Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 


The super $ \mathcal{W}_{1+\infty}$ algebra with integral central charge

Authors: Thomas Creutzig and Andrew R. Linshaw
Journal: Trans. Amer. Math. Soc. 367 (2015), 5521-5551
MSC (2010): Primary 17B69
Published electronically: February 3, 2015
MathSciNet review: 3347182
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Lie superalgebra $ \mathcal {S}\mathcal {D}$ of regular differential operators on the super circle has a universal central extension $ \widehat {\mathcal {S}\mathcal {D}}$. For each $ c\in \mathbb{C}$, the vacuum module $ \mathcal {M}_c(\widehat {\mathcal {S}\mathcal {D}})$ of central charge $ c$ admits a vertex superalgebra structure, and $ \mathcal {M}_c(\widehat {\mathcal {S}\mathcal {D}}) \cong \mathcal {M}_{-c}(\widehat {\mathcal {S}\mathcal {D}})$. The irreducible quotient $ \mathcal {V}_c(\widehat {\mathcal {S}\mathcal {D}})$ 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 {S}\mathcal {D}})$ 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\vert n)$. Finally, we realize $ \mathcal {V}_{n}(\widehat {\mathcal {S}\mathcal {D}})$ 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.

References [Enhancements On Off] (What's this?)

  • [AFMO] Hidetoshi Awata, Masafumi Fukuma, Yutaka Matsuo, and Satoru Odake, Quasifinite highest weight modules over the super $ {\mathcal {W}}_{1+\infty }$ algebra, Comm. Math. Phys. 170 (1995), no. 1, 151-179. MR 1331695 (96k:17041)
  • [ASV] M. Adler, T. Shiota, and P. van Moerbeke, From the $ w_\infty $-algebra to its central extension: a $ \tau $-function approach, Phys. Lett. A 194 (1994), no. 1-2, 33-43. MR 1299512 (95k:58067),
  • [BBSS] F. A. Bais, P. Bouwknegt, M. Surridge, and K. Schoutens, Coset construction for extended Virasoro algebras, Nuclear Phys. B 304 (1988), no. 2, 371-391. MR 952975 (90b:17031),
  • [B-H] R. Blumenhagen, W. Eholzer, A. Honecker, R. Hübel, and K. Hornfeck, Coset realization of unifying $ \mathcal {W}$ algebras, Internat. J. Modern Phys. A 10 (1995), no. 16, 2367-2430. MR 1334477 (96f:81039),
  • [BFH] J. de Boer, L. Fehér, and A. Honecker, A class of $ {\scr W}$-algebras with infinitely generated classical limit, Nuclear Phys. B 420 (1994), no. 1-2, 409-445. MR 1282655 (95f:81035),
  • [B] Richard E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 10, 3068-3071. MR 843307 (87m:17033),
  • [BG] Peter Bowcock and Peter Goddard, Coset constructions and extended conformal algebras, Nuclear Phys. B 305 (1988), no. 4, FS23, 685-709. MR 984287 (90f:17034),
  • [BS] Peter Bouwknegt and Kareljan Schoutens, $ \mathcal {W}$ symmetry in conformal field theory, Phys. Rep. 223 (1993), no. 4, 183-276. MR 1208246 (94e:81096),
  • [CHR] Thomas Creutzig, Yasuaki Hikida, and Peter B. Rønne, Higher spin $ \rm AdS_3$ supergravity and its dual CFT, J. High Energy Phys. 2 (2012), 109, front matter+33. MR 2996096
  • [CTZ] A. Cappelli, C. Trugenberger and G. Zemba, Classifications of quantum Hall universality classes by $ \mathcal {W}_{1+\infty }$ symmetry, Phys. Rev. Lett. 72 (1994), 1902-1905.
  • [CW] Shun-Jen Cheng and Weiqiang Wang, Lie subalgebras of differential operators on the super circle, Publ. Res. Inst. Math. Sci. 39 (2003), no. 3, 545-600. MR 2001187 (2004f:17032)
  • [FBZ] Edward Frenkel and David Ben-Zvi, Vertex algebras and algebraic curves, Mathematical Surveys and Monographs, vol. 88, American Mathematical Society, Providence, RI, 2001. MR 1849359 (2003f:17036)
  • [FF1] Boris Feigin and Edward Frenkel, Quantization of the Drinfeld-Sokolov reduction, Phys. Lett. B 246 (1990), no. 1-2, 75-81. MR 1071340 (92g:17029),
  • [FF2] Boris L. Feigin and Edward V. Frenkel, Representations of affine Kac-Moody algebras, bosonization and resolutions, Lett. Math. Phys. 19 (1990), no. 4, 307-317. MR 1051813 (91m:17037),
  • [FHL] Igor B. Frenkel, Yi-Zhi Huang, and James Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104 (1993), no. 494, viii+64. MR 1142494 (94a:17007),
