In this paper, we use free field realisations of the A-type principal, or Casimir, WN algebras to derive explicit formulae for singular vectors in Fock modules. These singular vectors are constructed by applying screening operators to Fock module highest weight vectors. The action of the screening operators is then explicitly evaluated in terms of Jack symmetric functions and their skew analogues. The resulting formulae depend on sequences of pairs of integers that completely determine the Fock module as well as the Jack symmetric functions.
REFERENCES
1.
Arakawa
, T.
, Creutzig
, T.
, and Linshaw
, A.
, “W-algebras as coset vertex algebras
,” e-print arXiv:1801.03822 [math.QA].2.
Arakawa
, T.
and Jiang
, C.
, “Coset vertex operator algebras and -algebras of A-type
,” Sci. China Math.
61
, 191
–206
(2018
).3.
Awata
, H.
, Matsuo
, Y.
, Odake
, S.
, and Shiraishi
, J.
, “Excited states of the Calogero-Sutherland model and singular vectors of the Wn algebra
,” Nucl. Phys. B
449
, 347
–374
(1995
); e-print arXiv:hep-th/9503043.4.
Bais
, F.
, Bouwknegt
, P.
, Schoutens
, K.
, and Surridge
, M.
, “Coset construction for extended Virasoro algebras
,” Nucl. Phys. B
304
, 371
–391
(1988
).5.
Bais
, F.
, Bouwknegt
, P.
, Surridge
, M.
, and Schoutens
, K.
, “Extensions of the Virasoro algebra constructed from Kac-Moody algebras using higher order Casimir invariants
,” Nucl. Phys. B
304
, 348
–370
(1988
).6.
Belavin
, A.
, Polyakov
, M.
, and Zamolodchikov
, A.
, “Infinite conformal symmetry in two-dimensional quantum field theory
,” Nucl. Phys. B
241
, 333
–380
(1984
).7.
Blondeau-Fournier
, O.
, Mathieu
, P.
, Ridout
, D.
, and Wood
, S.
, “The super-Virasoro singular vectors and Jack superpolynomials relationship revisited
,” Nucl. Phys. B
913
, 34
–63
(2016
); e-print arXiv:1605.08621 [math-ph].8.
Blondeau-Fournier
, O.
, Mathieu
, P.
, Ridout
, D.
, and Wood
, S.
, “Superconformal minimal models and admissible Jack polynomials
,” Adv. Math.
314
, 71
–123
(2017
); e-print arXiv:1606.04187 [hep-th].9.
Bouwknegt
, P.
, MacCarthy
, J.
, and Pilch
, K.
, The Algebra. Modules, Semi-Infinite Cohomology and BV Algebras, Lecture Notes in Physics (Springer
, Heidelberg
, 1996
).10.
Bouwknegt
, P.
and Schoutens
, K.
, “W-symmetry in conformal field theory
,” Phys. Rep.
223
, 183
–276
(1993
); e-print arXiv:hep-th/9210010.11.
de Boer
, J.
and Tjin
, T.
, “The relation between quantum W-algebras and Lie algebras
,” Commun. Math. Phys.
160
, 317
–332
(1994
); e-print arXiv:hep-th/9302006.12.
Desrosiers
, P.
, Lapointe
, L.
, and Mathieu
, P.
, “Supersymmetric Calogero-Moser-Sutherland models and Jack superpolynomials
,” Nucl. Phys. B
606
, 547
–582
(2001
); e-print arXiv:hep-th/0103178.13.
Desrosiers
, P.
, Lapointe
, L.
, and Mathieu
, P.
, “Superconformal field theory and Jack superpolynomials
,” J. High Energy Phys.
2012
, 37
; e-print arXiv:1205.0784 [hep-th].14.
Dotsenko
, V.
and Fateev
, V.
, “Conformal algebra and multipoint correlation functions in 2D statistical models
,” Nucl. Phys. B
240
, 312
–348
(1984
).15.
Fateev
, V.
and Zamolodchikov
, A.
, “Nonlocal (parafermion) currents in two-dimensional conformal quantum field theory and self-dual critical points in ZN-symmetric statistical systems
,” Sov. Phys. JETP
62
, 215
–225
(1985
), http://www.jetp.ac.ru/cgi-bin/e/index/e/62/2/p215?a=list.16.
