×

The super \(\mathcal{W}_{1+\infty}\) algebra with integral central charge. (English) Zbl 1312.17020

Summary: The Lie superalgebra \(\mathcal{SD}\) of regular differential operators on the super circle has a universal central extension \(\widehat{\mathcal{SD}}\). For each \(c\in \mathbb{C}\), the vacuum module \(\mathcal {M}_c(\widehat {\mathcal {SD}})\) of central charge \( c\) admits a vertex superalgebra structure, and \( \mathcal {M}_c(\widehat {\mathcal {SD}}) \cong \mathcal {M}_{-c}(\widehat {\mathcal {SD}})\). The irreducible quotient \( \mathcal {V}_c(\widehat {\mathcal {SD}})\) of the vacuum module is known as the super \(\mathcal {W}_{1+\infty }\) algebra. We show that for each integer \(n>0\), \( \mathcal {V}_{n}(\widehat {\mathcal {SD}})\) has a minimal strong generating set consisting of \(4n\) fields, and we identify it with a \( \mathcal {W}\)-algebra associated to the purely odd simple root system of \(\mathfrak{g} \mathfrak{l}(n| n)\). Finally, we realize \( \mathcal {V}_n(\widehat {\mathcal {SD}})\) as the limit of a family of commutant vertex algebras that generically have the same graded character and possess a minimal strong generating set of the same cardinality.

MSC:

