Twisted modules of $\frac{1}{2}\mathbb{Z}$-graded modular vertex superalgebras

TL;DR

This paper develops a theory for twisted modules of $rac{1}{2}bZ$-graded modular vertex superalgebras over fields of prime characteristic, extending Zhu's algebra and classifying irreducible twisted modules for specific superalgebras.

Contribution

It introduces a twisted Zhu algebra for modular vertex superalgebras and establishes a correspondence with twisted modules, including explicit classifications for affine Lie superalgebras and the Neveu-Schwarz algebra.

Findings

Constructed twisted Zhu algebra $A_g(V)$ for prime characteristic fields.

Established a bijection between simple $A_g(V)$-modules and twisted $V$-modules.

Classified irreducible twisted modules for the modular Neveu-Schwarz algebra.

Abstract

In this paper, we investigate the theory of $g$-twisted modules for modular $\frac{1}{2}\mathbb{Z}$-graded vertex superalgebras over an algebraically closed field $\mathbb{F}$ of prime characteristic $p>2$. For a $\frac{1}{2}\mathbb{Z}$-graded vertex superalgebra $V$ and an automorphism $g$ of $V$ of finite order $T$ relatively prime to $p$, we give a twisted version of Zhu's associative algebra, denoted by $A_g(V)$. We prove that there is a one-to-one correspondence between the set of equivalence classes of simple $A_g(V)$-modules and the set of equivalence classes of simple $\frac{1}{T_0}\mathbb{N}$-graded $g$-twisted $V$-modules, where $T_0$ is the order of the automorphism $g\sigma$ with $\sigma$ the parity automorphism. As an application, we study twisted modules for modular vertex superalgebras associated to the affine Lie superalgebras and determine the corresponding twisted Zhu algebra. We also compute the twisted Zhu algebra for the modular Neveu-Schwarz vertex superalgebra and classify its irreducible twisted modules.