Abstract
We compute the superconformal characters of various short multiplets in 4d \( \mathcal{N}=2 \) superconformal algebra, from which selection rules for operator products are obtained. Combining with the superconformal index, we show that a particular short multiplet appearing in the n-fold product of stress-tensor multiplet is absent in the (A1, A2n) Argyres-Douglas (AD) theory. This implies that certain operator product expansion (OPE) coefficients involving this multiplet vanish whenever the central charge c is identical to that of the AD theory. Similarly, by considering the n-th power of the current multiplet, we show that a particular short multiplet and OPE coefficients vanish for a class of AD theories with ADE flavor symmetry. We also consider the generalized AD theory of type (Ak−1, An−1) for coprime k, n and compute its Macdonald index using the associated W -algebra under a mild assumption. This allows us to show that a number of short multiplets and OPE coefficients vanish in this theory. We also provide a Mathematica file along with this paper, where we implement the algorithm by Cordova-Dumitrescu-Intriligator to compute the spectrum of 4d\( \mathcal{N}=2 \) superconformal multiplets as well as their superconformal character.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
V.K. Dobrev and V.B. Petkova, All positive energy unitary irreducible representations of extended conformal supersymmetry, Phys. Lett. 162B (1985) 127 [INSPIRE].
F.A. Dolan and H. Osborn, On short and semi-short representations for four-dimensional superconformal symmetry, Annals Phys. 307 (2003) 41 [hep-th/0209056] [INSPIRE].
C. Cordova, T.T. Dumitrescu and K. Intriligator, Multiplets of superconformal symmetry in diverse dimensions, JHEP 03 (2019) 163 [arXiv:1612.00809] [INSPIRE].
C. Cordova, T.T. Dumitrescu and K. Intriligator, Deformations of superconformal theories, JHEP 11 (2016) 135 [arXiv:1602.01217] [INSPIRE].
C. Beem et al., Infinite chiral symmetry in four dimensions, Commun. Math. Phys. 336 (2015) 1359 [arXiv:1312.5344] [INSPIRE].
M. Lemos and P. Liendo, \( \mathcal{N}=2 \) central charge bounds from 2d chiral algebras, JHEP 04 (2016) 004 [arXiv:1511.07449] [INSPIRE].
P. Liendo, I. Ramirez and J. Seo, Stress-tensor OPE in \( \mathcal{N}=2 \) superconformal theories, JHEP 02 (2016) 019 [arXiv:1509.00033] [INSPIRE].
P.C. Argyres and M.R. Douglas, New phenomena in SU(3) supersymmetric gauge theory, Nucl. Phys. B 448 (1995) 93 [hep-th/9505062] [INSPIRE].
P.C. Argyres, M.R. Plesser, N. Seiberg and E. Witten, New N = 2 superconformal field theories in four-dimensions, Nucl. Phys. B 461 (1996) 71 [hep-th/9511154] [INSPIRE].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, Gauge theories and Macdonald polynomials, Commun. Math. Phys. 319 (2013) 147 [arXiv:1110.3740] [INSPIRE].
J. Song, Superconformal indices of generalized Argyres-Douglas theories from 2d TQFT, JHEP 02 (2016) 045 [arXiv:1509.06730] [INSPIRE].
S. Cecotti, A. Neitzke and C. Vafa, R-twisting and 4d/2d correspondences, arXiv:1006.3435 [INSPIRE].
D. Xie, General Argyres-Douglas theory, JHEP 01 (2013) 100 [arXiv:1204.2270] [INSPIRE].
S. Cecotti and M. Del Zotto, Infinitely many N = 2 SCFT with ADE flavor symmetry, JHEP 01 (2013) 191 [arXiv:1210.2886] [INSPIRE].
S. Cecotti, M. Del Zotto and S. Giacomelli, More on the N = 2 superconformal systems of type D p(G), JHEP 04 (2013) 153 [arXiv:1303.3149] [INSPIRE].
