Transposing Noninvertible Polynomials

TL;DR

This paper disproves conjectured isomorphisms between certain algebraic models in Landau-Ginzburg mirror symmetry when the polynomial W has more monomials than variables, revealing limitations of existing conjectures.

Contribution

It demonstrates that the previously conjectured isomorphisms between A and B models do not hold in the noninvertible case, advancing understanding of algebraic structures in mirror symmetry.

Findings

Conjectured isomorphisms are false for noninvertible polynomials.

Provides new insights into group actions on graded algebras.

Clarifies limitations of mirror symmetry conjectures.

Abstract

Landau-Ginzburg mirror symmetry predicts isomorphisms between graded Frobenius algebras (denoted $\mathcal{A}$ and $\mathcal{B}$) that are constructed from a nondegenerate quasihomogeneous polynomial $W$ and a related group of symmetries $G$. Duality between $\mathcal{A}$ and $\mathcal{B}$ models has been conjectured for particular choices of $W$ and $G$. These conjectures have been proven in many instances where $W$ is restricted to having the same number of monomials as variables (called $\textit{invertible}$). Some conjectures have been made regarding isomorphisms between $\mathcal{A}$ and $\mathcal{B}$ models when $W$ is allowed to have more monomials than variables. In this paper we show these conjectures are false; that is, the conjectured isomorphisms do not exist. Insight into this problem will not only generate new results for Landau-Ginzburg mirror symmetry, but will also be interesting from a purely algebraic standpoint as a result about groups acting on graded algebras.