Glasnik Matematicki, Vol. 55, No. 2 (2020), 277-300.

PERMUTATION ORBIFOLDS OF 𝔰𝔩2 VERTEX OPERATOR ALGEBRAS

Antun Milas and Michael Penn

Department of Mathematics and Statistics, SUNY-Albany, Albany NY 12222, USA
e-mail: amilas@albany.edu

Mathematics Department, Randolph College, Lynchburg VA 24503, USA
e-mail: mpenn@randolphcollege.edu

Abstract.   We analyze two types of permutation orbifolds: (i) S2-orbifolds of the universal level k vertex operator algebra Vk(𝔰𝔩2) and of its simple quotient Lk(𝔰𝔩2), and (ii) the S3-orbifold of the level one simple vertex operator algebra L1(𝔰𝔩2). We determine their structures and discuss related W-algebras.

2010 Mathematics Subject Classification.   17B69

Key words and phrases.   Permutation orbifolds, vertex algebrass

Full text (PDF) (access from subscribing institutions only)

https://doi.org/10.3336/gm.55.2.08

References:

  1. D. Adamović, C.H. Lam, V. Pedić and N. Yu, On irreducibility of modules of Whittaker type for cyclic orbifold vertex algebras, J. Algebra 539 (2019), 1-23.
    MathSciNet     CrossRef

  2. T. Abe, C2-cofiniteness of 2-cyclic permutation orbifold models, Comm. Math. Phys. 317 (2013), 425-445.
    MathSciNet     CrossRef

  3. M. Al-Ali, The ℤ2-orbifold of the universal affine vertex algebra, J. Pure Appl. Algebra 223 (2019), 5430-5443.
    MathSciNet     CrossRef

  4. M. Al-Ali and A. Linshaw, The ℤ 2 -orbifold of the 𝓦 3 -algebra, Comm. Math. Phys. 35 (2017), 1129-1150.
    MathSciNet     CrossRef

  5. P. Bantay, Characters and modular properties of permutation orbifolds, Phys. Lett. B 419, 175-178.
    MathSciNet     CrossRef

  6. K. Barron and N. Vander Werf, Permutation-twisted modules for even order cycles acting on tensor product vertex operator superalgebras, Internat. J. Math. 25 (2014), 1450018, 35 pp.
    MathSciNet     CrossRef

  7. K. Barron, C. Dong and G. Mason, Twisted sectors for tensor product vertex operator algebras associated to permutation groups, Comm. Math. Phys. 227 (2002), 349-384.
    MathSciNet     CrossRef

  8. K. Barron, Y.-Z. Huang and J. Lepowsky, An equivalence of two constructions of permutation-twisted modules for lattice vertex operator algebras, J. Pure Appl. Algebra 210 (2007), 797-826.
    MathSciNet     CrossRef

  9. A. Belin, C. Keller and A. Maloney, String universality for permutation orbifolds, Phys. Rev. D 91 (2015), 106005, 11 pp.
    MathSciNet     CrossRef

  10. P. Bouwknegt, Extended conformal algebras from Kac-Moody algebras, In: Infinite-dimensional Lie algebras and groups, World Scientific, Teaneck, 1989, 527-555.
    MathSciNet    

  11. S. Carnahan, and M. Miyamoto, Regularity of fixed-point vertex operator subalgebras, arXiv preprint arXiv:1603.05645 (2016).

  12. C. Dong and G. Mason, On quantum Galois theory, Duke Math. J. 86 (1997), 305-321.
    MathSciNet     CrossRef

  13. C. Dong and C. Jiang, A characterization of vertex operator algebra VL+, in: Conformal field theory, automorphic forms and related topics, Springer, Heidelberg, 2014, 55-74.
    MathSciNet    

  14. C. Dong, C-H. Lam, K. Tanabe, H. Yamada and K. Yokoyama. ℤ3-symmetry and W3 algebra in lattice vertex operator algebras, Pacific J. Math. 215 (2004), 245-296.
    MathSciNet     CrossRef

  15. C. Dong, L. Ren and F. Xu, On orbifold theory, Adv. Math. 321 (2017), 1-30.
    MathSciNet     CrossRef

  16. C. Dong, F. Xu and N. Yu, The 3-permutation orbifold of a lattice vertex operator algebra, J. Pure Appl. Algebra 222 (2018), 1316-1336.
    CrossRef

  17. C. Dong, F. Xu and N. Yu, 2-permutations of lattice vertex operator algebras: higher rank, J. Algebra 476 (2017), 1-25.
    MathSciNet     CrossRef

  18. C.-H. Lam and H. Yamada, Decomposition of the lattice vertex operator algebra V√2Al, J. Algebra 272 (2004), 614-624.
    MathSciNet     CrossRef

  19. D. Graybill, A. Linshaw, M. Penn and J. Quintero,Strong finite generation of the ℤn orbifolds of the rank 2 Heisenberg system, in preparation.

  20. P. Goddard, A. Kent and D. Olive, Unitary representations of the Virasoro and super-Virasoro algebras, Comm. Math. Phys. 103 (1986), 105-119.
    MathSciNet     CrossRef

  21. H. Li, A. Milas and J. Wauchope, S2-orbifolds of N=1 and N=2 superconformal vertex algebras and W-algebras, to appear in Communications in Algebra.

  22. A. Linshaw, Invariant subalgebras of affine vertex algebras, Adv. Math. 234 (2013), 61-84.
    MathSciNet     CrossRef

  23. C. Jiang and Z. Lin, The commutant of L𝔰𝔩(2)(n, 0) in the vertex operator algebra L𝔰𝔩(2)(1,0)⊗ n, Adv. Math. 301 (2016), 227-257.
    MathSciNet     CrossRef

  24. H. Li, Equivariant oriented cohomology and associated schemes of vertex superalgebras, PhD thesis (UAlbany), in progress.

  25. A. Milas, M. Penn and C. Sadowski, Permutation orbifolds of the Virasoro vertex algebra and W-algebras, to appear in Journal of Algebra; arXiv:2005.08398.

  26. A. Milas, M. Penn and H. Shao, Permutation orbifolds of the Heisenberg vertex algebra 𝓗(3), J. Math. Phys. 60 (2019), 021703, 17 pp.
    MathSciNet     CrossRef

  27. A. Milas, M. Penn and J. Wauchope, Permutation orbifolds of rank three fermionic vertex superalgebras, in: D. Adamović, P. Papi (eds.), Affine, Vertex and W-algebras, Springer, Cham, 2019, 183-202.
    CrossRef

  28. A. Linshaw, A Hilbert theorem for vertex algebras, Transform. Groups 15 (2010), 427-448.
    MathSciNet     CrossRef

  29. M. Penn and J. Quiento, Finite group orbifolds of Heisenberg vertex algebras, in preparation.

  30. J. Wauchope, Permutation orbifolds of fermionic vertex algebras, PhD thesis, UAlbany, 2020.
    MathSciNet    
Glasnik Matematicki Home Page