Abstract
We explore the geometrical structure of Higgs branches of quantum field theories with 8 supercharges in 3, 4, 5 and 6 dimensions. They are symplectic singularities, and as such admit a decomposition (or foliation ) into so-called symplectic leaves, which are related to each other by transverse slices. We identify this foliation with the pattern of partial Higgs mechanism of the theory and, using brane systems and recently introduced notions of magnetic quivers and quiver subtraction, we formalise the rules to obtain the Hasse diagram which encodes the structure of the foliation. While the unbroken gauge symmetry and the number of flat directions are obtainable by classical field theory analysis for Lagrangian theories, our approach allows us to characterise the geometry of the Higgs branch by a Hasse diagram with symplectic leaves and transverse slices, thus refining the analysis and extending it to non-Lagrangian theories. Most of the Hasse diagrams we obtain extend beyond the cases of nilpotent orbit closures known in the mathematics literature. The geometric analysis developed in this paper is applied to Higgs branches of several Lagrangian gauge theories, Argyres-Douglas theories, five dimensional SQCD theories at the conformal fixed point, and six dimensional SCFTs.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
F. Englert and R. Brout, Broken symmetry and the mass of gauge vector mesons, Phys. Rev. Lett.13 (1964) 321 [INSPIRE].
P.W. Higgs, Broken symmetries and the masses of gauge bosons, Phys. Rev. Lett.13 (1964) 508 [INSPIRE].
G.S. Guralnik, C.R. Hagen and T.W.B. Kibble, Global conservation laws and massless particles, Phys. Rev. Lett.13 (1964) 585 [INSPIRE].
T.W.B. Kibble, Symmetry breaking in nonAbelian gauge theories, Phys. Rev.155 (1967) 1554 [INSPIRE].
N.J. Hitchin, A. Karlhede, U. Lindström and M. Roček, Hyper-Kähler metrics and supersymmetry, Commun. Math. Phys.108 (1987) 535 [INSPIRE].
A. Beauville, Symplectic singularities, Invent. Math.139 (2000) 541 [math/9903070].
E. Brieskorn, Singular elements of semi-simple algebraic groups, Act. Congr.Int. Math.2 (1970) 279.
P. Slodowy, Simple singularities and simple algebraic groups, Lecture Notes in Mathematics volume 815, Spinger, Germany (1980).
H. Kraft and C. Procesi, Minimal singularities in GLn, Inv. Math.62 (1980) 503.
H. Kraft and C. Procesi, On the geometry of conjugacy classes in classical groups, Comment. Mat. Helv.57 (1982) 539.
B. Fu, D. Juteau, P. Levy and E. Sommers, Generic singularities of nilpotent orbit closures, Adv. Math.305 (2017) 1 [arXiv:1502.05770].
S. Cabrera and A. Hanany, Branes and the Kraft-Procesi transition, JHEP11 (2016) 175 [arXiv:1609.07798] [INSPIRE].
S. Cabrera and A. Hanany, Branes and the Kraft-Procesi transition: classical case, JHEP04 (2018) 127 [arXiv:1711.02378] [INSPIRE].
S. Cabrera and A. Hanany, Quiver subtractions, JHEP09 (2018) 008 [arXiv:1803.11205] [INSPIRE].
M. Del Zotto and A. Hanany, Complete graphs, Hilbert series and the Higgs branch of the 4d \( \mathcal{N} \) = 2(An, Am) SCFTs, Nucl. Phys.B 894 (2015) 439 [arXiv:1403.6523] [INSPIRE].
S. Cremonesi, G. Ferlito, A. Hanany and N. Mekareeya, Instanton operators and the Higgs branch at infinite coupling, JHEP04 (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, JHEP07 (2018) 061 [arXiv:1712.06604] [INSPIRE].
S. Cabrera, A. Hanany and F. Yagi, Tropical geometry and five dimensional Higgs branches at infinite coupling, JHEP01 (2019) 068 [arXiv:1810.01379] [INSPIRE].
S. Cabrera, A. Hanany and M. Sperling, Magnetic quivers, Higgs branches and 6d N = (1, 0) theories, JHEP06 (2019) 071 [Erratum ibid.07 (2019) 137] [arXiv:1904.12293] [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].
A. Braverman, M. Finkelberg and H. Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional \( \mathcal{N} \) = 4 gauge theories, II, Adv. Theor. Math. Phys.22 (2018) 1071 [arXiv:1601.03586] [INSPIRE].
B. Assel and S. Cremonesi, The infrared physics of bad theories, SciPost Phys.3 (2017) 024 [arXiv:1707.03403] [INSPIRE].
B. Assel and S. Cremonesi, The infrared fixed points of 3d \( \mathcal{N} \) = 4 USp(2N) SQCD theories, SciPost Phys.5 (2018) 015 [arXiv:1802.04285] [INSPIRE].
J.J. Heckman, T. Rudelius and A. Tomasiello, 6D RG flows and nilpotent hierarchies, JHEP07 (2016) 082 [arXiv:1601.04078] [INSPIRE].
J.J. Heckman and T. Rudelius, Top down approach to 6D SCFTs, J. Phys.A 52 (2019) 093001 [arXiv:1805.06467] [INSPIRE].
F. Hassler et al., T-branes, string junctions and 6D SCFTs, arXiv:1907.11230 [INSPIRE].
J. Rogers and R. Tatar, Moduli space singularities for 3d \( \mathcal{N} \) = 4 circular quiver gauge theories, JHEP11 (2018) 022 [arXiv:1807.01754] [INSPIRE].
