Twisted Representations of the Extended Heisenberg-Virasoro Vertex Operator Algebra

Abstract

In this paper, we study simple weak and ordinary twisted modules of the extended Heisenberg-Virasoro vertex operator algebra $V_{\tilde{\mathcal {L}}_{F}}(\ell _{123},0)$. We first determine the full automorphism groups of $V_{\tilde{\mathcal {L}}_{F}}(\ell _{123},0)$ for all $\ell _{1}, \ell _{2},\ell _{3},F\in {\mathbb C}$. They are isomorphic to certain subgroups of the general linear group $\text {GL}_{2}({\mathbb C})$. Then for a family of finite order automorphisms $\sigma _{r_{1},r_{2}}$ of $V_{\tilde{\mathcal {L}}_{F}}(\ell _{123},0)$, we show that weak $\sigma _{r_{1},r_{2}}$-twisted $V_{\tilde{\mathcal {L}}_{F}}(\ell _{123},0)$-modules are in one-to-one correspondence with restricted modules of certain Lie algebras of level $\ell _{123}$, where $r_{1}, r_2\in {\mathbb N}$. By this identification and vertex algebra theory, we give complete lists of simple ordinary $\sigma _{r_{1},r_{2}}$-twisted modules over $V_{\tilde{\mathcal {L}}_{F}}(\ell _{123},0)$. The results depend on whether F or $\ell _{2}$ is zero or not. Furthermore, simple weak $\sigma _{r_{1},r_{2}}$-twisted $V_{\tilde{\mathcal {L}}_{F}}(\ell _{123},0)$-modules are also investigated. For this, we introduce and study restricted modules (including Whittaker modules) of a new Lie algebra $\mathcal {L}_{r_{1},r_{2}}$ which is related to the mirror Heisenberg-Virasoro algebra.