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$p$-adic Berglund-Hübsch Duality

Marco Aldi, Andrija Peruničić

Abstract

Berglund-Hübsch duality is an example of mirror symmetry between orbifold Landau-Ginzburg models. In this paper we study a D-module-theoretic variant of Borisov's proof of Berglund-Hübsch duality. In the $p$-adic case, the D-module approach makes it possible to endow the orbifold chiral rings with the action of a non-trivial Frobenius endomorphism. Our main result is that the Frobenius endomorphism commutes with Berglund-Hübsch duality up to an explicit diagonal operator.

$p$-adic Berglund-Hübsch Duality

Abstract

Berglund-Hübsch duality is an example of mirror symmetry between orbifold Landau-Ginzburg models. In this paper we study a D-module-theoretic variant of Borisov's proof of Berglund-Hübsch duality. In the -adic case, the D-module approach makes it possible to endow the orbifold chiral rings with the action of a non-trivial Frobenius endomorphism. Our main result is that the Frobenius endomorphism commutes with Berglund-Hübsch duality up to an explicit diagonal operator.

Paper Structure

This paper contains 10 sections, 19 theorems, 103 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Invertible Polynomials
  3. Exterior Operators
  4. The Basic Complex
  5. Bigrading
  6. Unprojected Orbifold de Rham Cohomology
  7. Unprojected Duality
  8. Overconvergent Power Series
  9. The Frobenius Endomorphism
  10. Examples

Key Result

Proposition 2.3

Let $A \in \textnormal{BH}_n(\mathbb F)$ be irreducible. Then there exists $S\in \mathcal{W}_n$ such that $W_{AS}(x)$ is in one of the following canonical forms:

Figures (1)

  • Figure 1: Eigenspaces of $T_{A,2}$ in $F^2$ for $W_A = x_1^2 + x_1 x_2^3$, with eigenvalues designated along the left. Each point represents $\gamma$ in $x^\gamma$.

Theorems & Definitions (44)

  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3: Kre
  • Corollary 2.4
  • Corollary 2.5
  • proof
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Lemma 4.1
  • ...and 34 more