Orbifold Saito theory of A and D type singularities

TL;DR

This paper develops Saito theory for A and D type Landau-Ginzburg orbifolds, constructing orbifold Brieskorn lattices and Gauss-Manin connections explicitly for these singularities.

Contribution

It introduces the first explicit construction of Saito theory for A and D type orbifold Landau-Ginzburg models, including orbifold Brieskorn lattices and Gauss-Manin connections.

Findings

Constructed orbifold Brieskorn lattices for A and D types.

Explicit formulas for Gauss-Manin connections in these cases.

Extended Saito theory to orbifold Landau-Ginzburg models.

Abstract

Saito theory associates to an isolated singularity rich structure that plays an important role in mirror symmetry. In this note we construct Saito theory for A and D type Landau--Ginzburg orbifolds. Namely, for the pairs $(f,G)$, where $f$ defines an isolated singularity of A and D type and $G$ is a group of symmetries of $f$. In total we consider five families of such pairs. In particular, we construct the orbifold versions of Brieskorn lattice and the Gauss--Manin connection computing them explicitly for the A and D type Landau--Ginzburg orbifolds.