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

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.