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Mirror Map for Landau-Ginzburg models with nonabelian groups

Annabelle Clawson, Drew Johnson, Duncan Morais, Nathan Priddis, Caroline B. White

Abstract

BHK mirror symmetry as introduced by Berglund--Hübsch and Marc Krawitz between Landau--Ginzburg (LG) models has been the topic of much study in recent years. An LG model is determined by a potential function and a group of symmetries. BHK mirror symmetry is only valid when the group of symmetries is comprised of the so-called diagonal symmetries. Recently, an extension to BHK mirror symmetry to include nonabelian symmetry groups has been conjectured. In this article, we provide a mirror map at the level of state spaces between the LG A-model state space and the LG B-model state space for the mirror model predicted by the BHK mirror symmetry extension for nonabelian LG models. We introduce two technical conditions, the Diagonal Scaling Condition, and the Equivariant $Φ$ condition, under which a bi-degree preserving isomorphism of state spaces (the mirror map) is guaranteed to exist, and we prove that the condition is always satisfied if the permutation part of the group is cyclic of prime order.

Mirror Map for Landau-Ginzburg models with nonabelian groups

Abstract

BHK mirror symmetry as introduced by Berglund--Hübsch and Marc Krawitz between Landau--Ginzburg (LG) models has been the topic of much study in recent years. An LG model is determined by a potential function and a group of symmetries. BHK mirror symmetry is only valid when the group of symmetries is comprised of the so-called diagonal symmetries. Recently, an extension to BHK mirror symmetry to include nonabelian symmetry groups has been conjectured. In this article, we provide a mirror map at the level of state spaces between the LG A-model state space and the LG B-model state space for the mirror model predicted by the BHK mirror symmetry extension for nonabelian LG models. We introduce two technical conditions, the Diagonal Scaling Condition, and the Equivariant condition, under which a bi-degree preserving isomorphism of state spaces (the mirror map) is guaranteed to exist, and we prove that the condition is always satisfied if the permutation part of the group is cyclic of prime order.
Paper Structure (36 sections, 52 theorems, 220 equations, 1 table)

This paper contains 36 sections, 52 theorems, 220 equations, 1 table.

Table of Contents

  1. Introduction
  2. Invertible Polynomials and their Symmetries
  3. The Polynomial W
  4. The Group G
  5. Diagonal Symmetries
  6. Permutations
  7. Permutations of variables in invertible polynomials
  8. Semidirect Products
  9. A basis for $\mathop{\mathrm{Fix}}\nolimits(\sigma\lambda)$
  10. State Spaces
  11. Milnor Rings
  12. The Definition of State Spaces
  13. State space computation example
  14. Relation to other work
  15. BHK Mirror Symmetry for Diagonal Groups
  16. ...and 21 more sections

Key Result

Corollary 1

Let $(W,G)$ be a pair consisting of an invertible polynomial $W$ and a group $G=S\rtimes H$ with $S$ generated by a permutation of odd prime order. Then there is a bidegree-preserving isomorphism between the state space of the LG A-model for $(W,G)$ and the LG B-model state space for the mirror pair

Theorems & Definitions (122)

  • Corollary
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Kreuzer and Skarke KS
  • Example 2.5
  • Example 2.6
  • Remark
  • Definition 2.7: Mukai M
  • Definition 2.8
  • Definition 2.9: Mukai M
  • ...and 112 more