It was conjectured by Belavin et al. [J. High Energy Phys. 2013(3), 35] that bosonization of a singular vector (in the Neveu–Schwarz sector) of the super analog of the Virasoro algebra can be identified with the Uglov symmetric function. In this paper, we prove this conjecture. We also extend this result to the Ramond sector of the super-Virasoro algebra.
REFERENCES
1.
K.
Mimachi
and Y.
Yamada
, “Singular vectors of the Virasoro algebra in terms of Jack symmetric polynomials
,” Commun. Math. Phys.
174
(2
), 447
–455
(1995
).2.
H.
Awata
, Y.
Matsuo
, S.
Odake
, and J.
Shiraishi
, “Excited states of the Calogero-Sutherland model and singular vectors of the WN algebra
,” Nucl. Phys. B
449
(1–2
), 347
–374
(1995
); arXiv:hep-th/9503043.3.
J.
Shiraishi
, H.
Kubo
, H.
Awata
, and S.
Odake
, “A quantum deformation of the Virasoro algebra and the Macdonald symmetric functions
,” Lett. Math. Phys.
38
(1
), 33
–51
(1996
); arXiv:q-alg/9507034.4.
A. A.
Belavin
, M. A.
Bershtein
, and G. M.
Tarnopolsky
, “Bases in coset conformal field theory from AGT correspondence and Macdonald polynomials at the roots of unity
,” J. High Energy Phys.
2013
(3
), 19
; arXiv:1211.2788.5.
D.
Uglov
, “Yangian Gelfand-Zetlin bases, -Jack polynomials and computation of dynamical correlation functions in the spin Calogero-Sutherland model
,” Commun. Math. Phys.
191
(3
), 663
–696
(1998
); arXiv:hep-th/9702020.6.
In the preprint,19 the proof in the NS sector was suggested. However, that proof contains serious gaps, and we do not know how to fill them. Our proof is based on a different (but related) approach.
7.
H.
Itoyama
, T.
Oota
, and R.
Yoshioka
, “2d-4d connection between q-Virasoro/W block at root of unity limit and instanton partition function on ALE space
,” Nucl. Phys. B
877
(2
), 506
–537
(2013
); arXiv:1308.2068.8.
H.
Itoyama
, T.
Oota
, and R.
Yoshioka
, “q-Virasoro/W algebra at root of unity and parafermions
,” Nucl. Phys. B
889
, 25
–35
(2014
); arXiv:1408.4216.9.
O.
Khlaif
and T.
Kimura
, “Virasoro constraint for Uglov matrix model
,” J. High Energy Phys.
2022
(4
), 29
; arXiv:2201.06839.10.
P.
Desrosiers
, L.
Lapointe
, and P.
Mathieu
, “Superconformal field theory and Jack superpolynomials
,” J. High Energy Phys.
2012
(9
), 37
; arXiv:1508.06036.11.
L.
Alarie-Vézina
, P.
Desrosiers
, and P.
Mathieu
, “Ramond singular vectors and Jack superpolynomials
,” J. Phys. A: Math. Theor.
47
, 035202
(2013
); arXiv:1309.7965.12.
O.
Blondeau-Fournier
, P.
Mathieu
, D.
Ridout
, and S.
Wood
, “The super-Virasoro singular vectors and Jack superpolynomials relationship revisited
,” Nucl. Phys. B
913
, 34
–63
(2016
); arXiv:1605.08621.13.
I. G.
MacDonald
, Symmetric Functions and Hall Polynomials
, 2nd ed. (Oxford University Press
, 1998
).14.
K.
Iohara
and Y.
Koga
, “Representation theory of Neveu-Schwarz and Ramond algebras. I: Verma modules
,” Adv. Math.
178
(1
), 1
–65
(2003
).15.
V. G.
Kac
and M.
Wakimoto
, “Unitarizable highest weight representations of the Virasoro, Neveu-Schwarz and Ramond algebras
,” in Conformal Groups and Related Symmetries: Physical Results and Mathematical Background (Clausthal-Zellerfeld, 1985)
, Lecture Notes in Physics Vol. 261 (Springer
, Berlin
, 1986
), pp. 345
–371
.16.
G. M. T.
Watts
, “Null vectors of the superconformal algebra: The Ramond sector
,” Nucl. Phys. B
407
(1
), 213
–236
(1993
); arXiv:hep-th/9306034.17.
J. D.
Cohn
and D.
Friedan
, “Super characters and chiral asymmetry in superconformal field theory
,” Nucl. Phys. B
296
(4
), 779
–799
(1988
).18.
M.
Bershtein
, P.
Gavrylenko
, and A.
Marshakov
, “Twist-field representations of W-algebras, exact conformal blocks and character identities
,” J. High Energy Phys.
2018
(8
), 56
; arXiv:1705.00957.19.
S.
Yanagida
, “Singular vectors of N = 1 super Virasoro algebra via Uglov symmetric functions
,” arXiv:1508.06036 (2015
).20.
H.
Awata
, H.
Kubo
, S.
Odake
, and J.
Shiraishi
, “Quantum WN algebras and Macdonald polynomials
,” Commun. Math. Phys.
179
(2
), 401
–416
(1996
); arXiv:q-alg/9508011.© 2022 Author(s). Published under an exclusive license by AIP Publishing.
2022
Author(s)
You do not currently have access to this content.