Fateev
, V.
and Zamolodchikov
, A.
, “Conformal quantum field theory models in two-dimensions having Z3 symmetry
,” Nucl. Phys. B
280
, 644
–660
(1987
).17.
Feigin
, B.
and Frenkel
, E.
, “Quantization of the Drinfeld-Sokolov reduction
,” Phys. Lett. B
246
, 75
–81
(1990
).18.
Gaberdiel
, M.
and Gopakumar
, R.
, “Minimal model holography
,” J. Phys. A: Math. Theor.
46
, 214002
(2013
); e-print arXiv:1207.6697 [hep-th].19.
Goddard
, P.
, Kent
, A.
, and Olive
, D.
, “Virasoro algebras and coset space models
,” Nucl. Phys. B
152
, 88
–92
(1985
).20.
Jack
, H.
, “A class of symmetric polynomials with a parameter
,” Proc. - R. Soc. Edinburgh, Sect. A: Math. Phys. Sci.
69
, 1
–18
(1970
-1971
).21.
Kato
, M.
and Yamada
, Y.
, “Missing link between Virasoro and Kac-Moody algebras
,” Prog. Theor. Phys. Suppl.
110
, 291
–302
(1992
).22.
Lukyanov
, A.
, “Quantization of the Gel’fand-Dikii brackets
,” Funct. Anal. Appl.
22
, 255
–262
(1988
).23.
Macdonald
, I.
, Symmetric Functions and Hall Polynomials
, Oxford Mathematical Monographs, 2nd ed. (Clarendon Press
, Oxford
, 1995
).24.
Mimachi
, K.
and Yamada
, Y.
, “Singular vectors of the Virasoro algebra in terms of Jack symmetric polynomials
,” Commun. Math. Phys.
174
, 447
–455
(1995
).25.
Ridout
, D.
and Wood
, S.
, “From Jack polynomials to minimal model spectra
,” J. Phys. A: Math. Theor.
48
, 045201
(2015
); e-print arXiv:1409.4847 [hep-th].26.
Ridout
, D.
and Wood
, S.
, “Relaxed singular vectors, Jack symmetric functions and fractional level models
,” Nucl. Phys. B
894
, 621
–664
(2015
); e-print arXiv:1501.07318 [hep-th].27.
Tarasov
, V.
and Varchenko
, A.
, “Selberg type integrals associated with
,” Lett. Math. Phys.
65
, 173
–185
(2003
); e-print arXiv:math/0302148 [math.QA].28.
Tsuchiya
, A.
and Kanie
, Y.
, “Fock space representations of Virasoro algebra and intertwining operators
,” Proc. Jpn. Acad., Ser. A
62
, 12
–15
(1986
).29.
Tsuchiya
, A.
and Kanie
, Y.
, “Fock space representations of Virasoro algebra and intertwining operators
,” Publ. Res. Inst. Math. Sci.
22
, 259
–327
(1986
).30.
Tsuchiya
, A.
and Wood
, S.
, “On the extended W-algebra of type sl2 at positive rational level
,” Int. Math. Res. Not.
2015
, 5357
–5435
; e-print arXiv:1302.6435 [math.QA].31.
Vasiliev
, M.
, “Higher-spin gauge theories in four, three and two dimensions
,” Int. J. Mod. Phys. D
05
, 763
–797
(1996
); e-print arXiv:hep-th/9611024.32.
Wakimoto
, M.
and Yamada
, H.
, “Irreducible decompositions of Fock representations of the Virasoro algebra
,” Lett. Math. Phys.
7
, 513
–516
(1983
).33.
Warnaar
, S.
, “A Selberg integral for the Lie algebra An
,” Acta Math.
203
, 269
–304
(2009
); e-print arXiv:0708.1193 [math.CA].34.
35.
Yanagida
, S.
, “Singular vectors of N = 1 super Virasoro algebra via Uglov symmetric functions
,” e-print arXiv:1508.06036 [math.QA].36.
Zamolodchikov
, A.
, “Infinite additional symmetries in two-dimensional conformal quantum field theory
,” Theor. Math. Phys.
65
, 1205
–1213
(1985
).© 2018 Author(s).
2018
Author(s)
You do not currently have access to this content.
