Abstract
The minimal model \( \mathfrak{osp}\left(1|2\right) \) vertex operator superalgebras are the simple quotients of affine vertex operator superalgebras constructed from the affine Lie super algebra \( \hat{\mathfrak{osp}}\left(1\left|2\right.\right) \) at certain rational values of the level k. We classify all isomorphism classes of ℤ2-graded simple relaxed highest weight modules over the minimal model \( \mathfrak{osp}\left(1|2\right) \) vertex operator superalgebras in both the Neveu–Schwarz and Ramond sectors. To this end, we combine free field realisations, screening operators and the theory of symmetric functions in the Jack basis to compute explicit presentations for the Zhu algebras in both the Neveu–Schwarz and Ramond sectors. Two different free field realisations are used depending on the level. For k < −1, the free field realisation resembles the Wakimoto free field realisation of affine \( \mathfrak{sl}(2) \) and is originally due to Bershadsky and Ooguri. It involves 1 free boson (or rank 1 Heisenberg vertex algebra), one βγ bosonic ghost system and one bc fermionic ghost system. For k > −1, the argument presented here requires the bosonisation of the βγ system by embedding it into an indefinite rank 2 lattice vertex algebra.
Article PDF
References
V. Kac, Lie superalgebras, Adv. Math. 26 (1977), no. 1, 8–96.
D. Ridout, J. Snadden, S. Wood, An admissible level \( \hat{\mathfrak{osp}}\left(1\left|2\right.\right) \)-model: modular transformations and the Verlinde formula, Lett. Math. Phys. 108 (2018), no. 11, 2363–2423.
J.-B. Fan, M. Yu, Modules over affine Lie superalgebras, arXiv:hep-th/9304122 (1993).
I. P. Ennes, A. V. Ramallo, Fusion rules and singular vectors of the osp(1|2) current algebra, Nucl. Phys. B502 (1997), no. 3, 671–712.
V. Kac, W. Wang, Vertex operator superalgebras and their representations, in: Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups, Contemporary Mathematics, Vol. 175, American Mathematical Society, Providence, 1994, pp. 161–191.
T. Creutzig, J. Frohlich, S. Kanade, Representation theory of \( {L}_k\left(\mathfrak{osp}\left(1\left|2\right.\right)\right) \)from vertex tensor categories and Jacobi forms, Proc. Amer. Math. Soc. 146 (2018), no. 11, 4571–4589.
V. Kac, M. Wakimoto, Modular invariant representations of infinite-dimensional Lie algebras and superalgebras, Proc. Nat. Acad. Sci. USA 85 (1988), no. 14, 4956–4960.
K. Kawasetsu, D. Ridout, Relaxed highest-weight modules I: rank 1 cases, Comm. Math. Phys. 368 (2019), no. 2, 627–663.
D. Ridout, \( \hat{\mathfrak{sl}}{(2)}_{-1/2} \): A case study, Nucl. Phys. B814 (2009), no. 3, 485–521.
T. Creutzig, D. Ridout, Modular data and Verlinde formulae for fractional level WZW models I, Nucl. Phys. B865 (2012), no. 1, 83–114.
T. Creutzig, D. Ridout, Modular data and verlinde formulae for fractional level WZW models II, Nucl. Phys. B875 (2013), no. 2, 423–458.
S. Kanade, T. Liu, D. Ridout, Cosets, characters and fusion for the admissible level \( \mathfrak{osp}\left(1\left|2\right.\right) \)minimal models, Nucl. Phys. B938 (2019), no. , 22–55.
D. Adamović, Realizations of simple affine vertex algebras and their modules: the cases sl(2) and osp(1; 2), Comm. Math. Phys. 366 (2019), no. 3, 1025–1067.
B. Feigin, T. Nakanishi, H. Ooguri, The annihilating ideals of minimal models, Int. J. Mod. Phys. A7 (1992), no. 1, 217–238.
Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), no. 1, 237–302.
K. Iohara, Y. Koga, Enright functors for Kac–Moody superalgebra, Abh. Math. Semin. Univ. Hambg. 82 (2012), no. 2, 205–226.
Ф. Г. Маликов, Б. Л. Фейгин, Д. Б. Фукс, Особые векторы в модулях Верма над алгебрами Каца–Муди, Функц. анализ и его прил. 20 (1986), вып. 2, 25–37. Engl. transl.: F. G. Malikov, B. L. Feigin, D. B. Fuks, Singular vectors in Verma modules over Kac–Moody algebras, Funct. Anal. Appl. 20 (1986), no. 2, 103–113.
B. Feigin, D. Fuchs, Representations of the Virasoro algebra, in: Representation of Lie groups and Related Topics, Advanced Studies in Contemporary Mathematics, Vol. 7, Gordon and Breach, New York, 1990, pp. 465–554.
M. Wakimoto, Fock representations of the affine Lie algebra \( {A}_1^{(1)} \), Comm. Math. Phys. 104 (1986), no. 4, 605–609.
V. Dotsenko, V. Fateev, Conformal algebra and multipoint correlation functions in 2D statistical models, Nucl. Phys. B240 (1984), no. 3, 312–348.
A. Tsuchiya, Y. Kanie, Fock space representations of the Virasoro algebra—inter-twining operators, Publ. Res. Inst. Math. Sci. 22 (1986), no. 2, 259–327.
M. Wakimoto, Y. Yamada, The Fock representations of the Virasoro algebra and the Hirota equations of the modified KP hierarchies, Hiroshima Math. J. 16 (1986), no. 2, 427–441.
