Hochschild cohomology of Fermat type polynomials with non-abelian symmetries

TL;DR

This paper computes the Hochschild cohomology of categories of G-equivariant matrix factorizations for Fermat type polynomials with non-abelian symmetry groups, revealing Frobenius algebra structures.

Contribution

It explicitly calculates Hochschild cohomology for non-abelian symmetry groups of Fermat polynomials and introduces a Frobenius pairing on it.

Findings

Hochschild cohomology is explicitly computed for non-abelian symmetry groups.

The cohomology forms a Frobenius algebra with a new pairing.

Provides detailed algebraic structures of matrix factorizations under symmetries.

Abstract

For a polynomial $f = x_1^n + \dots + x_N^n$ 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.