Y. Wang and D. Xie, Classification of Argyres-Douglas theories from M 5 branes, Phys. Rev. D 94 (2016) 065012 [arXiv:1509.00847] [INSPIRE].
M. Buican and T. Nishinaka, On the superconformal index of Argyres-Douglas theories, J. Phys. A 49 (2016) 015401 [arXiv:1505.05884] [INSPIRE].
M. Buican and T. Nishinaka, Argyres-Douglas theories, the Macdonald index and an RG inequality, JHEP 02 (2016) 159 [arXiv:1509.05402] [INSPIRE].
M. Buican and T. Nishinaka, On irregular singularity wave functions and superconformal indices, JHEP 09 (2017) 066 [arXiv:1705.07173] [INSPIRE].
J. Song, D. Xie and W. Yan, Vertex operator algebras of Argyres-Douglas theories from M 5-branes, JHEP 12 (2017) 123 [arXiv:1706.01607] [INSPIRE].
A. Gadde, E. Pomoni, L. Rastelli and S.S. Razamat, S-duality and 2d topological QFT, JHEP 03 (2010) 032 [arXiv:0910.2225] [INSPIRE].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, The 4d superconformal index from q-deformed 2d Yang-Mills, Phys. Rev. Lett. 106 (2011) 241602 [arXiv:1104.3850] [INSPIRE].
D. Gaiotto, L. Rastelli and S.S. Razamat, Bootstrapping the superconformal index with surface defects, JHEP 01 (2013) 022 [arXiv:1207.3577] [INSPIRE].
L. Rastelli and S.S. Razamat, The superconformal index of theories of class \( \mathcal{S} \), arXiv:1412.7131 [INSPIRE].
C. Cordova and S.-H. Shao, Schur indices, BPS particles and Argyres-Douglas theories, JHEP 01 (2016) 040 [arXiv:1506.00265] [INSPIRE].
S. Cecotti, J. Song, C. Vafa and W. Yan, Superconformal index, BPS monodromy and chiral algebras, JHEP 11 (2017) 013 [arXiv:1511.01516] [INSPIRE].
D. Xie, W. Yan and S.-T. Yau, Chiral algebra of Argyres-Douglas theory from M 5 brane, arXiv:1604.02155 [INSPIRE].
M. Buican and T. Nishinaka, Conformal Manifolds in Four Dimensions and Chiral Algebras, J. Phys. A 49 (2016) 465401 [arXiv:1603.00887] [INSPIRE].
T. Creutzig, W-algebras for Argyres-Douglas theories, arXiv:1701.05926 [INSPIRE].
T. Creutzig, Logarithmic W-algebras and Argyres-Douglas theories at higher rank, JHEP 11 (2018) 188 [arXiv:1809.01725] [INSPIRE].
J. Song, Macdonald index and chiral algebra, JHEP 08 (2017) 044 [arXiv:1612.08956] [INSPIRE].
M. Fluder and J. Song, Four-dimensional lens space index from two-dimensional chiral algebra, JHEP 07 (2018) 073 [arXiv:1710.06029] [INSPIRE].
K. Maruyoshi and J. Song, Enhancement of supersymmetry via renormalization group flow and the superconformal index, Phys. Rev. Lett. 118 (2017) 151602 [arXiv:1606.05632] [INSPIRE].
K. Maruyoshi and J. Song, \( \mathcal{N}=1 \) deformations and RG flows of \( \mathcal{N}=2 \) SCFTs, JHEP 02 (2017) 075 [arXiv:1607.04281] [INSPIRE].
P. Agarwal, K. Maruyoshi and J. Song, \( \mathcal{N}=1 \) deformations and RG flows of \( \mathcal{N}=2 \) SCFTs, part II: non-principal deformations, JHEP 12 (2016) 103 [arXiv:1610.05311] [INSPIRE].
P. Agarwal, A. Sciarappa and J. Song, \( \mathcal{N}=1 \) Lagrangians for generalized Argyres-Douglas theories, JHEP 10 (2017) 211 [arXiv:1707.04751] [INSPIRE].
