Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 040, 16 pages      arXiv:1006.1752      https://doi.org/10.3842/SIGMA.2012.040

The Vertex Algebra $M(1)^+$ and Certain Affine Vertex Algebras of Level $-1$

Dražen Adamović and Ozren Perše
Faculty of Science, Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia

Received March 09, 2012, in final form July 01, 2012; Published online July 08, 2012

Abstract
We give a coset realization of the vertex operator algebra $M(1)^+$ with central charge $\ell$. We realize $M(1) ^+$ as a commutant of certain affine vertex algebras of level $-1$ in the vertex algebra $L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0) \otimes L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}\Lambda_0)$. We show that the simple vertex algebra $L_{C_{\ell} ^{(1)}}(-\Lambda_0)$ can be (conformally) embedded into $L_{A_{2 \ell -1} ^{(1)}} (-\Lambda_0)$ and find the corresponding decomposition. We also study certain coset subalgebras inside $L_{C_{\ell} ^{(1)}}(-\Lambda_0)$.

Key words: vertex operator algebra; affine Kac-Moody algebra; coset vertex algebra; conformal embedding; $\mathcal{W}$-algebra.

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References

  1. Adamović D., A construction of some ideals in affine vertex algebras, Int. J. Math. Math. Sci. (2003), 971-980, math.QA/0103006.
  2. Adamović D., Some rational vertex algebras, Glas. Mat. Ser. III 29(49) (1994), 25-40, q-alg/9502015.
  3. Adamović D., Milas A., On the triplet vertex algebra ${\mathcal W}(p)$, Adv. Math. 217 (2008), 2664-2699, arXiv:0707.1857.
  4. Adamović D., Milas A., The $N=1$ triplet vertex operator superalgebras, Comm. Math. Phys. 288 (2009), 225-270, arXiv:0712.0379.
  5. Adamović D., Milas A., Vertex operator algebras associated to modular invariant representations for $A^{(1)}_1$, Math. Res. Lett. 2 (1995), 563-575, q-alg/9509025.
  6. Adamović D., Perše O., Fusion rules and complete reducibility of certain modules for affine Lie algebras, in preparation.
  7. Adamović D., Perše O., On coset vertex algebras with central charge 1, Math. Commun. 15 (2010), 143-157.
  8. Adamović D., Perše O., Representations of certain non-rational vertex operator algebras of affine type, J. Algebra 319 (2008), 2434-2450, math.QA/0702018.
  9. Arakawa T., Representation theory of ${\mathcal W}$-algebras, Invent. Math. 169 (2007), 219-320, math.QA/0506056.
  10. Borcherds R.E., Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat. Acad. Sci. USA 83 (1986), 3068-3071.
  11. Bourbaki N., Éléments de mathématique. Fasc. XXXVIII: Groupes et algèbres de Lie, Actualités Scientifiques et Industrielles, Vol. 1364, Hermann, Paris, 1975.
  12. Dong C., Griess R.L., Rank one lattice type vertex operator algebras and their automorphism groups, J. Algebra 208 (1998), 262-275, q-alg/9710017.
  13. Dong C., Lam C.H., Yamada H., $\mathcal W$-algebras related to parafermion algebras, J. Algebra 322 (2009), 2366-2403, arXiv:0809.3630.
  14. Dong C., Mason G., On quantum Galois theory, Duke Math. J. 86 (1997), 305-321, hep-th/9412037.
  15. Dong C., Nagatomo K., Classification of irreducible modules for the vertex operator algebra $M(1)^+$, J. Algebra 216 (1999), 384-404, math.QA/9806051.
  16. Dong C., Nagatomo K., Classification of irreducible modules for the vertex operator algebra $M(1)^+$. II. Higher rank, J. Algebra 240 (2001), 289-325, math.QA/9905064.
  17. Feingold A.J., Frenkel I.B., Classical affine algebras, Adv. Math. 56 (1985), 117-172.
  18. Frenkel E., Ben-Zvi D., Vertex algebras and algebraic curves, Mathematical Surveys and Monographs, Vol. 88, American Mathematical Society, Providence, RI, 2001.
  19. Frenkel I.B., Huang Y.Z., Lepowsky J., On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104 (1993), no. 494.
  20. Frenkel I.B., Lepowsky J., Meurman A., Vertex operator algebras and the Monster, Pure and Applied Mathematics, Vol. 134, Academic Press Inc., Boston, MA, 1988.
  21. Frenkel I.B., Zhu Y., Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J. 66 (1992), 123-168.
  22. Goddard P., Kent A., Olive D., Virasoro algebras and coset space models, Phys. Lett. B 152 (1985), 88-92.
  23. Kac V.G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990.
  24. Kac V.G., Vertex algebras for beginners, University Lecture Series, Vol. 10, 2nd ed., American Mathematical Society, Providence, RI, 1998.
  25. Kac V.G., Wakimoto M., Modular invariant representations of infinite-dimensional Lie algebras and superalgebras, Proc. Nat. Acad. Sci. USA 85 (1988), 4956-4960.
  26. Kac V.G., Wakimoto M., On rationality of $W$-algebras, Transform. Groups 13 (2008), 671-713, arXiv:0711.2296.
  27. Lepowsky J., Li H., Introduction to vertex operator algebras and their representations, Progress in Mathematics, Vol. 227, Birkhäuser Boston Inc., Boston, MA, 2004.
  28. Li H.S., Local systems of vertex operators, vertex superalgebras and modules, J. Pure Appl. Algebra 109 (1996), 143-195, hep-th/9406185.
  29. Meurman A., Primc M., Annihilating fields of standard modules of ${\mathfrak{sl}}(2,{\mathbb C})^\sim$ and combinatorial identities, Mem. Amer. Math. Soc. 137 (1999), no. 652.
  30. Perše O., Vertex operator algebras associated to certain admissible modules for affine Lie algebras of type $A$, Glas. Mat. Ser. III 43(63) (2008), 41-57, arXiv:0707.4129.
  31. Perše O., Vertex operator algebras associated to type $B$ affine Lie algebras on admissible half-integer levels, J. Algebra 307 (2007), 215-248, math.QA/0512129.

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