17B69 Vertex operators; vertex operator algebras and related structures
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Awata, Hidetoshi; Fukuma, Masafumi; Matsuo, Yutaka; Odake, Satoru, Quasifinite highest weight modules over the super \({\mathcal{W}}_{1+\infty }\) algebra, Comm. Math. Phys., 170, 1, 151-179 (1995) · Zbl 0832.17026
[2] Adler, M.; Shiota, T.; van Moerbeke, P., From the \(w_\infty \)-algebra to its central extension: a \(\tau \)-function approach, Phys. Lett. A, 194, 1-2, 33-43 (1994) · Zbl 0925.58031
[3] Bais, F. A.; Bouwknegt, P.; Surridge, M.; Schoutens, K., Coset construction for extended Virasoro algebras, Nuclear Phys. B, 304, 2, 371-391 (1988) · Zbl 0675.17010
[4] Blumenhagen, R.; Eholzer, W.; Honecker, A.; H{\"u}bel, R.; Hornfeck, K., Coset realization of unifying \(\mathcal{W}\) algebras, Internat. J. Modern Phys. A, 10, 16, 2367-2430 (1995) · Zbl 1044.81594
[5] de Boer, J.; Feh{\'e}r, L.; Honecker, A., A class of \({\scr W} \)-algebras with infinitely generated classical limit, Nuclear Phys. B, 420, 1-2, 409-445 (1994) · Zbl 0990.81525
[6] Borcherds, Richard E., Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat. Acad. Sci. U.S.A., 83, 10, 3068-3071 (1986) · Zbl 0613.17012
[7] Bowcock, Peter; Goddard, Peter, Coset constructions and extended conformal algebras, Nuclear Phys. B, 305, 4, FS23, 685-709 (1988) · Zbl 0689.17021
[8] Bouwknegt, Peter; Schoutens, Kareljan, \( \mathcal{W}\) symmetry in conformal field theory, Phys. Rep., 223, 4, 183-276 (1993)
[9] Creutzig, Thomas; Hikida, Yasuaki; R{\o }nne, Peter B., Higher spin \(\rm AdS_3\) supergravity and its dual CFT, J. High Energy Phys., 2, 109, front matter+33 pp. (2012) · Zbl 1309.81151
[10] [CTZ] A. Cappelli, C. Trugenberger and G. Zemba, Classifications of quantum Hall universality classes by \(\mathcalW_1+\infty\) symmetry, Phys. Rev. Lett. 72 (1994), 1902-1905.
[11] Cheng, Shun-Jen; Wang, Weiqiang, Lie subalgebras of differential operators on the super circle, Publ. Res. Inst. Math. Sci., 39, 3, 545-600 (2003) · Zbl 1049.17021
[12] Frenkel, Edward; Ben-Zvi, David, Vertex algebras and algebraic curves, Mathematical Surveys and Monographs 88, xii+348 pp. (2001), American Mathematical Society, Providence, RI · Zbl 0981.17022
[13] Feigin, Boris; Frenkel, Edward, Quantization of the Drinfel\cprime {d}-Sokolov reduction, Phys. Lett. B, 246, 1-2, 75-81 (1990) · Zbl 1242.17023
[14] Feigin, Boris L.; Frenkel, Edward V., Representations of affine Kac-Moody algebras, bosonization and resolutions, Lett. Math. Phys., 19, 4, 307-317 (1990) · Zbl 0711.17012
[15] Frenkel, Igor B.; Huang, Yi-Zhi; Lepowsky, James, On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc., 104, 494, viii+64 pp. (1993) · Zbl 0789.17022
[16] Frenkel, Edward; Kac, Victor; Radul, Andrey; Wang, Weiqiang, \( \mathcal{W}_{1+\infty }\) and \(\mathcal{W}(\mathfrak{g}\mathfrak{l}_N)\) with central charge \(N\), Comm. Math. Phys., 170, 2, 337-357 (1995) · Zbl 0838.17028
[17] Frenkel, Igor; Lepowsky, James; Meurman, Arne, Vertex operator algebras and the Monster, Pure and Applied Mathematics 134, liv+508 pp. (1988), Academic Press, Inc., Boston, MA · Zbl 0674.17001
[18] Friedan, Daniel; Martinec, Emil; Shenker, Stephen, Conformal invariance, supersymmetry and string theory, Nuclear Phys. B, 271, 1, 93-165 (1986)
[19] [GG] M. R. Gaberdiel and R. Gopakumar, An \(AdS_3\) Dual for Minimal Model CFTs, Phys. Rev. D 83 (2011), 066007.
[20] Goddard, Peter; Schwimmer, Adam, Unitary construction of extended conformal algebras, Phys. Lett. B, 206, 1, 62-70 (1988)
[21] Gorelik, Maria; Kac, Victor, On simplicity of vacuum modules, Adv. Math., 211, 2, 621-677 (2007) · Zbl 1112.17023
[22] [I] K. Ito, The W algebra structure of \(N=2 CP(n)\) coset models, hep-th/9210143.
[23] Kac, Victor, Vertex algebras for beginners, University Lecture Series 10, vi+201 pp. (1998), American Mathematical Society, Providence, RI · Zbl 0924.17023
[24] Kac, Victor G.; Peterson, Dale H., Spin and wedge representations of infinite-dimensional Lie algebras and groups, Proc. Nat. Acad. Sci. U.S.A., 78, 6, 3308-3312 (1981) · Zbl 0469.22016
[25] Kac, Victor; Radul, Andrey, Quasifinite highest weight modules over the Lie algebra of differential operators on the circle, Comm. Math. Phys., 157, 3, 429-457 (1993) · Zbl 0826.17027
[26] Kac, Victor; Radul, Andrey, Representation theory of the vertex algebra \(W_{1+\infty } \), Transform. Groups, 1, 1-2, 41-70 (1996) · Zbl 0862.17023
[27] Kac, Victor; Roan, Shi-Shyr; Wakimoto, Minoru, Quantum reduction for affine superalgebras, Comm. Math. Phys., 241, 2-3, 307-342 (2003) · Zbl 1106.17026
[28] Li, Hai-Sheng, Local systems of vertex operators, vertex superalgebras and modules, J. Pure Appl. Algebra, 109, 2, 143-195 (1996) · Zbl 0854.17035
[29] Li, Haisheng, Vertex algebras and vertex Poisson algebras, Commun. Contemp. Math., 6, 1, 61-110 (2004) · Zbl 1050.17024
[30] Lian, Bong H.; Linshaw, Andrew R., Howe pairs in the theory of vertex algebras, J. Algebra, 317, 1, 111-152 (2007) · Zbl 1221.17034
[31] Lian, Bong H.; Zuckerman, Gregg J., Commutative quantum operator algebras, J. Pure Appl. Algebra, 100, 1-3, 117-139 (1995) · Zbl 0838.17029
[32] Linshaw, Andrew R., Invariant theory and the \(\mathcal{W}_{1+\infty }\) algebra with negative integral central charge, J. Eur. Math. Soc. (JEMS), 13, 6, 1737-1768 (2011) · Zbl 1244.17016
[33] Linshaw, Andrew R., A Hilbert theorem for vertex algebras, Transform. Groups, 15, 2, 427-448 (2010) · Zbl 1268.17031
[34] Wang, Weiqiang, \( \mathcal{W}_{1+\infty }\) algebra, \( \mathcal{W}_3\) algebra, and Friedan-Martinec-Shenker bosonization, Comm. Math. Phys., 195, 1, 95-111 (1998) · Zbl 0980.17014
[35] Weyl, Hermann, The classical groups, Princeton Landmarks in Mathematics, xiv+320 pp. (1997), Princeton University Press, Princeton, NJ · Zbl 1024.20501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.