Abstract
We consider the 6d superconformal field theory realized on M5-branes probing the E 8 end-of-the-world brane on the deformed and resolved ℂ 2/ℤ k singularity. We give an explicit algorithm which determines, for arbitrary holonomy at infinity, the 6d quiver gauge theory on the tensor branch, the type-A class S description of the T 2 compactification, and the star-shaped quiver obtained as the mirror of the T 3 compactification.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld and Yu. I. Manin, Construction of Instantons, Phys. Lett. A 65 (1978) 185 [INSPIRE].
P.B. Kronheimer and H. Nakajima, Yang-Mills instantons on ALE gravitational instantons, Math. Ann. 288 (1990) 263.
M. Bianchi, F. Fucito, G. Rossi and M. Martellini, Explicit construction of Yang-Mills instantons on ALE spaces, Nucl. Phys. B 473 (1996) 367 [hep-th/9601162] [INSPIRE].
E. Witten, Small instantons in string theory, Nucl. Phys. B 460 (1996) 541 [hep-th/9511030] [INSPIRE].
M.R. Douglas and G.W. Moore, D-branes, quivers and ALE instantons, hep-th/9603167 [INSPIRE].
P.S. Aspinwall and D.R. Morrison, Point-like instantons on K3 orbifolds, Nucl. Phys. B 503 (1997) 533 [hep-th/9705104] [INSPIRE].
D. Gaiotto, \( \mathcal{N}=2 \) dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
S. Cremonesi, A. Hanany and A. Zaffaroni, Monopole operators and Hilbert series of Coulomb branches of 3d \( \mathcal{N}=4 \) gauge theories, JHEP 01 (2014) 005 [arXiv:1309.2657] [INSPIRE].
J.J. Heckman, D.R. Morrison and C. Vafa, On the Classification of 6D SCFTs and Generalized ADE Orbifolds, JHEP 05 (2014) 028 [Erratum ibid. 1506 (2015) 017] [arXiv:1312.5746] [INSPIRE].
M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6d Conformal Matter, JHEP 02 (2015) 054 [arXiv:1407.6359] [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].
G. Zafrir, Brane webs, 5d gauge theories and 6d \( \mathcal{N}=\left(1,0\right) \) SCFT’s, JHEP 12 (2015) 157 [arXiv:1509.02016] [INSPIRE].
K. Ohmori and H. Shimizu, S 1 /T 2 compactifications of 6d \( \mathcal{N}=\left(1,0\right) \) theories and brane webs, JHEP 03 (2016) 024 [arXiv:1509.03195] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, 6d SCFTs, 5d Dualities and Tao Web Diagrams, arXiv:1509.03300 [INSPIRE].
K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, 6d \( \mathcal{N}=\left(1,0\right) \) theories on T 2 and class S theories: Part I, JHEP 07 (2015) 014 [arXiv:1503.06217] [INSPIRE].
P.B. Kronheimer, Instantons and the geometry of the nilpotent variety, J. Diff. Geom. 32 (1990) 473 [INSPIRE].
Y. Tachikawa, Moduli spaces of SO(8) instantons on smooth ALE spaces as Higgs branches of 4d N = 2 supersymmetric theories, JHEP 06 (2014) 056 [arXiv:1402.4200] [INSPIRE].
F. Benini, Y. Tachikawa and D. Xie, Mirrors of 3d Sicilian theories, JHEP 09 (2010) 063 [arXiv:1007.0992] [INSPIRE].
H. Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional \( \mathcal{N}=4 \) gauge theories, I, Adv. Theor. Math. Phys. 20 (2016) 595 [arXiv:1503.03676] [INSPIRE].
H. Nakajima, Questions on provisional Coulomb branches of 3-dimensional \( \mathcal{N}=4 \) gauge theories, arXiv:1510.03908 [INSPIRE].
S. Cremonesi, G. Ferlito, A. Hanany and N. Mekareeya, Coulomb Branch and The Moduli Space of Instantons, JHEP 12 (2014) 103 [arXiv:1408.6835] [INSPIRE].
N. Mekareeya, The moduli space of instantons on an ALE space from 3d N = 4 field theories, JHEP 12 (2015) 174 [arXiv:1508.06813] [INSPIRE].
