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.
Similar content being viewed by others
Data Availability
No datasets were generated or analysed during the current study.
References
Dijkgraaf, R., Vafa, C., Verlinde, E., Verlinde, H.: The operator algebra of orbifold models. Comm. Math. Phys. 123(3), 485–526 (1989)
Dixon, L., Harvey, J., Vafa, C., Witten, E.: Strings on orbifolds. Nucl. Phys. B 261(4), 678–686 (1985). II. Nucl. Phys. B 274(2), 285–314 (1986)
Dong, C.-Y.: Twisted modules for vertex algebras associated with even lattices. J. Algebra 165(1), 91–112 (1994)
Dong, C.-Y., Li, H.-S., Mason, G.: Twisted representations of vertex operator algebras. Math. Ann. 310(3), 571–600 (1998)
Dong, C.-Y., Li, H.-S., Mason, G.: Twisted representations of vertex operator algebras and associative algebras. Int. Math. Res. Notices (8), 389–397 (1998)
Dong, C.-Y., Nagatomo, K.: Automorphism groups and twisted modules for lattice vertex operator algebras. Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998). Contemp. Math. 248, 117–133. Amer. Math. Soc., Providence, RI (1999)
Dong, C.-Y., Ren, L., Xu, F.: On orbifold theory. Adv. Math. 321, 1–30 (2017)
Feingold, A., Frenkel, I., Reis, J.: Spinor construction of vertex operator algebras, triality, and \(E_{8}^{(1)}\). Contemp. Math. vol. 121. American Mathematical Society, Providence, RI, x\(+\)146 (1991)
Frenkel, E., Szczesny, M.: Twisted modules over vertex algebras on algebraic curves. Adv. Math. 187(1), 195–227 (2004)
Frenkel, I., Lepowsky, J., Meurman, A.: Vertex operator algebras and the monster. Pure Appl. Math. 134. Academic Press, Massachusetts, liv\(+\)508 (1988)
Guo, H.-Y., Li, H.-M.: Restricted modules and associated vertex algebras of extended Heisenberg-Virasoro algebra. J. Algebra 635, 463–485 (2023)
Guo, H.-Y., Qi, Q.-X.: Whittaker modules of the extended mirror Heisenberg-Virasoro algebra, in preparation
Guo, H.-Y., Xu, C.-K.: Restricted modules for gap-\(p\) Virasoro algebra and twisted modules for certain vertex algebras. J. Pure Appl. Algebra 227(7), 107322, 17 (2023)
Lepowsky, J., Li, H.-S.: Introduction to vertex operator algebras and their representations. Progr. Math. 227. Birkhäuser, Boston, xiv\(+\)318 (2004)
Li, H.-S.: Local systems of vertex operators, vertex superalgebras and modules. J. Pure Appl. Algebra 109(2), 143–195 (1996)
Li, H.-S.: Local systems of twisted vertex operators, vertex operator superalgebras and twisted modules. Moonshine, the Monster, and related topics. Contemp. Math. 193, 203–236. Amer. Math. Soc., Providence, RI (1996)
Mazorchuk, V., Zhao, K.-M.: Simple Virasoro modules which are locally finite over a positive part. Selecta Math. (N.S.) 20(3), 839–854 (2014)
Xu, X.-P.: Intertwining operators for twisted modules of a colored vertex operator superalgebra. J. Algebra 175(1), 241–273 (1995)
Yang, J.-W.: Twisted representations of vertex operator algebras associated to affine Lie algebras. J. Algebra 484, 88–108 (2017)
Funding
H. Guo is partially supported by National Natural Science Foundation of China (No. 11901224) and the Fundamental Research Funds for the Central Universities (No. CCNU22QN002).
Author information
Authors and Affiliations
Contributions
All authors discussed the results and contributed to the final manuscript.
Corresponding author
Ethics declarations
Ethical Approval
This declaration is not applicable.
Competing interests
The authors declare no competing interests.
Additional information
Presented by: Anne Moreau
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Guo, H., Li, H. Twisted Representations of the Extended Heisenberg-Virasoro Vertex Operator Algebra. Algebr Represent Theor 27, 1563–1580 (2024). https://doi.org/10.1007/s10468-024-10270-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-024-10270-0
Keywords
- Extended Heisenberg-Virasoro algebra
- Vertex operator algebra
- Weak twisted module
- Ordinary twisted module
- Restricted module