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.
, βW-algebras as coset vertex algebras
,β e-print arXiv:1801.03822 [math.QA].2.
Arakawa
, T.
and Jiang
, C.
, βCoset vertex operator algebras and -algebras of A-type
,β Sci. China Math.
β61
, 191
β206
(2018
).3.
Awata
, H.
, Matsuo
, Y.
, Odake
, S.
, and Shiraishi
, J.
, βExcited states of the Calogero-Sutherland model and singular vectors of the Wn algebra
,β Nucl. Phys. B
β449
, 347
β374
(1995
); e-print arXiv:hep-th/9503043.4.
Bais
, F.
, Bouwknegt
, P.
, Schoutens
, K.
, and Surridge
, M.
, βCoset construction for extended Virasoro algebras
,β Nucl. Phys. B
β304
, 371
β391
(1988
).5.
Bais
, F.
, Bouwknegt
, P.
, Surridge
, M.
, and Schoutens
, K.
, βExtensions of the Virasoro algebra constructed from Kac-Moody algebras using higher order Casimir invariants
,β Nucl. Phys. B
β304
, 348
β370
(1988
).6.
Belavin
, A.
, Polyakov
, M.
, and Zamolodchikov
, A.
, βInfinite conformal symmetry in two-dimensional quantum field theory
,β Nucl. Phys. B
β241
, 333
β380
(1984
).7.
Blondeau-Fournier
, O.
, Mathieu
, P.
, Ridout
, D.
, and Wood
, S.
, βThe super-Virasoro singular vectors and Jack superpolynomials relationship revisited
,β Nucl. Phys. B
β913
, 34
β63
(2016
); e-print arXiv:1605.08621 [math-ph].8.
Blondeau-Fournier
, O.
, Mathieu
, P.
, Ridout
, D.
, and Wood
, S.
, βSuperconformal minimal models and admissible Jack polynomials
,β Adv. Math.
β314
, 71
β123
(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.
, βW-symmetry in conformal field theory
,β Phys. Rep.
β223
, 183
β276
(1993
); e-print arXiv:hep-th/9210010.11.
de Boer
, J.
and Tjin
, T.
, βThe relation between quantum W-algebras and Lie algebras
,β Commun. Math. Phys.
β160
, 317
β332
(1994
); e-print arXiv:hep-th/9302006.12.
Desrosiers
, P.
, Lapointe
, L.
, and Mathieu
, P.
, βSupersymmetric Calogero-Moser-Sutherland models and Jack superpolynomials
,β Nucl. Phys. B
β606
, 547
β582
(2001
); e-print arXiv:hep-th/0103178.13.
Desrosiers
, P.
, Lapointe
, L.
, and Mathieu
, P.
, βSuperconformal field theory and Jack superpolynomials
,β J. High Energy Phys.
β2012
, 37
; e-print arXiv:1205.0784 [hep-th].14.
Dotsenko
, V.
and Fateev
, V.
, βConformal algebra and multipoint correlation functions in 2D statistical models
,β Nucl. Phys. B
β240
, 312
β348
(1984
).15.
Fateev
, V.
and Zamolodchikov
, A.
, βNonlocal (parafermion) currents in two-dimensional conformal quantum field theory and self-dual critical points in ZN-symmetric statistical systems
,β Sov. Phys. JETP
β62
, 215
β225
(1985
), http://www.jetp.ac.ru/cgi-bin/e/index/e/62/2/p215?a=list.16.
Fateev
, V.
and Zamolodchikov
, A.
, βConformal quantum field theory models in two-dimensions having Z3 symmetry
,β Nucl. Phys. B
β280
, 644
β660
(1987
).17.
Feigin
, B.
and Frenkel
, E.
, βQuantization of the Drinfeld-Sokolov reduction
,β Phys. Lett. B
β246
, 75
β81
(1990
).18.
Gaberdiel
, M.
and Gopakumar
, R.
