Twisted Representations of the Extended Heisenberg-Virasoro Vertex Operator Algebra

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.