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

SIGMA 11 (2015), 019, 16 pages      arXiv:1503.01542      https://doi.org/10.3842/SIGMA.2015.019
Contribution to the Special Issue on New Directions in Lie Theory

Vertex Algebras $\mathcal{W}(p)^{A_m}$ and $\mathcal{W}(p)^{D_m}$ and Constant Term Identities

Dražen Adamović a, Xianzu Lin b and Antun Milas c
a) Department of Mathematics, University of Zagreb, Bijenicka 30, 10000 Zagreb, Croatia
b) College of Mathematics and Computer Science, Fujian Normal University, Fuzhou, 350108, China
c) Department of Mathematics and Statistics, SUNY-Albany, 1400 Washington Avenue, Albany 12222,USA

Received October 03, 2014, in final form February 25, 2015; Published online March 05, 2015

Abstract
We consider $AD$-type orbifolds of the triplet vertex algebras $\mathcal{W}(p)$ extending the well-known $c=1$ orbifolds of lattice vertex algebras. We study the structure of Zhu's algebras $A(\mathcal{W}(p)^{A_m})$ and $A(\mathcal{W}(p)^{D_m})$, where $A_m$ and $D_m$ are cyclic and dihedral groups, respectively. A combinatorial algorithm for classification of irreducible $\mathcal{W}(p)^\Gamma$-modules is developed, which relies on a family of constant term identities and properties of certain polynomials based on constant terms. All these properties can be checked for small values of $m$ and $p$ with a computer software. As a result, we argue that if certain constant term properties hold, the irreducible modules constructed in [Commun. Contemp. Math. 15 (2013), 1350028, 30 pages; Internat. J. Math. 25 (2014), 1450001, 34 pages] provide a complete list of irreducible $\mathcal{W}(p)^{A_m}$ and $\mathcal{W}(p)^{D_m}$-modules. This paper is a continuation of our previous work on the $ADE$ subalgebras of the triplet vertex algebra $\mathcal{W}(p)$.

Key words: $C_{2}$-cofiniteness, triplet vertex algebra, orbifold subalgebra, constant term identities.

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References

  1. Adamović D., Classification of irreducible modules of certain subalgebras of free boson vertex algebra, J. Algebra 270 (2003), 115-132, math.QA/0207155.
  2. Adamović D., Lin X., Milas A., ADE subalgebras of the triplet vertex algebra ${\mathcal W}(p)$: $A$-series, Commun. Contemp. Math. 15 (2013), 1350028, 30 pages, arXiv:1212.5453.
  3. Adamović D., Lin X., Milas A., ADE subalgebras of the triplet vertex algebra ${\mathcal W}(p)$: $D$-series, Internat. J. Math. 25 (2014), 1450001, 34 pages, arXiv:1304.5711.
  4. Adamović D., Milas A., Logarithmic intertwining operators and ${\mathcal W}(2,2p-1)$ algebras, J. Math. Phys. 48 (2007), 073503, 20 pages, math.QA/0702081.
  5. Adamović D., Milas A., On the triplet vertex algebra ${\mathcal W}(p)$, Adv. Math. 217 (2008), 2664-2699, arXiv:0707.1857.
  6. Adamović D., Milas A., The $N=1$ triplet vertex operator superalgebras: twisted sector, SIGMA 4 (2008), 087, 24 pages, arXiv:0806.3560.
  7. Adamović D., Milas A., On $W$-algebras associated to $(2,p)$ minimal models and their representations, Int. Math. Res. Not. 2010 (2010), 3896-3934, arXiv:0908.4053.
  8. Adamović D., Milas A., On ${\mathcal W}$-algebra extensions of $(2,p)$ minimal models: $p$>$3$, J. Algebra 344 (2011), 313-332, arXiv:1101.0803.
  9. Adamović D., Milas A., The structure of Zhu's algebras for certain ${\mathcal W}$-algebras, Adv. Math. 227 (2011), 2425-2456, arXiv:1006.5134.
  10. Dong C., Griess Jr. R.L., Rank one lattice type vertex operator algebras and their automorphism groups, J. Algebra 208 (1998), 262-275, q-alg/9710017.
  11. Forrester P.J., Warnaar S.O., The importance of the Selberg integral, Bull. Amer. Math. Soc. 45 (2008), 489-534, arXiv:0710.3981.
  12. Frenkel I., Lepowsky J., Meurman A., Vertex operator algebras and the Monster, Pure and Applied Mathematics, Vol. 134, Academic Press, Inc., Boston, MA, 1988.
  13. Ginsparg P., Curiosities at $c=1$, Nuclear Phys. B 295 (1988), 153-170.
  14. Lepowsky J., Li H., Introduction to vertex operator algebras and their representations, Progress in Mathematics, Vol. 227, Birkhäuser Boston, Inc., Boston, MA, 2004.
  15. Zhu Y., Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), 237-302.

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