Abstract
We give a general, direct and explicit construction of lower-bounded generalized twisted modules satisfying a universal property for a grading-restricted vertex (super)algebra V associated to an automorphism g of V. In particular, when g is the identity, we obtain lower-bounded generalized V-modules satisfying a universal property. Let W be a lower-bounded graded vector space equipped with a set of “generating twisted fields” and a set of “generator twist fields” satisfying a weak commutativity for generating twisted fields, a generalized weak commutativity for one generating twisted field and one generator twist field and some other properties that are relatively easy to verify. We first prove the convergence and commutativity of products of an arbitrary number of generating twisted fields, one twist generator field and an arbitrary number of generating fields for V. Then using the convergence and commutativity, we define a twisted vertex operator map for W and prove that W equipped with this twisted vertex operator map is a lower-bounded generalized g-twisted V-module. Using this result, we give an explicit construction of lower-bounded generalized g-twisted V-modules satisfying a universal property starting from vector spaces graded by weights, \({\mathbb {Z}}_{2}\)-fermion numbers and g-weights (eigenvalues of g) and real numbers corresponding to the lower bounds of the weights of the modules to be constructed. In particular, every lower-bounded generalized g-twisted V-module (every lower-bounded generalized V-module when g is the identity) is a quotient of such a universal lower-bounded generalized g-twisted V-module (a universal lower-bounded generalized V-module).
Similar content being viewed by others
References
Bakalov, B.: Twisted logarithmic modules of vertex algebras. Commun. Math. Phys. 345, 355–383 (2015)
Barron, K., Dong, C., Mason, G.: Twisted sectors for tensor products vertex operator algebras associated to permutation groups. Commun. Math. Phys. 227, 349–384 (2002)
Barron, K., Huang, Y.-Z., Lepowsky, J.: An equivalence of two constructions of permutation-twisted modules for lattice vertex operator algebras. J. Pure Appl. Algebra 210, 797–826 (2007)
Dong, C.: Twisted modules for vertex algebras associated with even lattice. J. Algebra 165, 91–112 (1994)
Dong, C., Lepowsky, J.: The algebraic structure of relative twisted vertex operators. J. Pure Appl. Algebra 110, 259–295 (1996)
Dong, C., Li, H., Mason, G.: Twisted representations of vertex operator algebras. Math. Ann. 310, 571–600 (1998)
Dong, C., Li, H., Mason, G.: Modular invariance of trace functions in orbifold theory and generalized moonshine. Commun. Math. Phys. 214, 1–56 (2000)
Doyon, B., Lepowsky, J., Milas, A.: Twisted modules for vertex operator algebras and Bernoulli polynomials. Int. Math. Res. Not. 44, 2391–2408 (2003)
Doyon, B., Lepowsky, J., Milas, A.: Twisted vertex operators and Bernoulli polynomials. Commun. Contemp. Math. 8, 247–307 (2006)
Frenkel, I., Lepowsky, J., Meurman, A.: A natural representation of the Fischer-Griess Monster with the modular function \(J\) as character. Proc. Natl. Acad. Sci. USA 81, 3256–3260 (1984)
Frenkel, I., Lepowsky, J., Meurman, A.: Vertex operator calculus. In: Yau, S.-T. (ed.) Mathematical Aspects of String Theory, Proc. 1986 Conference, San Diego, pp. 150–188. World Scientific, Singapore (1987)
Frenkel, I., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster. Pure and Applied Math., vol. 134. Academic Press, Cambridge (1988)
Huang, Y.-Z.: Generalized twisted modules associated to general automorphisms of a vertex operator algebra. Commun. Math. Phys. 298, 265–292 (2010)
Huang, Y.-Z.: Two constructions of grading-restricted vertex (super)algebras. J. Pure Appl. Algebra 220, 3628–3649 (2016)
Huang, Y.-Z.: Some open problems in mathematical two-dimensional conformal field theory. In: Barron, K., Jurisich, E., Li, H., Milas, A., Misra, K.C. (eds.) Proceedings of the Conference on Lie Algebras, Vertex Operator Algebras, and Related Topics, held at University of Notre Dame, Notre Dame, 14–18 Aug, 2015. Contemp. Math, Vol. 695. American Mathematical Society, Providence, pp. 123–138 (2017)
Huang, Y.-Z.: On the applicability of logarithmic tensor category theory, to appear. arXiv:1702.00133
Huang, Y.-Z.: Intertwining operators among twisted modules associated to not-necessarily-commuting automorphisms. J. Algebra 493, 346–380 (2018)
Huang, Y.-Z.: Twist vertex operators for twisted modules. J. Algebra 539, 53–83 (2019)
Huang, Y.-Z., Lepowsky, J., Zhang, L.: Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, V: convergence condition for intertwining maps and the corresponding compatibility condition. arXiv: 1012.4199
Huang, Y.-Z., Yang, J.: Associative algebras for (logarithmic) twisted modules for a vertex operator algebra. Trans. Am. Math. Soc. 371, 3747–3786 (2019)
Lepowsky, J.: Calculus of twisted vertex operators. Proc. Natl. Acad. Sci. USA 82, 8295–8299 (1985)
Lepowsky, J.: Perspectives on vertex operators and the Monster. In: Proc. 1987 Symposium on the Mathematical Heritage of Hermann Weyl, Duke Univ., Proc. Symp. Pure. Math., American Math. Soc., vol. 48, pp. 181–197 (1988)
Li, H.: Local systems of twisted vertex operators, vertex operator superalgebras and twisted modules. In: Dong, C., Mason, G. (eds.) Moonshine, the Monster, and Related Topics Mount Holyoke, 1994. Contemporary Math., vol. 193. Amer. Math. Soc., Providence, pp. 203–236 (1996)
Yang, J.: Twisted representations of vertex operator algebras associated to affine Lie algebras. J. Algebra 484, 88–108 (2017)
Acknowledgements
The author is grateful to Jason Saied for questions on the construction of modules for grading-restricted vertex algebras using the approach in [H2].
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. Schweigert
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Huang, YZ. A Construction of Lower-Bounded Generalized Twisted Modules for a Grading-Restricted Vertex (Super)Algebra. Commun. Math. Phys. 377, 909–945 (2020). https://doi.org/10.1007/s00220-019-03582-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-019-03582-6