Semisimple FJRW theory of polynomials with two variables

TL;DR

This paper investigates the structure of Dubrovin-Frobenius manifolds in FJRW theory for two-variable polynomials, confirming semisimplicity in many cases and supporting related conjectures.

Contribution

It proves the semisimplicity of Dubrovin-Frobenius manifolds for certain Landau-Ginzburg pairs with two variables, extending known results to a broader class.

Findings

Semisimplicity confirmed for simple singularities.

Semisimplicity confirmed for almost all Brieskorn-Pham polynomials.

Supports Dubrovin type and Virasoro conjectures in these cases.

Abstract

We study the Dubrovin-Frobenius manifold in the Fan-Jarvis-Ruan-Witten theory of Landau-Ginzburg pairs $(W, \<J\>)$, where $W$ is an invertible nondegenerate quasihomogeneous polynomial with two variables and $\<J\>$ is the minimal admissible group of $W$. We conjecture that the Dubrovin-Frobenius manifolds from these FJRW theory are semisimple. We show the conjecture holds true for simple singularities and almost all Brieskorn-Pham polynomials. For Brieskorn-Pham polynomials, the result follows from the calculation of a quantum Euler class in the FJRW theory. As a consequence, our result shows that for the FJRW theory of these Landau-Ginzburg pairs, both a Dubrovin type conjecture and a Virasoro conjecture hold true.