J. Rogers and R. Tatar, DnDynkin quiver moduli spaces, J. Phys.A 52 (2019) 425401 [arXiv:1902.10019] [INSPIRE].
N. Yamatsu, Finite-dimensional Lie algebras and their representations for unified model building, arXiv:1511.08771.
A. Hanany and N. Mekareeya, The small E8instanton and the Kraft Procesi transition, JHEP07 (2018) 098 [arXiv:1801.01129] [INSPIRE].
W. Hesselink, Singularities in the nilpotent scheme of a classical group, Trans. Amer. Math. Soc.222 (1976) 1.
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].
G. Ferlito and A. Hanany, A tale of two cones: the Higgs Branch of Sp(n) theories with 2n flavours, arXiv:1609.06724 [INSPIRE].
A. Bourget et al., Brane webs and magnetic quivers for SQCD, arXiv:1909.00667 [INSPIRE].
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].
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].
O.J. Ganor and A. Hanany, Small E8instantons and tensionless noncritical strings, Nucl. Phys.B 474 (1996) 122 [hep-th/9602120] [INSPIRE].
N. Seiberg and E. Witten, Comments on string dynamics in six-dimensions, Nucl. Phys.B 471 (1996) 121 [hep-th/9603003] [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].
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].
A. Hanany and A. Zaffaroni, Branes and six-dimensional supersymmetric theories, Nucl. Phys.B 529 (1998) 180 [hep-th/9712145] [INSPIRE].
N. Mekareeya, T. Rudelius and A. Tomasiello, T-branes, anomalies and moduli spaces in 6D SCFTs, JHEP10 (2017) 158 [arXiv:1612.06399] [INSPIRE].
N. Mekareeya, K. Ohmori, H. Shimizu and A. Tomasiello, Small instanton transitions for M5 fractions, JHEP10 (2017) 055 [arXiv:1707.05785] [INSPIRE].
M. Del Zotto and G. Lockhart, Universal features of BPS strings in six-dimensional SCFTs, JHEP08 (2018) 173 [arXiv:1804.09694] [INSPIRE].
G. Zafrir, Brane webs, 5d gauge theories and 6d \( \mathcal{N} \) = (1, 0) SCFT’s, JHEP12 (2015) 157 [arXiv:1509.02016] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, 6d SCFTs, 5d dualities and Tao web diagrams, JHEP05 (2019) 203 [arXiv:1509.03300] [INSPIRE].
N. Mekareeya, K. Ohmori, Y. Tachikawa and G. Zafrir, E8instantons on type-A ALE spaces and supersymmetric field theories, JHEP09 (2017) 144 [arXiv:1707.04370] [INSPIRE].
A. Hanany and G. Zafrir, Discrete gauging in six dimensions, JHEP07 (2018) 168 [arXiv:1804.08857] [INSPIRE].
M. Bershadsky et al., Geometric singularities and enhanced gauge symmetries, Nucl. Phys.B 481 (1996) 215 [hep-th/9605200] [INSPIRE].
J. Distler and A. Karch, N = 1 dualities for exceptional gauge groups and quantum global symmetries, Fortsch. Phys.45 (1997) 517 [hep-th/9611088] [INSPIRE].
S. Cecotti, A. Neitzke and C. Vafa, R-Twisting and 4d/2d correspondences, arXiv:1006.3435 [INSPIRE].
D. Xie, General Argyres-Douglas theory, JHEP01 (2013) 100 [arXiv:1204.2270] [INSPIRE].
P. Boalch, Irregular connections and Kac-Moody root systems, arXiv:0806.1050.
P. Boalch, Simply-laced isomonodromy systems, Publ. Math. IHES116 (2012) 1.
A. Hanany and A. Zajac, Ungauging schemes and Coulomb branches of non-simply laced quiver gauge theories, to appear.
O. Aharony and A. Hanany, Branes, superpotentials and superconformal fixed points, Nucl. Phys.B 504 (1997) 239 [hep-th/9704170] [INSPIRE].
O. Aharony, A. Hanany and B. Kol, Webs of (p, q) five-branes, five-dimensional field theories and grid diagrams, JHEP01 (1998) 002 [hep-th/9710116] [INSPIRE].
I. Brunner and A. Karch, Branes and six-dimensional fixed points, Phys. Lett.B 409 (1997) 109 [hep-th/9705022] [INSPIRE].
A. Hanany and A. Zaffaroni, Chiral symmetry from type IIA branes, Nucl. Phys.B 509 (1998) 145 [hep-th/9706047] [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].
P. Vanhaecke, Integrable systems in the realm of algebraic geometry, Springer, Germany (2001).
I. Vaisman, Lectures on the geometry of Poisson manifolds, Birkhäuser, Switzerland (2012).
R.L. Fernandes and I. Marcut, Lectures on Poisson geometry, (2014).
Y. Namikawa, Extension of 2-forms and symplectic varieties, math/0010114.
B. Fu, A survey on symplectic singularities and symplectic resolutions, math/0510346.
G. Bellamy, Symplectic singularities and their quantization, https://www.maths.gla.ac.uk/∼gbellamy/Padova.pdf
D. Kaledin, Symplectic singularities from the Poisson point of view, J. Reine Angew. Math.2006 (2006) 135 [math/0310186].
H. Nakajima, Questions on provisional Coulomb branches of 3-dimensional \( \mathcal{N} \) = 4 gauge theories, arXiv:1510.03908 [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: 1908.04245
Imperial/TP/19/AH/02 (Amihay Hanany)
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Bourget, A., Cabrera, S., Grimminger, J.F. et al. The Higgs mechanism — Hasse diagrams for symplectic singularities. J. High Energ. Phys. 2020, 157 (2020). https://doi.org/10.1007/JHEP01(2020)157
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2020)157