, βMinimal model holography
,β J. Phys. A: Math. Theor.
β46
, 214002
(2013
); e-print arXiv:1207.6697 [hep-th].19.
Goddard
, P.
, Kent
, A.
, and Olive
, D.
, βVirasoro algebras and coset space models
,β Nucl. Phys. B
β152
, 88
β92
(1985
).20.
Jack
, H.
, βA class of symmetric polynomials with a parameter
,β Proc. - R. Soc. Edinburgh, Sect. A: Math. Phys. Sci.
β69
, 1
β18
(1970
-1971
).21.
Kato
, M.
and Yamada
, Y.
, βMissing link between Virasoro and Kac-Moody algebras
,β Prog. Theor. Phys. Suppl.
β110
, 291
β302
(1992
).22.
Lukyanov
, A.
, βQuantization of the Gelβfand-Dikii brackets
,β Funct. Anal. Appl.
β22
, 255
β262
(1988
).23.
Macdonald
, I.
, Symmetric Functions and Hall Polynomials
, Oxford Mathematical Monographs, 2nd ed. (Clarendon Press
, Oxford
, 1995
).24.
Mimachi
, K.
and Yamada
, Y.
, βSingular vectors of the Virasoro algebra in terms of Jack symmetric polynomials
,β Commun. Math. Phys.
β174
, 447
β455
(1995
).25.
Ridout
, D.
and Wood
, S.
, βFrom Jack polynomials to minimal model spectra
,β J. Phys. A: Math. Theor.
β48
, 045201
(2015
); e-print arXiv:1409.4847 [hep-th].26.
Ridout
, D.
and Wood
, S.
, βRelaxed singular vectors, Jack symmetric functions and fractional level models
,β Nucl. Phys. B
β894
, 621
β664
(2015
); e-print arXiv:1501.07318 [hep-th].27.
Tarasov
, V.
and Varchenko
, A.
, βSelberg type integrals associated with
,β Lett. Math. Phys.
β65
, 173
β185
(2003
); e-print arXiv:math/0302148 [math.QA].28.
Tsuchiya
, A.
and Kanie
, Y.
, βFock space representations of Virasoro algebra and intertwining operators
,β Proc. Jpn. Acad., Ser. A
β62
, 12
β15
(1986
).29.
Tsuchiya
, A.
and Kanie
, Y.
, βFock space representations of Virasoro algebra and intertwining operators
,β Publ. Res. Inst. Math. Sci.
β22
, 259
β327
(1986
).30.
Tsuchiya
, A.
and Wood
, S.
, βOn the extended W-algebra of type sl2 at positive rational level
,β Int. Math. Res. Not.
β2015
, 5357
β5435
; e-print arXiv:1302.6435 [math.QA].31.
Vasiliev
, M.
, βHigher-spin gauge theories in four, three and two dimensions
,β Int. J. Mod. Phys. D
β05
, 763
β797
(1996
); e-print arXiv:hep-th/9611024.32.
Wakimoto
, M.
and Yamada
, H.
, βIrreducible decompositions of Fock representations of the Virasoro algebra
,β Lett. Math. Phys.
β7
, 513
β516
(1983
).33.
Warnaar
, S.
, βA Selberg integral for the Lie algebra An
,β Acta Math.
β203
, 269
β304
(2009
); e-print arXiv:0708.1193 [math.CA].34.
Warnaar
, S.
, βThe Selberg integral
,β Adv. Math.
β224
, 499
β524
(2010
); e-print arXiv:0901.4176.35.
Yanagida
, S.
, βSingular vectors of N = 1 super Virasoro algebra via Uglov symmetric functions
,β e-print arXiv:1508.06036 [math.QA].36.
Zamolodchikov
, A.
, βInfinite additional symmetries in two-dimensional conformal quantum field theory
,β Theor. Math. Phys.
β65
, 1205
β1213
(1985
).Β© 2018 Author(s).
2018
Author(s)
You do not currently have access to this content.