A level-rank duality for parafermion vertex operator algebras of type A

Lam, Ching Hung

doi:10.1090/S0002-9939-2014-12167-8

A level-rank duality for parafermion vertex operator algebras of type A
HTML articles powered by AMS MathViewer

by Ching Hung Lam PDF

Proc. Amer. Math. Soc. 142 (2014), 4133-4142 Request permission

Abstract:

We show that the tensor product of the parafermion vertex operator algebras $K(sl_{k+1},n+1) \otimes K(sl_{n+1}, k+1)$ can be embedded as a full subVOA into the lattice VOA $V_{A_n\otimes A_k}$. The decomposition of $V_{A_n\otimes A_k}$ as a direct sum of irreducible $K(sl_{k+1},n+1) \otimes K(sl_{n+1}, k+1)$-modules is also obtained. In addition, we show that the parafermion VOA $K(sl_{n}, k)$ contains a full subVOA isomorphic to a tensor product of $W$-algebras $W_{sl_{k}}(1,1)\otimes W_{sl_{k}}(1,2) \otimes \cdots \otimes W_{sl_{k}}(1,n-1)$.

References

Daniel Altschüler, Michel Bauer, and Claude Itzykson, The branching rules of conformal embeddings, Comm. Math. Phys. 132 (1990), no. 2, 349–364. MR 1069826
Chongying Dong, Ching Hung Lam, and Hiromichi Yamada, $W$-algebras related to parafermion algebras, J. Algebra 322 (2009), no. 7, 2366–2403. MR 2553685, DOI 10.1016/j.jalgebra.2009.03.034
Chongying Dong, Ching Hung Lam, Qing Wang, and Hiromichi Yamada, The structure of parafermion vertex operator algebras, J. Algebra 323 (2010), no. 2, 371–381. MR 2564844, DOI 10.1016/j.jalgebra.2009.08.003
Chongying Dong and James Lepowsky, Generalized vertex algebras and relative vertex operators, Progress in Mathematics, vol. 112, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1233387, DOI 10.1007/978-1-4612-0353-7
Chongying Dong and Qing Wang, The structure of parafermion vertex operator algebras: general case, Comm. Math. Phys. 299 (2010), no. 3, 783–792. MR 2718932, DOI 10.1007/s00220-010-1114-8
Igor Frenkel, James Lepowsky, and Arne Meurman, Vertex operator algebras and the Monster, Pure and Applied Mathematics, vol. 134, Academic Press, Inc., Boston, MA, 1988. MR 996026
Igor B. Frenkel and Yongchang Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J. 66 (1992), no. 1, 123–168. MR 1159433, DOI 10.1215/S0012-7094-92-06604-X
P. Goddard, A. Kent, and D. Olive, Virasoro algebras and coset space models, Phys. Lett. B 152 (1985), no. 1-2, 88–92. MR 778819, DOI 10.1016/0370-2693(85)91145-1
Robert L. Griess Jr. and Ching Hung Lam, $EE_8$-lattices and dihedral groups, Pure Appl. Math. Q. 7 (2011), no. 3, Special Issue: In honor of Jacques Tits, 621–743. MR 2848589, DOI 10.4310/PAMQ.2011.v7.n3.a6
Koji Hasegawa, Spin module versions of Weyl’s reciprocity theorem for classical Kac-Moody Lie algebras—an application to branching rule duality, Publ. Res. Inst. Math. Sci. 25 (1989), no. 5, 741–828. MR 1031225, DOI 10.2977/prims/1195172705
V. G. Kac and A. K. Raina, Bombay lectures on highest weight representations of infinite-dimensional Lie algebras, Advanced Series in Mathematical Physics, vol. 2, World Scientific Publishing Co., Inc., Teaneck, NJ, 1987. MR 1021978
Victor G. Kac and Minoru Wakimoto, Modular and conformal invariance constraints in representation theory of affine algebras, Adv. in Math. 70 (1988), no. 2, 156–236. MR 954660, DOI 10.1016/0001-8708(88)90055-2
Tomoki Nakanishi and Akihiro Tsuchiya, Level-rank duality of WZW models in conformal field theory, Comm. Math. Phys. 144 (1992), no. 2, 351–372. MR 1152377
C. Pauly, Strange duality revisited, arXiv:1204.1186v1.