Abstract
When n M5 branes coincide on an A type singularity, ℂ2/ℤk, there is a multitude of tensionless strings which arise in the spectrum. The low energy theory when all M5 branes are separated at the singularity is given by a linear quiver with parameters n and k. The theory has a multitude of phases, as many as partitions of n, each characterized by a different Higgs branch. Each such Higgs branch can be described by a Coulomb branch of a 3d \( \mathcal{N}=4 \) quiver. For example, at finite coupling, when all branes are separated, the quiver has a bouquet of n U(1) nodes connected to a single node. There is a natural discrete non Abelian Sn global symmetry which acts in the theory by permuting n identical objects. It acts in particular on the Higgs branch at the above finite coupling phase. It is conjectured that at the coincident point this discrete Sn flavor symmetry is gauged, and at partial coincidence the corresponding subgroup of Sn is gauged. This elegant and simple effect solves several problems which are raised recently on the physics of multiple M5 branes on an A type singularity. Similar results on multitude of phases are concluded for a system of n M5 branes on an A type singularity next to an M9 plane.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6D conformal matter, JHEP 02 (2015) 054 [arXiv:1407.6359] [INSPIRE].
B. Haghighat, C. Kozcaz, G. Lockhart and C. Vafa, Orbifolds of M-strings, Phys. Rev. D 89 (2014) 046003 [arXiv:1310.1185] [INSPIRE].
S. Cremonesi, G. Ferlito, A. Hanany and N. Mekareeya, Instanton operators and the Higgs branch at infinite coupling, JHEP 04 (2017) 042 [arXiv:1505.06302] [INSPIRE].
G. Ferlito, A. Hanany, N. Mekareeya and G. Zafrir, 3d Coulomb branch and 5d Higgs branch at infinite coupling, JHEP 07 (2018) 061 [arXiv:1712.06604] [INSPIRE].
A. Hanany and N. Mekareeya, The Small E 8 Instanton and the Kraft Procesi Transition, arXiv:1801.01129 [INSPIRE].
M. Del Zotto and A. Hanany, Complete graphs, Hilbert series and the Higgs branch of the 4d \( \mathcal{N}=2\left({A}_n,{A}_m\right) \) SCFTs, Nucl. Phys. B 894 (2015) 439 [arXiv:1403.6523] [INSPIRE].
O.J. Ganor and A. Hanany, Small E 8 instantons and tensionless noncritical strings, Nucl. Phys. B 474 (1996) 122 [hep-th/9602120] [INSPIRE].
S. Cabrera and A. Hanany, Branes and the Kraft-Procesi transition, JHEP 11 (2016) 175 [arXiv:1609.07798] [INSPIRE].
S. Cabrera and A. Hanany, Branes and the Kraft-Procesi transition: classical case, JHEP 04 (2018) 127 [arXiv:1711.02378] [INSPIRE].
S. Cabrera and A. Hanany, Quiver subtractions, arXiv:1803.11205 [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].
I. Bah et al., 4d \( \mathcal{N}=1 \) from 6d \( \mathcal{N}=\left(1,0\right) \) on a torus with fluxes, JHEP 06 (2017) 022 [arXiv:1702.04740] [INSPIRE].
K.A. Intriligator, RG fixed points in six-dimensions via branes at orbifold singularities, Nucl. Phys. B 496 (1997) 177 [hep-th/9702038] [INSPIRE].
I. Brunner and A. Karch, Branes and six-dimensional fixed points, Phys. Lett. B 409 (1997) 109 [hep-th/9705022] [INSPIRE].
J.D. Blum and K.A. Intriligator, New phases of string theory and 6D RG fixed points via branes at orbifold singularities, Nucl. Phys. B 506 (1997) 199 [hep-th/9705044] [INSPIRE].
K.A. Intriligator, New string theories in six-dimensions via branes at orbifold singularities, Adv. Theor. Math. Phys. 1 (1998) 271 [hep-th/9708117] [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].
A. Hanany and R. Kalveks, Highest weight generating functions for Hilbert series, JHEP 10 (2014) 152 [arXiv:1408.4690] [INSPIRE].
S. Benvenuti, A. Hanany and N. Mekareeya, The Hilbert series of the one instanton moduli space, JHEP 06 (2010) 100 [arXiv:1005.3026] [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].
A. Dancer, F. Kirwan and A. Swann, Implosion for hyper-Kähler manifolds, arXiv:1209.1578 [INSPIRE].
A. Kapustin and M.J. Strassler, On mirror symmetry in three-dimensional Abelian gauge theories, JHEP 04 (1999) 021 [hep-th/9902033].
E. Witten, SL(2, Z) action on three-dimensional conformal field theories with Abelian symmetry, hep-th/0307041.
N. Mekareeya, K. Ohmori, Y. Tachikawa and G. Zafrir, E 8 instantons on type-A ALE spaces and supersymmetric field theories, JHEP 09 (2017) 144 [arXiv:1707.04370] [INSPIRE].
A. Hanany and A. Zajac, Discrete gauging in coulomb branches of three dimensional \( \mathcal{N}=4 \) supersymmetric gauge theories, in preparation.
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].
R. Brylinski and B. Kostant, Nilpotent orbits, normality, and Hamiltonian group actions, math/9204227.
A. Hanany and R. Kalveks, Quiver theories for moduli spaces of classical group nilpotent orbits, JHEP 06 (2016) 130 [arXiv:1601.04020] [INSPIRE].
S. Benvenuti, B. Feng, A. Hanany and Y.-H. He, Counting BPS operators in gauge theories: quivers, syzygies and plethystics, JHEP 11 (2007) 050 [hep-th/0608050] [INSPIRE].
A. Hanany and N. Mekareeya, Tri-vertices and SU(2)’s, JHEP 02 (2011) 069 [arXiv:1012.2119] [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].
A. Hanany and R. Kalveks, Quiver theories and formulae for nilpotent orbits of exceptional algebras, JHEP 11 (2017) 126 [arXiv:1709.05818] [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].
A. Hanany and A. Zaffaroni, Issues on orientifolds: on the brane construction of gauge theories with SO(2N) global symmetry, JHEP 07 (1999) 009 [hep-th/9903242] [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].
H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, 6D SCFTs, 5D dualities and Tao web diagrams, arXiv:1509.03300 [INSPIRE].
S. Cabrera, A. Hanany and A. Zajac, Minimally unbalanced quivers, in preparation.
U.H. Danielsson, G. Ferretti, J. Kalkkinen and P. Stjernberg, Notes on supersymmetric gauge theories in five-dimensions and six-dimensions, Phys. Lett. B 405 (1997) 265 [hep-th/9703098] [INSPIRE].
W.H. Hesselink, Polarization in the classical groups, Math. Z. 160 (1978) 217.
F. Benini, Y. Tachikawa and D. Xie, Mirrors of 3d Sicilian theories, JHEP 09 (2010) 063 [arXiv:1007.0992] [INSPIRE].
A. Kapustin, B. Willett and I. Yaakov, Tests of Seiberg-like duality in three dimensions, arXiv:1012.4021 [INSPIRE].
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [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].
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].
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].
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].
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: 1804.08857
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
Hanany, A., Zafrir, G. Discrete gauging in six dimensions. J. High Energ. Phys. 2018, 168 (2018). https://doi.org/10.1007/JHEP07(2018)168
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2018)168