Twisted Zhu algebras

TL;DR

This paper studies twisted Zhu algebras of vertex superalgebras, proving they have PBW bases, are isomorphic to universal enveloping algebras of non-linear Lie superalgebras, and applying these results to affine superalgebras and W-algebras.

Contribution

It establishes the structure of twisted Zhu algebras as PBW bases and their isomorphism to universal enveloping algebras, extending prior results to twisted cases.

Findings

Twisted Zhu algebras have PBW bases.

They are isomorphic to universal enveloping algebras of non-linear Lie superalgebras.

Computed twisted Zhu algebras for affine vertex superalgebras and W-algebras.

Abstract

Let $V$ be a freely generated pregraded vertex superalgebra, $H$ a Hamiltonian operator of $V$, and $g$ a diagonalizable automorphism of V commuting with $H$ with modulus $1$ eigenvalues. We prove that the $(g, H)$-twisted Zhu algebra of $V$ has a PBW basis, is isomorphic to the universal enveloping algebra of some non-linear Lie superalgebra, and satisfies the commutativity of BRST cohomology functors, which generalizes results of De Sole and Kac. As applications, we compute the twisted Zhu algebras of affine vertex superalgebras and affine $W$-algebras.