Abstract
We explicitly construct and list all unitary superconformal multiplets, along with their index contributions, in five and six dimensions. From this data, we uncover various unifying themes in the representation theory of five- and six-dimensional superconformal field theories. At the same time, we provide a detailed argument for the complete classification of unitary irreducible representations in five dimensions using a combination of physical and mathematical techniques.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Montonen and D.I. Olive, Magnetic Monopoles as Gauge Particles?, Phys. Lett. B 72 (1977) 117 [INSPIRE].
P. Goddard, J. Nuyts and D.I. Olive, Gauge Theories and Magnetic Charge, Nucl. Phys. B 125 (1977) 1 [INSPIRE].
E. Witten and D.I. Olive, Supersymmetry Algebras That Include Topological Charges, Phys. Lett. B78 (1978) 97.
N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [INSPIRE].
P.C. Argyres and N. Seiberg, S-duality in N = 2 supersymmetric gauge theories, JHEP 12 (2007) 088 [arXiv:0711.0054] [INSPIRE].
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
D. Green, Z. Komargodski, N. Seiberg, Y. Tachikawa and B. Wecht, Exactly Marginal Deformations and Global Symmetries, JHEP 06 (2010) 106 [arXiv:1005.3546] [INSPIRE].
N. Seiberg, Electric-magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].
K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].
W. Nahm, Supersymmetries and their Representations, Nucl. Phys. B 135 (1978) 149 [INSPIRE].
C. Romelsberger, Counting chiral primaries in N = 1, D = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [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].
J. Bhattacharya, S. Bhattacharyya, S. Minwalla and S. Raju, Indices for Superconformal Field Theories in 3, 5 and 6 Dimensions, JHEP 02 (2008) 064 [arXiv:0801.1435] [INSPIRE].
C. Beem, L. Rastelli and B.C. van Rees, \( \mathcal{W} \) symmetry in six dimensions, JHEP 05 (2015) 017 [arXiv:1404.1079] [INSPIRE].
C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli and B.C. van Rees, Infinite Chiral Symmetry in Four Dimensions, Commun. Math. Phys. 336 (2015) 1359 [arXiv:1312.5344] [INSPIRE].
C. Cordova, T.T. Dumitrescu and K. Intriligator, Anomalies, Renormalization Group Flows and the a-Theorem in Six-Dimensional (1, 0) Theories, JHEP 10 (2016) 080 [arXiv:1506.03807] [INSPIRE].
J. Louis and S. Lüst, Supersymmetric AdS 7 backgrounds in half-maximal supergravity and marginal operators of (1, 0) SCFTs, JHEP 10 (2015) 120 [arXiv:1506.08040] [INSPIRE].
C. Cordova, T.T. Dumitrescu and K. Intriligator, Deformations of Superconformal Theories, arXiv:1602.01217 [INSPIRE].
C. Beem, M. Lemos, L. Rastelli and B.C. van Rees, The (2, 0) superconformal bootstrap, Phys. Rev. D 93 (2016) 025016 [arXiv:1507.05637] [INSPIRE].
S. Minwalla, Restrictions imposed by superconformal invariance on quantum field theories, Adv. Theor. Math. Phys. 2 (1998) 781 [hep-th/9712074] [INSPIRE].
V.K. Dobrev and V.B. Petkova, All Positive Energy Unitary Irreducible Representations of Extended Conformal Supersymmetry, Phys. Lett. B 162 (1985) 127 [INSPIRE].
V.K. Dobrev and V.B. Petkova, ON the group theoretical approach to extended conformal supersymmetry: classification of multiplets, Lett. Math. Phys. 9 (1985) 287 [INSPIRE].
V.K. Dobrev and V.B. Petkova, Group Theoretical Approach to Extended Conformal Supersymmetry: Function Space Realizations and Invariant Differential Operators, Fortsch. Phys. 35 (1987) 537 [INSPIRE].
V.K. Dobrev, Positive energy unitary irreducible representations of D = 6 conformal supersymmetry, J. Phys. A 35 (2002) 7079 [hep-th/0201076] [INSPIRE].
E. Witten, Some comments on string dynamics, in proceedings of Future perspectives in string theory (Strings’95), Los Angeles, U.S.A., 13-18 March 1995, hep-th/9507121 [INSPIRE].
N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [INSPIRE].
D.R. Morrison and N. Seiberg, Extremal transitions and five-dimensional supersymmetric field theories, Nucl. Phys. B 483 (1997) 229 [hep-th/9609070] [INSPIRE].
M.R. Douglas, S.H. Katz and C. Vafa, Small instantons, Del Pezzo surfaces and type-I-prime theory, Nucl. Phys. B 497 (1997) 155 [hep-th/9609071] [INSPIRE].
K.A. Intriligator, D.R. Morrison and N. Seiberg, Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces, Nucl. Phys. B 497 (1997) 56 [hep-th/9702198] [INSPIRE].
J.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa, Atomic Classification of 6D SCFTs, Fortsch. Phys. 63 (2015) 468 [arXiv:1502.05405] [INSPIRE].
L. Bhardwaj, Classification of 6d \( \mathcal{N}=\left(1,0\right) \) gauge theories, JHEP 11 (2015) 002 [arXiv:1502.06594] [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].
J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a slightly broken higher spin symmetry, Class. Quant. Grav. 30 (2013) 104003 [arXiv:1204.3882] [INSPIRE].
V. Alba and K. Diab, Constraining conformal field theories with a higher spin symmetry in d >3 dimensions, JHEP 03 (2016) 044 [arXiv:1510.02535] [INSPIRE].