  • [FKRW] Edward Frenkel, Victor Kac, Andrey Radul, and Weiqiang Wang, $ \mathcal {W}_{1+\infty }$ and $ \mathcal {W}(\mathfrak{g}\mathfrak{l}_N)$ with central charge $ N$, Comm. Math. Phys. 170 (1995), no. 2, 337-357. MR 1334399 (96i:17024)
  • [FLM] Igor Frenkel, James Lepowsky, and Arne Meurman, Vertex operator algebras and the Monster, Pure and Applied Mathematics, vol. 134, Academic Press, Inc., Boston, MA, 1988. MR 996026 (90h:17026)
  • [FMS] Daniel Friedan, Emil Martinec, and Stephen Shenker, Conformal invariance, supersymmetry and string theory, Nuclear Phys. B 271 (1986), no. 1, 93-165. MR 845945 (87i:81202),
  • [GG] M. R. Gaberdiel and R. Gopakumar, An AdS$ _3$ Dual for Minimal Model CFTs, Phys. Rev. D 83 (2011), 066007.
  • [GS] Peter Goddard and Adam Schwimmer, Unitary construction of extended conformal algebras, Phys. Lett. B 206 (1988), no. 1, 62-70. MR 942173 (89g:81075),
  • [GK] Maria Gorelik and Victor Kac, On simplicity of vacuum modules, Adv. Math. 211 (2007), no. 2, 621-677. MR 2323540 (2008e:17024),
  • [I] K. Ito, The W algebra structure of $ N=2 CP(n)$ coset models, hep-th/9210143.
  • [K] Victor Kac, Vertex algebras for beginners, 2nd ed., University Lecture Series, vol. 10, American Mathematical Society, Providence, RI, 1998. MR 1651389 (99f:17033)
  • [KP] Victor G. Kac and Dale H. Peterson, Spin and wedge representations of infinite-dimensional Lie algebras and groups, Proc. Nat. Acad. Sci. U.S.A. 78 (1981), no. 6, 3308-3312. MR 619827 (82j:17019)
  • [KRI] Victor Kac and Andrey Radul, Quasifinite highest weight modules over the Lie algebra of differential operators on the circle, Comm. Math. Phys. 157 (1993), no. 3, 429-457. MR 1243706 (95f:81036)
  • [KRII] Victor Kac and Andrey Radul, Representation theory of the vertex algebra $ W_{1+\infty }$, Transform. Groups 1 (1996), no. 1-2, 41-70. MR 1390749 (97f:17033),
  • [KRW] Victor Kac, Shi-Shyr Roan, and Minoru Wakimoto, Quantum reduction for affine superalgebras, Comm. Math. Phys. 241 (2003), no. 2-3, 307-342. MR 2013802 (2004h:17024)
  • [LiI] Hai-Sheng Li, Local systems of vertex operators, vertex superalgebras and modules, J. Pure Appl. Algebra 109 (1996), no. 2, 143-195. MR 1387738 (97d:17016),
  • [LiII] Haisheng Li, Vertex algebras and vertex Poisson algebras, Commun. Contemp. Math. 6 (2004), no. 1, 61-110. MR 2048777 (2005d:17036),
  • [LL] Bong H. Lian and Andrew R. Linshaw, Howe pairs in the theory of vertex algebras, J. Algebra 317 (2007), no. 1, 111-152. MR 2360143 (2008j:17055),
  • [LZ] Bong H. Lian and Gregg J. Zuckerman, Commutative quantum operator algebras, J. Pure Appl. Algebra 100 (1995), no. 1-3, 117-139. MR 1344847 (97a:17020),
  • [LI] Andrew R. Linshaw, Invariant theory and the $ \mathcal {W}_{1+\infty }$ algebra with negative integral central charge, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 6, 1737-1768. MR 2835328 (2012k:81136),
  • [LII] Andrew R. Linshaw, A Hilbert theorem for vertex algebras, Transform. Groups 15 (2010), no. 2, 427-448. MR 2657448 (2011e:17046),
  • [W] Weiqiang Wang, $ \mathcal {W}_{1+\infty }$ algebra, $ \mathcal {W}_3$ algebra, and Friedan-Martinec-Shenker bosonization, Comm. Math. Phys. 195 (1998), no. 1, 95-111. MR 1637409 (2000d:81051),
  • [We] Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. MR 1488158 (98k:01049)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 17B69

Retrieve articles in all journals with MSC (2010): 17B69

Additional Information

Thomas Creutzig
Affiliation: Department of Mathematics, University of Alberta, 116 St. and 85 Ave., Edmonton, AB T6G 2R3, Canada

Andrew R. Linshaw
Affiliation: Department of Mathematics, University of Denver, 2199 S. University Blvd., Denver, Colorado 80208

Received by editor(s): September 28, 2012
Received by editor(s) in revised form: June 10, 2013
Published electronically: February 3, 2015
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society