T and dual group G^T.