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

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.