Abstract
We propose two novel methods for computing the superconformal index of 5d superconformal field theories that cannot be described by conventional Lagrangian descriptions under mass deformations. The first approach involves the use of Higgs branch flows from UV Lagrangian theories, guided by transitions in 5-brane webs in Type IIB string theory. The second method employs the relationship between O7+-plane and O7−-plane with eight D7-branes, which applies to particular non-Lagrangian theories realized by brane configurations involving an O7+-plane. As a concrete application of our method, we compute the superconformal indices for all known rank-1 non-Lagrangian theories, which we also use to identify flavor symmetries and their global forms at the conformal field theory (CFT) fixed points.
Article PDF
Similar content being viewed by others
References
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].
A. Gadde, S.S. Razamat and B. Willett, “Lagrangian” for a Non-Lagrangian Field Theory with \(\mathcal{N}\) = 2 Supersymmetry, Phys. Rev. Lett. 115 (2015) 171604 [arXiv:1505.05834] [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].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [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. Bhardwaj, On the classification of 5d SCFTs, JHEP 09 (2020) 007 [arXiv:1909.09635] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, Seiberg-Witten curves with O7±-planes, JHEP 11 (2023) 178 [arXiv:2306.11631] [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].
A. Hanany and E. Witten, Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics, Nucl. Phys. B 492 (1997) 152 [hep-th/9611230] [INSPIRE].
H.-C. Kim, M. Kim, S.-S. Kim and K.-H. Lee, Bootstrapping BPS spectra of 5d/6d field theories, JHEP 04 (2021) 161 [arXiv:2101.00023] [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].
H. Nakajima and K. Yoshioka, Instanton counting on blowup. I. Dimensional pure gauge theory, Invent. Math. 162 (2005) 313 [math/0306198] [INSPIRE].
H. Nakajima and K. Yoshioka, Instanton counting on blowup. II. K-theoretic partition function, math/0505553 [INSPIRE].
D. Bashkirov, A comment on the enhancement of global symmetries in superconformal SU(2) gauge theories in 5D, arXiv:1211.4886 [INSPIRE].
O. Bergman, D. Rodríguez-Gómez and G. Zafrir, 5-Brane Webs, Symmetry Enhancement, and Duality in 5d Supersymmetric Gauge Theory, JHEP 03 (2014) 112 [arXiv:1311.4199] [INSPIRE].
P. Benetti Genolini and L. Tizzano, Instantons, symmetries and anomalies in five dimensions, JHEP 04 (2021) 188 [arXiv:2009.07873] [INSPIRE].
F. Apruzzi, S. Schafer-Nameki, L. Bhardwaj and J. Oh, The Global Form of Flavor Symmetries and 2-Group Symmetries in 5d SCFTs, SciPost Phys. 13 (2022) 024 [arXiv:2105.08724] [INSPIRE].
L. Bhardwaj, Flavor symmetry of 5d SCFTs. Part II. Applications, JHEP 04 (2021) 221 [arXiv:2010.13235] [INSPIRE].
S.-S. Kim, X. Li, S. Nawata and F. Yagi, Freezing and BPS jumping, arXiv:2403.12525.
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].
M.R. Douglas, On D = 5 super Yang-Mills theory and (2, 0) theory, JHEP 02 (2011) 011 [arXiv:1012.2880] [INSPIRE].
N. Lambert, C. Papageorgakis and M. Schmidt-Sommerfeld, M5-Branes, D4-Branes and Quantum 5D super-Yang-Mills, JHEP 01 (2011) 083 [arXiv:1012.2882] [INSPIRE].
Y. Tachikawa, On S-duality of 5d super Yang-Mills on S1, JHEP 11 (2011) 123 [arXiv:1110.0531] [INSPIRE].
C. Cordova, T.T. Dumitrescu and K. Intriligator, Multiplets of Superconformal Symmetry in Diverse Dimensions, JHEP 03 (2019) 163 [arXiv:1612.00809] [INSPIRE].
Y. Tachikawa, Instanton operators and symmetry enhancement in 5d supersymmetric gauge theories, PTEP 2015 (2015) 043B06 [arXiv:1501.01031] [INSPIRE].
N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [INSPIRE].
Acknowledgments
We are grateful to Hirotaka Hayashi, Kimyeong Lee, Xiaobin Li, Satoshi Nawata, Jaewon Song, Yi-Nan Wang, and Futoshi Yagi for their insightful discussions and comments. SK thanks the hospitality of POSTECH where part of this work was done. HK and MK are supported by Samsung Science and Technology Foundation under Project Number SSTF-BA2002-05 and by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2023R1A2C1006542). SK is supported by the NSFC grant No. 12250610188.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2307.03231
Rights and permissions
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.
About this article
Cite this article
Kim, HC., Kim, M., Kim, SS. et al. Superconformal indices for non-Lagrangian theories in five dimensions. J. High Energ. Phys. 2024, 164 (2024). https://doi.org/10.1007/JHEP03(2024)164
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2024)164