V.G. Kac, Infinite Dimensional Lie Algebras, Cambridge University Press, Cambridge U.K. (1994).
H. Nakajima, Moduli spaces of anti-self-dual connections on ALE gravitational instantons, Invent. Math. 102 (1990) 267.
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].
K. Ohmori, H. Shimizu and Y. Tachikawa, Anomaly polynomial of E-string theories, JHEP 08 (2014) 002 [arXiv:1404.3887] [INSPIRE].
I. Brunner and A. Karch, Branes at orbifolds versus Hanany Witten in six-dimensions, JHEP 03 (1998) 003 [hep-th/9712143] [INSPIRE].
A. Hanany and A. Zaffaroni, Branes and six-dimensional supersymmetric theories, Nucl. Phys. B 529 (1998) 180 [hep-th/9712145] [INSPIRE].
O. Bergman and D. Rodriguez-Gomez, 5d quivers and their AdS 6 duals, JHEP 07 (2012) 171 [arXiv:1206.3503] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee, M. Taki and F. Yagi, A new 5d description of 6d D-type minimal conformal matter, JHEP 08 (2015) 097 [arXiv:1505.04439] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee, M. Taki and F. Yagi, More on 5d descriptions of 6d SCFTs, JHEP 10 (2016) 126 [arXiv:1512.08239] [INSPIRE].
O. Bergman and G. Zafrir, 5d fixed points from brane webs and O7-planes, JHEP 12 (2015) 163 [arXiv:1507.03860] [INSPIRE].
F. Benini, S. Benvenuti and Y. Tachikawa, Webs of five-branes and N = 2 superconformal field theories, JHEP 09 (2009) 052 [arXiv:0906.0359] [INSPIRE].
O. Chacaltana and J. Distler, Tinkertoys for Gaiotto Duality, JHEP 11 (2010) 099 [arXiv:1008.5203] [INSPIRE].
Y. Tachikawa, A review of the T N theory and its cousins, PTEP 2015 (2015) 11B102 [arXiv:1504.01481] [INSPIRE].
D. Gaiotto and E. Witten, S-duality of Boundary Conditions In N = 4 Super Yang-Mills Theory, Adv. Theor. Math. Phys. 13 (2009) 721 [arXiv:0807.3720] [INSPIRE].
M. Bullimore, T. Dimofte and D. Gaiotto, The Coulomb Branch of 3d \( \mathcal{N}=4 \) Theories, Commun. Math. Phys. 354 (2017) 671 [arXiv:1503.04817] [INSPIRE].
K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, 6d \( \mathcal{N}=\left(1,0\right) \) theories on S 1 /T 2 and class S theories: part II, JHEP 12 (2015) 131 [arXiv:1508.00915] [INSPIRE].
D. Gaiotto and S.S. Razamat, Exceptional Indices, JHEP 05 (2012) 145 [arXiv:1203.5517] [INSPIRE].
S. Cremonesi, A. Hanany, N. Mekareeya and A. Zaffaroni, Coulomb branch Hilbert series and Three Dimensional Sicilian Theories, JHEP 09 (2014) 185 [arXiv:1403.2384] [INSPIRE].
I. Bah, A. Passias and A. Tomasiello, AdS5 compactifications with punctures in massive IIA supergravity, arXiv:1704.07389 [INSPIRE].
Y. Tachikawa, \( \mathcal{N}=2 \) Supersymmetric Dynamics for Pedestrians, Lect. Notes Phys. 890 (2013) 2014 [arXiv:1312.2684] [INSPIRE].
J.J. Heckman, T. Rudelius and A. Tomasiello, 6D RG Flows and Nilpotent Hierarchies, JHEP 07 (2016) 082 [arXiv:1601.04078] [INSPIRE].
N. Mekareeya, T. Rudelius and A. Tomasiello, T-branes, Anomalies and Moduli Spaces in 6D SCFTs, arXiv:1612.06399 [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: 1707.04370
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
Mekareeya, N., Ohmori, K., Tachikawa, Y. et al. E 8 instantons on type-A ALE spaces and supersymmetric field theories. J. High Energ. Phys. 2017, 144 (2017). https://doi.org/10.1007/JHEP09(2017)144
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2017)144