S. Benvenuti and S. Giacomelli, Lagrangians for generalized Argyres-Douglas theories, JHEP 10 (2017) 106 [arXiv:1707.05113] [INSPIRE].
O. Aharony and Y. Tachikawa, A holographic computation of the central charges of d = 4, N = 2 SCFTs, JHEP 01 (2008) 037 [arXiv:0711.4532] [INSPIRE].
A.D. Shapere and Y. Tachikawa, Central charges of N = 2 superconformal field theories in four dimensions, JHEP 09 (2008) 109 [arXiv:0804.1957] [INSPIRE].
V.K. Dobrev, Characters of the positive energy UIRs of D = 4 conformal supersymmetry, Phys. Part. Nucl. 38 (2007) 564 [hep-th/0406154] [INSPIRE].
V.K. Dobrev, Explicit character formulae for positive energy unitary irreducible representations of D = 4 conformal supersymmetry, J. Phys. A 46 (2013) 405202 [arXiv:1208.6250] [INSPIRE].
C. Beem and L. Rastelli, Vertex operator algebras, Higgs branches and modular differential equations, JHEP 08 (2018) 114 [arXiv:1707.07679] [INSPIRE].
F. Bonetti, C. Meneghelli and L. Rastelli, VOAs labelled by complex reflection groups and 4d SCFTs, JHEP 05 (2019) 155 [arXiv:1810.03612] [INSPIRE].
G. Mack, All unitary ray representations of the conformal group SU(2, 2) with positive energy, Commun. Math. Phys. 55 (1977) 1 [INSPIRE].
S. Minwalla, Restrictions imposed by superconformal invariance on quantum field theories, Adv. Theor. Math. Phys. 2 (1998) 783 [hep-th/9712074] [INSPIRE].
J. Kinney, J.M. Maldacena, S. Minwalla and S. Raju, An index for 4 dimensional super conformal theories, Commun. Math. Phys. 275 (2007) 209 [hep-th/0510251] [INSPIRE].
I.A. Ramírez, Mixed OPEs in \( \mathcal{N}=2 \) superconformal theories, JHEP 05 (2016) 043 [arXiv:1602.07269] [INSPIRE].
J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a higher spin symmetry, J. Phys. A 46 (2013) 214011 [arXiv:1112.1016] [INSPIRE].
G. Andrews, A. Schilling and S. Warnaar, An A 2 Bailey lemma and Rogers-Ramanujan-type identities, J. Amer. Math. Soc. 12 (1999) 677.
M. Buican, T. Nishinaka and C. Papageorgakis, Constraints on chiral operators in \( \mathcal{N}=2 \) SCFTs, JHEP 12 (2014) 095 [arXiv:1407.2835] [INSPIRE].
C. Beem, M. Lemos, P. Liendo, L. Rastelli and B.C. van Rees, The \( \mathcal{N}=2 \) superconformal bootstrap, JHEP 03 (2016) 183 [arXiv:1412.7541] [INSPIRE].
M. Lemos and P. Liendo, Bootstrapping \( \mathcal{N}=2 \) chiral correlators, JHEP 01 (2016) 025 [arXiv:1510.03866] [INSPIRE].
M. Lemos, P. Liendo, C. Meneghelli and V. Mitev, Bootstrapping \( \mathcal{N}=3 \) superconformal theories, JHEP 04 (2017) 032 [arXiv:1612.01536] [INSPIRE].
M. Cornagliotto, M. Lemos and V. Schomerus, Long multiplet bootstrap, JHEP 10 (2017) 119 [arXiv:1702.05101] [INSPIRE].
M. Cornagliotto, M. Lemos and P. Liendo, Bootstrapping the (A 1 , A 2) Argyres-Douglas theory, JHEP 03 (2018) 033 [arXiv:1711.00016] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1812.04743
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 https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Agarwal, P., Lee, S. & Song, J. Vanishing OPE coefficients in 4d \( \mathcal{N}=2 \) SCFTs. J. High Energ. Phys. 2019, 102 (2019). https://doi.org/10.1007/JHEP06(2019)102
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2019)102