M. Buican, S. Giacomelli, T. Nishinaka and C. Papageorgakis, Argyres-Douglas Theories and S-duality, JHEP 02 (2015) 185 [arXiv:1411.6026] [INSPIRE].
M. Del Zotto, C. Vafa and D. Xie, Geometric engineering, mirror symmetry and \( 6{\mathrm{d}}_{\left(1,0\right)}\to\ 4{\mathrm{d}}_{\left(\mathcal{N}=2\right)} \), JHEP 11 (2015) 123 [arXiv:1504.08348] [INSPIRE].
M. Buican and T. Nishinaka, Conformal Manifolds in Four Dimensions and Chiral Algebras, J. Phys. A 49 (2016) 465401 [arXiv:1603.00887] [INSPIRE].
C. Córdova, Deformations of Superconformal Field Theories, Autumn Symposium on String/M Theory 2014, Princeton University Seminar 2014, http://media.kias.re.kr/detailPage.do?pro_seq=564&type=p.
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].
F.A. Dolan, Character formulae and partition functions in higher dimensional conformal field theory, J. Math. Phys. 47 (2006) 062303 [hep-th/0508031] [INSPIRE].
J. Penedones, E. Trevisani and M. Yamazaki, Recursion Relations for Conformal Blocks, JHEP 09 (2016) 070 [arXiv:1509.00428] [INSPIRE].
M. Yamazaki, Comments on Determinant Formulas for General CFTs, JHEP 10 (2016) 035 [arXiv:1601.04072] [INSPIRE].
Y. Oshima and M. Yamazaki, Determinant Formula for Parabolic Verma Modules of Lie Superalgebras, arXiv:1603.06705 [INSPIRE].
C. Cordova, T.T. Dumitrescu and K. Intriligator, Multiplets of superconformal symmetry in diverse dimensions, to appear.
K. Intriligator, Anomalies, RG flows, and the a-theorem in six-dimensional (1, 0) theories, in proceedings of Strings 2015, https://strings2015.icts.res.in.
T. Dumitrescu, Anomalies, RG Flows, and the a-theorem in 6d — Part I, in proceedings of 2015 Simons Summer Workshop, http://scgp.stonybrook.edu/archives/category/videos.
C. Córdova, Anomalies, RG Flows, and the a-theorem in 6d — Part II, in proceedings of 2015 Simons Summer Workshop, http://scgp.stonybrook.edu/archives/category/videos.
C. Córdova, Anomalies RG-Flows and the a-Theorem in Six-Dimensions, London Triangle Seminar, December 2015.
M. Bianchi, F.A. Dolan, P.J. Heslop and H. Osborn, N = 4 superconformal characters and partition functions, Nucl. Phys. B 767 (2007) 163 [hep-th/0609179] [INSPIRE].
H.-C. Kim, S.-S. Kim and K. Lee, 5-dim Superconformal Index with Enhanced En Global Symmetry, JHEP 10 (2012) 142 [arXiv:1206.6781] [INSPIRE].
D. Rodr´ıguez-Gómez and G. Zafrir, On the 5d instanton index as a Hilbert series, Nucl. Phys. B 878 (2014) 1 [arXiv:1305.5684] [INSPIRE].
Y. Tachikawa, Instanton operators and symmetry enhancement in 5d supersymmetric gauge theories, PTEP 2015 (2015) 043B06 [arXiv:1501.01031] [INSPIRE].
V.G. Kac, A Sketch of Lie Superalgebra Theory, Commun. Math. Phys. 53 (1977) 31 [INSPIRE].
L. Frappat, P. Sorba and A. Sciarrino, Dictionary on Lie superalgebras, hep-th/9607161 [INSPIRE].
C. Hwang, J. Kim, S. Kim and J. Park, General instanton counting and 5d SCFT, JHEP 07 (2015) 063 [Addendum ibid. 04 (2016) 094] [arXiv:1406.6793] [INSPIRE].
A. Passias and A. Tomasiello, Spin-2 spectrum of six-dimensional field theories, arXiv:1604.04286 [INSPIRE].
P.S. Howe and A. Umerski, Anomaly multiplets in six-dimensions and ten-dimensions, Phys. Lett. B 198 (1987) 57 [INSPIRE].
S.M. Kuzenko, J. Novak and I.B. Samsonov, The anomalous current multiplet in 6D minimal supersymmetry, JHEP 02 (2016) 132 [arXiv:1511.06582] [INSPIRE].
M. Buican, Minimal Distances Between SCFTs, JHEP 01 (2014) 155 [arXiv:1311.1276] [INSPIRE].
M. Buican, T. Nishinaka and C. Papageorgakis, Constraints on chiral operators in \( \mathcal{N}=2 \) SCFTs, JHEP 12 (2014) 095 [arXiv:1407.2835] [INSPIRE].
K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, Anomaly polynomial of general 6d SCFTs, PTEP 2014 (2014) 103B07 [arXiv:1408.5572] [INSPIRE].
K. Intriligator, 6d, \( \mathcal{N}=\left(1,0\right) \) Coulomb branch anomaly matching, JHEP 10 (2014) 162 [arXiv:1408.6745] [INSPIRE].
J. Fuchs and C. Schweigert, Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists, Cambridge University Press (2003).
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: 1606.00810
Electronic supplementary material
Below is the link to the electronic supplementary material.
ESM 1
(NB 174 kb)
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Buican, M., Hayling, J. & Papageorgakis, C. Aspects of superconformal multiplets in D > 4. J. High Energ. Phys. 2016, 91 (2016). https://doi.org/10.1007/JHEP11(2016)091
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2016)091