Abstract
The Zamolodchikov \({\mathcal{W}_3}\)-algebra \({\mathcal{W}^c_3}\) with central charge c has full automorphism group \({\mathbb{Z}_2}\). It was conjectured in the physics literature over 20 years ago that the orbifold \({(\mathcal{W}^c_3)^{\mathbb{Z}_2}}\) is of type \({\mathcal{W}(2,6,8,10,12)}\) for generic values of c. We prove this conjecture for all \({c \neq \frac{559 \pm 7 \sqrt{76657}}{95}}\), and we show that for these two values, the orbifold is of type \({\mathcal{W}(2,6,8,10,12,14)}\). This paper is part of a larger program of studying orbifolds and cosets of vertex algebras that depend continuously on a parameter. Minimal strong generating sets for orbifolds and cosets are often easy to find for generic values of the parameter, but determining which values are generic is a difficult problem. In the example of \({(\mathcal{W}^c_3)^{\mathbb{Z}_2}}\), we solve this problem using tools from algebraic geometry.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arakawa T.: Rationality of \({\mathcal{W}}\)-algebras: principal nilpotent cases. Ann. Math. 182(2), 565–604 (2015)
Arakawa, T., Creutzig, T., Linshaw, A.: Cosets of Bershadsky–Polyakov algebras and rational \({\mathcal{W}}\)-algebras of type A. arXiv:1511.09143
Arakawa, T., Creutzig, T., Kawasetsu, K., Linshaw, A.: Orbifolds and cosets of minimal \({\mathcal{W}}\)-algebras. arXiv:1610.09348
Blumenhagen R., Eholzer W., Honecker A., Hornfeck K., Hubel R.: Coset realizations of unifying \({\mathcal{W}}\)-algebras. Int. J. Mod. Phys. Lett. A 10, 2367–2430 (1995)
Borcherds R.: Vertex operator algebras, Kac–Moody algebras and the monster. Proc. Natl. Acad. Sci. USA 83, 3068–3071 (1986)
Bouwknegt P., Schoutens K.: \({\mathcal{W}}\)-symmetry in conformal field theory. Phys. Rep. 223, 183–276 (1993)
Carnahan, S., Miyamoto, M.: Rationality of fixed-point vertex operator algebras. arXiv:1603.05645
Creutzig, T., Linshaw, A.: Cosets of affine vertex algebras inside larger structures. arXiv:1407.8512v3
Dijkgraaf R., Vafa C., Verlinde E., Verlinde H.: The operator algebra of orbifold models. Commun. Math. Phys. 123, 485–526 (1989)
Dixon L., Harvey J., Vafa C., Witten E.: Strings on orbifolds. Nucl. Phys. B 261, 678–686 (1985)
Dong C., Mason G.: On quantum Galois theory. Duke Math. J. 86, 305–321 (1997)
Dong C., Li H., Mason G.: Compact automorphism groups of vertex operator algebras. Int. Math. Res. Not. 18, 913–921 (1996)
Dong C., Li H., Mason G.: Twisted representations of vertex operator algebras. Math. Ann. 310, 571–600 (1998)
Dong, C., Ren, L., Xu, F.: On orbifold theory. arXiv:1507.03306
Frenkel E., Ben-Zvi D.: Vertex Algebras and Algebraic Curves. Mathematical Surveys and Monographs, vol. 88. American Mathematical Society, Providence (2001)
Frenkel I.B., Lepowsky J., Meurman A.: Vertex Operator Algebras and the Monster. Academic Press, New York (1988)
Frenkel I.B., Zhu Y.C.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66(1), 123–168 (1992)
Kac V.: Vertex Algebras for Beginners, University Lecture Series, vol. 10. American Mathematical Society, Providence (1998)
Li H.: Local systems of vertex operators, vertex superalgebras and modules. J. Pure Appl. Algebra 109(2), 143–195 (1996)
Li H.: Vertex algebras and vertex Poisson algebras. Commun. Contemp. Math. 6, 61–110 (2004)
Lian B., Zuckerman G.: Commutative quantum operator algebras. J. Pure Appl. Algebra 100(1–3), 117–139 (1995)
Lian B., Linshaw A.: Howe pairs in the theory of vertex algebras. J. Algebra 317, 111–152 (2007)
Linshaw A.: Invariant subalgebras of affine vertex algebras. Adv. Math. 234, 61–84 (2013)
Miyamoto, M.: C 2-cofiniteness of cyclic orbifold models. Commun. Math. Phys. 335, 1279–1286 (2015)
Zamolodchikov, A.B.: Infinite extra symmetries in two-dimensional conformal quantum field theory (Russian). Teoret. Mat. Fiz. 65, 347–359 (1985) [English translation, Theoret. Math. Phys. 65, 1205–1213 (1985)]
Zhu Y.: Modular invariants of characters of vertex operators. J. Am. Math. Soc. 9, 237–302 (1996)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Kawahigashi
Rights and permissions
About this article
Cite this article
Al-Ali, M., Linshaw, A.R. The \({\mathbb{Z}_2}\)-Orbifold of the \({\mathcal{W}_3}\)-Algebra. Commun. Math. Phys. 353, 1129–1150 (2017). https://doi.org/10.1007/s00220-016-2812-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-016-2812-7