Landau-Ginzburg/Conformal Field Theory Correspondence for $x^d$ and Module Tensor Categories

TL;DR

This paper establishes a tensor equivalence between categories of matrix factorisations of the polynomial $x^d$ and categories from $N=2$ supersymmetric conformal field theories for all $d$, extending previous odd-$d$ results.

Contribution

We generalize the Landau-Ginzburg/Conformal Field Theory correspondence to include even degrees $d$, using module tensor categories over $ ext{Vec}_{bZ_d}$ to construct the equivalence.

Findings

Established tensor equivalence for all $d$, including even cases.

Used module tensor categories to describe the CFT side as generated by a single object.

Provided an explicit functor realizing the tensor equivalence.

Abstract

The Landau-Ginzburg/Conformal Field Theory correspondence predicts tensor equivalences between categories of matrix factorisations of certain polynomials and categories associated to the $N=2$ supersymmetric conformal field theories. We realise this correspondence for $x^d$ for any $d$, where previous results were limited to odd $d$. Our proof uses the fact that both sides of the correspondence carry the structure of module tensor categories over the category of $\mathbb{Z}_d$-graded vector spaces equipped with a non-degenerate braiding. This allows us to describe the CFT side as generated by a single object, and use this to efficiently provide a functor realising the tensor equivalence.