K. Mimachi, Y. Yamada, Singular vectors of the Virasoro algebra in terms of Jack symmetric polynomials, Comm. Math. Phys. 174 (1995), no. 2, 447–455.
M. Kato,Y. Yamada, Missing link between Virasoro and \( \hat{sl(2)} \)Kac–Moody algebras, Progr. Theoret. Phys. Suppl. 110 (1992), no. 110, 291–302.
D. Ridout, S. Wood, Relaxed singular vectors, Jack symmetric functions and fractional level \( \hat{\mathfrak{sl}}(2) \)models, Nucl. Phys. B894 (2015), 621–664.
P. Desrosiers, L. Lapointe, P. Mathieu, Supersymmetric Calogero–Moser–Sutherland models and Jack superpolynomials, Nucl. Phys. B606 (2001), no. 3, 547–582.
S. Yanagida, Singular vectors of N = 1 super Virasoro algebra via Uglov symmetric functions, arXiv:1508.06036 (2015).
O. Blondeau-Fournier, P. Mathieu, D. Ridout, S. Wood, The super-Virasoro singular vectors and Jack superpolynomials relationship revisited, Nucl. Phys. B913 (2016), 34–63.
H. Awata, Y. Matsuo, S. Odake, J. Shiraishi, Excited states of the Calogero–Sutherland model and singular vectors of the Wn algebra, Nucl. Phys. B449 (1995), no. 1–2 ,347–374.
D. Ridout, S. Siu, S. Wood, Singular vectors for the WN algebras, J. Math. Phys., 59 (2018), no. 3, 031701.
B. Feigin, M. Jimbo, T. Miwa, E. Mukhin, A differential ideal of symmetric polynomials spanned by Jack polynomials at β = − (r − 1)/(k + 1), Int. Math. Res. Not. 2002 (2002), no. 23, 1223–1237.
A. Tsuchiya, S. Wood, On the extended W-algebra of type \( {\mathfrak{sl}}_2 \)at positive rational level, Int. Math. Res. Not. 2015 (2015), no. 14, 5357–5435.
D. Ridout, S. Wood, From Jack polynomials to minimal model spectra, J. Phys. A48 (2015), no. 4, 045201.
O. Blondeau-Fournier, P. Mathieu, D. Ridout, S. Wood, Superconformal minimal models and admissible Jack polynomials, Adv. Math. 314 (2017), 71–123.
S.-J. Chen, W. Wang, Dualities and Representations of Lie Superalgebras, Graduate Studies in Mathematics, Vol. 144, American Mathematical Society, Providence, 2012.
G. Pinczon, The enveloping algebra of the lie superalgebra osp(1|2), J. Algebra 132 (1990), no. 1, 219–242.
A. Leśniewski, A remark on the Casimir elements of Lie superalgebras and quantized Lie superalgebras, J. Math. Phys. 36 (1995), no. 3, 1457–1461.
R. Block, Classification of the irreducible representations of \( \mathfrak{sl}\left(2,\mathrm{\mathbb{C}}\right) \), Bull. Amer. Math. Soc. 1 (1972), no. 1, 247–250.
R. Block, The irreducible representations of the Weyl algebra A1, Lecture Notes in Mathematics 740 (1979), 69–79.
V. Mazorchuk, Lectures on \( \mathfrak{sl}\left(2,\mathrm{\mathbb{C}}\right) \)-Modules, Imperial College Press, London, 2010.
D. Ridout, S. Wood, Bosonic ghosts at c = 2 as a logarithmic CFT, Lett. Math. Phys. 105 (2015), no. 2, 279–307.
E. Frenkel, D. Ben-Zvi, Vertex Algebras and Algebraic Curves, Mathematical Surveys and Monographs, Vol. 88, American Mathematical Society, Providence, 2001.
M. Gorelik, V. Kac, On simplicity of vacuum modules, Adv. Math. 211 (2007), no. 2, 621–677.
C. Dong, H. Li, G. Mason, Twisted representations of vertex operator algebras, Math. Ann. 310 (1998), no. 3, 571–600.
I. Frenkel, Y. Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke. Math. J. 66 (1992), no. 1, 123–168.
K. Iohara, Y. Koga, Fusion algebras for N = 1 superconformal field theories through coinvariants I: \( \hat{\mathfrak{osp}}\left(1\left|2\right.\right) \)-symmetry, J. reine angew. Math. 531 (2001), 1–34.
V. Kac, Vertex Algebras for Beginners, 2nd edition, University Lecture Series, Vol. 10, American Mathematical Society, Providence, 1998.
C. Dong, J. Lepowsky, Generalized Vertex Algebras and Relative Vertex Operators, Progress in Mathematics, Vol. 112, Birkhäuser, Boston, 1993.
M. Bershadsky, H. Ooguri, Hidden OSp(N, 2) symmetries in superconformal field theories, Phys. Lett. B229 (1989), no. 4, 374–378.
M. Bershadsky, H. Ooguri, Hidden SL(n) symmetry in conformal field theories, Comm. Math. Phys. 126 (1989), no. 1, 49–83.
J. L. Petersen, J. Rasmussen, M. Yu, Conformal blocks for admissible representations in SL(2) current algebra, Nucl. Phys. B458 (1995), no. 1–2, 309–342.
I. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1995.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
WOOD, S. ADMISSIBLE LEVEL \( \mathfrak{osp}\left(1\left|2\right.\right) \) MINIMAL MODELS AND THEIR RELAXED HIGHEST WEIGHT MODULES. Transformation Groups 25, 887–943 (2020). https://doi.org/10.1007/s00031-020-09567-3
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00031-020-09567-3