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

SIGMA 4 (2008), 087, 24 pages      arXiv:0806.3560      https://doi.org/10.3842/SIGMA.2008.087
Contribution to the Special Issue on Kac-Moody Algebras and Applications

The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector

Drazen Adamovic a and Antun Milas b
a) Department of Mathematics, University of Zagreb, Croatia
b) Department of Mathematics and Statistics, University at Albany (SUNY), Albany, NY 12222, USA

Received August 31, 2008, in final form December 05, 2008; Published online December 13, 2008

Abstract
We classify irreducible σ-twisted modules for the N = 1 super triplet vertex operator superalgebra SW(m) introduced recently [Adamovic D., Milas A., Comm. Math. Phys., to appear, arXiv:0712.0379]. Irreducible graded dimensions of σ-twisted modules are also determined. These results, combined with our previous work in the untwisted case, show that the SL(2,Z)-closure of the space spanned by irreducible characters, irreducible supercharacters and σ-twisted irreducible characters is (9m + 3)-dimensional. We present strong evidence that this is also the (full) space of generalized characters for SW(m). We are also able to relate irreducible SW(m) characters to characters for the triplet vertex algebra W(2m + 1), studied in [Adamovic D., Milas A., Adv. Math. 217 (2008), 2664-2699, arXiv:0707.1857].

Key words: vertex operator superalgebras; Ramond twisted representations.

pdf (362 kb)   ps (237 kb)   tex (24 kb)

References

  1. Abe T., Buhl G., Dong C., Rationality, regularity, and C2-cofiniteness, Trans. Amer. Math. Soc. 356 (2004), 3391-3402, math.QA/0204021.
  2. Adamovic D., Classification of irreducible modules of certain subalgebras of free boson vertex algebra, J. Algebra 270 (2003), 115-132, math.QA/0207155.
  3. Adamovic D., Milas A., Logarithmic intertwining operators and W(2,2p-1)-algebras, J. Math. Phys. 48(2007), 073503, 20 pages, math.QA/0702081.
  4. Adamovic D., Milas A., On the triplet vertex algebra W(p), Adv. Math. 217 (2008), 2664-2699, arXiv:0707.1857.
  5. Adamovic D., Milas A., The N = 1 triplet vertex operator superalgebras, Comm. Math. Phys., to appear, arXiv:0712.0379.
  6. Adamovic D., Milas A., An analogue of modular BPZ-equations in logarithmic (super)conformal field theory, submitted.
  7. Adamovic D., Milas A., Lattice construction of logarithmic modules, in preparation.
  8. Arakawa T., Representation theory of W-algebras, II: Ramond twisted representations, arXiv:0802.1564.
  9. Bakalov B., Kac V., Twisted modules over lattice vertex algebras, in Lie Theory and Its Applications in Physics V, World Sci. Publ., River Edge, NJ, 2004, 3-26, math.QA/0402315.
  10. Dong C., Twisted modules for vertex algebras associated with even lattices, J. Algebra 165 (1994), 91-112.
  11. Dong C., Zhao Z., Modularity in orbifold theory for vertex operator superalgebras, Comm. Math. Phys. 260 (2005), 227-256, math.QA/0411524.
  12. Dörrzapf M., Highest weight representations of the N=1 Ramond algebra, Nuclear Phys. B 595 (2001), 605-653.
  13. Feingold A.J., Frenkel I.B., Ries J.F.X., Spinor construction of vertex operator algebras, triality, and E8 (1), Contemporary Mathematics, Vol. 121, American Mathematical Society, Providence, RI, 1991.
  14. Feingold A.J., Ries J.F.X., Weiner M., Spinor construction of the c = 1/2 minimal model, in Moonshine, the Monster and Related Topics (South Hadley, MA, 1994), Editors C. Dong and G. Mason, Contemporary Mathematics, Vol. 193, American Mathematical Society, Providence, RI, 1996, 45-92, hep-th/9501114.
  15. Feigin B.L., Ga nutdinov A.M., Semikhatov A.M., Tipunin I.Yu., Modular group representations and fusion in logarithmic conformal field theories and in the quantum group center, Comm. Math. Phys. 265 (2006), 47-93, hep-th/0504093.
  16. 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.
  17. Huang Y.-Z., Milas A., Intertwining operator superalgebras and vertex tensor categories for superconformal algebras. I, Commun. Contemp. Math. 4 (2002), 327-355, math.QA/9909039.
  18. Iohara K., Koga Y., Fusion algebras for N = 1 superconformal field theories through coinvariants. II. N = 1 super-Virasoro-symmetry, J. Lie Theory 11 (2001), 305-337.
  19. Iohara K., Koga Y., Representation theory of the Neveu-Schwarz and Ramond algebras. I. Verma modules, Adv. Math. 178 (2003), 1-65.
  20. Iohara K., Koga Y., Representation theory of the Neveu-Schwarz and Ramond algebras. II. Fock modules, Ann. Inst. Fourier (Grenoble) 53 (2003), 1755-1818.
  21. Kac V.G., Vertex algebras for beginners, 2nd ed., University Lecture Series, Vol. 10, American Mathematical Society, Providence, RI, 1998.
  22. Kac V., Wakimoto M., Unitarizable highest weight representations of the Virasoro, Neveu-Schwarz and Ramond algebras, in Conformal Groups and Related Symmetries: Physical Results and Mathematical Background (Clausthal-Zellerfeld, 1985), Lecture Notes in Phys., Vol. 261, Springer, Berlin, 1986, 345-371.
  23. Kac V., Wakimoto M., Quantum reduction in the twisted case, in Infinite Dimensional Algebras and Quantum Integrable Systems, Progr. Math., Vol. 237, Birkhäuser, Basel, 2005, 89-131, math-ph/0404049.
  24. Li H., Local systems of twisted vertex operators, vertex operator superalgebras and twisted modules, in Moonshine, the Monster, and Related Topics (South Hadley, MA, 1994), Editors C. Dong and G. Mason, Contemporary Mathematics, Vol. 193, American Mathematical Society, Providence, RI, 1996, 203-236, q-alg/9504022.
  25. Meurman A., Rocha-Caridi A., Highest weight representations of the Neveu-Schwarz and Ramond algebras, Comm. Math. Phys. 107 (1986), 263-294.
  26. Milas A., Fusion rings for degenerate minimal models, J. Algebra 254 (2002), 300-335, math.QA/0003225.
  27. Milas A., Characters, supercharacters and Weber modular functions, J. Reine Angew. Math. 608 (2007), 35-64.
  28. Miyamoto M., Modular invariance of vertex operator algebras satisfying C2-cofiniteness, Duke Math. J. 122 (2004), 51-91, math.QA/020910.
  29. Xu X., Introduction to vertex operator superalgebras and their modules, Mathematics and Its Applications, Vol. 456, Kluwer Academic Publishers, Dordrecht, 1998.
  30. Zhu Y.-C., Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), 237-302.

Previous article   Next article   Contents of Volume 4 (2008)