Hochschild cohomology of Fermat type polynomials with non-abelian symmetries

Abstract

For a polynomial $f=x_1^n+\cdots +x_N^n$ let $G_f$ be the non--abelian maximal group of symmetries of $f$. This is a group generated by all $g\in \mathrm{GL}(N,\mathbb{C})$, rescaling and permuting the variables, so that $f(\mathbf{x})=f(g\cdot \mathbf{x})$. For any $G\subseteq G_f$ we compute explicitly Hochschild cohomology of the category of $G$--equivarint matrix factorizations of $f$. We introduce the pairing on it showing that it is a Frobenius algebra.