ADMISSIBLE LEVEL osp 1 2 $$ \mathfrak{osp}\left(1\left|2\right.\right) $$ MINIMAL MODELS AND THEIR RELAXED HIGHEST WEIGHT MODULES

WOOD, SIMON

doi:10.1007/s00031-020-09567-3

Open Access
Published: 04 May 2020

ADMISSIBLE LEVEL $ \mathfrak{osp}\left(1\left|2\right.\right) $ MINIMAL MODELS AND THEIR RELAXED HIGHEST WEIGHT MODULES

SIMON WOOD¹

Transformation Groups volume 25, pages 887–943 (2020)Cite this article

252 Accesses
4 Citations
Metrics details

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.

References

V. Kac, Lie superalgebras, Adv. Math. 26 (1977), no. 1, 8–96.
MATH Google Scholar
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.
MathSciNet MATH Google Scholar
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.
MathSciNet MATH Google Scholar
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.
MathSciNet MATH Google Scholar
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.
MathSciNet MATH Google Scholar
K. Kawasetsu, D. Ridout, Relaxed highest-weight modules I: rank 1 cases, Comm. Math. Phys. 368 (2019), no. 2, 627–663.
MathSciNet MATH Google Scholar
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.
MathSciNet MATH Google Scholar
T. Creutzig, D. Ridout, Modular data and verlinde formulae for fractional level WZW models II, Nucl. Phys. B875 (2013), no. 2, 423–458.
MathSciNet MATH Google Scholar
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.
MathSciNet MATH Google Scholar
B. Feigin, T. Nakanishi, H. Ooguri, The annihilating ideals of minimal models, Int. J. Mod. Phys. A7 (1992), no. 1, 217–238.
MathSciNet MATH Google Scholar
Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), no. 1, 237–302.
MathSciNet MATH Google Scholar
K. Iohara, Y. Koga, Enright functors for Kac–Moody superalgebra, Abh. Math. Semin. Univ. Hambg. 82 (2012), no. 2, 205–226.
MathSciNet MATH Google Scholar
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.
MathSciNet MATH Google Scholar
V. Dotsenko, V. Fateev, Conformal algebra and multipoint correlation functions in 2D statistical models, Nucl. Phys. B240 (1984), no. 3, 312–348.
Google Scholar
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.
MathSciNet MATH Google Scholar
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.
MathSciNet MATH Google Scholar
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.
MathSciNet MATH Google Scholar
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.
MathSciNet MATH Google Scholar
P. Desrosiers, L. Lapointe, P. Mathieu, Supersymmetric Calogero–Moser–Sutherland models and Jack superpolynomials, Nucl. Phys. B606 (2001), no. 3, 547–582.
MathSciNet MATH Google Scholar
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.
MATH Google Scholar
H. Awata, Y. Matsuo, S. Odake, J. Shiraishi, Excited states of the Calogero–Sutherland model and singular vectors of the W_n algebra, Nucl. Phys. B449 (1995), no. 1–2 ,347–374.
MathSciNet MATH Google Scholar
D. Ridout, S. Siu, S. Wood, Singular vectors for the W_N 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.
MathSciNet MATH Google Scholar
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.
MATH Google Scholar
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.
MathSciNet MATH Google Scholar
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.
MathSciNet MATH Google Scholar
A. Leśniewski, A remark on the Casimir elements of Lie superalgebras and quantized Lie superalgebras, J. Math. Phys. 36 (1995), no. 3, 1457–1461.
MathSciNet MATH Google Scholar
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.
Google Scholar
V. Mazorchuk, Lectures on $ \mathfrak{sl}\left(2,\mathrm{\mathbb{C}}\right) $-Modules, Imperial College Press, London, 2010.
Google Scholar
D. Ridout, S. Wood, Bosonic ghosts at c = 2 as a logarithmic CFT, Lett. Math. Phys. 105 (2015), no. 2, 279–307.
MathSciNet MATH Google Scholar
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.
MathSciNet MATH Google Scholar
C. Dong, H. Li, G. Mason, Twisted representations of vertex operator algebras, Math. Ann. 310 (1998), no. 3, 571–600.
MathSciNet MATH Google Scholar
I. Frenkel, Y. Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke. Math. J. 66 (1992), no. 1, 123–168.
MathSciNet MATH Google Scholar
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.
MathSciNet MATH Google Scholar
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.
MathSciNet MATH Google Scholar
M. Bershadsky, H. Ooguri, Hidden SL(n) symmetry in conformal field theories, Comm. Math. Phys. 126 (1989), no. 1, 49–83.
MathSciNet MATH Google Scholar
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.
MathSciNet MATH Google Scholar
I. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1995.

Download references

Author information

Authors and Affiliations

School of Mathematics, Cardiff University, Cardiff, United Kingdom, CF24 4AG
SIMON WOOD

Authors

SIMON WOOD
View author publications
You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to SIMON WOOD.

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/.

Reprints and Permissions

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

Download citation

Published: 04 May 2020
Issue Date: September 2020
DOI: https://doi.org/10.1007/s00031-020-09567-3