Abstract
We consider mixed branches of 3d \( \mathcal{N} \) = 4 T [SU(N )] theory. We compute the Hilbert series of the Coulomb branch part of the mixed branch from a restriction rule acting on the Hilbert series of the full Coulomb branch that will truncate the magnetic charge summation only to the subset of BPS dressed monopole operators that arise in the Coulomb branch sublocus where the mixed branch stems. This restriction can be understood directly from the type IIB brane picture by a relation between the magnetic charges of the monopoles and brane position moduli. We also apply the restriction rule to the Higgs branch part of a given mixed branch by exploiting 3d mirror symmetry. Both ccases show complete agreement with the results calculated by different methods.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
K.A. Intriligator and N. Seiberg, Lectures on supersymmetric gauge theories and electric-magnetic duality, Nucl. Phys. Proc. Suppl. 45BC (1996) 1 [hep-th/9509066] [INSPIRE].
M.J. Strassler, An unorthodox introduction to supersymmetric gauge theory, in the proceedings of Strings, Branes and Extra Dimensions (TASI 2001), June 4-29, Boulder, U.S.A. (2003), hep-th/0309149 [INSPIRE].
K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].
J. de Boer, K. Hori, H. Ooguri and Y. Oz, Mirror symmetry in three-dimensional gauge theories, quivers and D-branes, Nucl. Phys. B 493 (1997) 101 [hep-th/9611063] [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].
J. de Boer, K. Hori, H. Ooguri, Y. Oz and Z. Yin, Mirror symmetry in three-dimensional theories, \( \mathrm{S}\mathrm{L}\left(2,\mathbb{Z}\right) \) and D-brane moduli spaces, Nucl. Phys. B 493 (1997) 148 [hep-th/9612131] [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].
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].
M. Bullimore, T. Dimofte and D. Gaiotto, The Coulomb branch of 3D \( \mathcal{N} \) = 4 theories, arXiv:1503.04817 [INSPIRE].
A. Braverman, M. Finkelberg and H. Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional \( \mathcal{N} \) = 4 gauge theories, II, arXiv:1601.03586 [INSPIRE].
M. Bullimore, T. Dimofte, D. Gaiotto, J. Hilburn and H.-C. Kim, Vortices and vermas, arXiv:1609.04406 [INSPIRE].
G. ’t Hooft, On the phase transition towards permanent quark confinement, Nucl. Phys. B 138 (1978) 1 [INSPIRE].
V. Borokhov, A. Kapustin and X.-k. Wu, Topological disorder operators in three-dimensional conformal field theory, JHEP 11 (2002) 049 [hep-th/0206054] [INSPIRE].
V. Borokhov, A. Kapustin and X.-k. Wu, Monopole operators and mirror symmetry in three-dimensions, JHEP 12 (2002) 044 [hep-th/0207074] [INSPIRE].
V. Borokhov, Monopole operators in three-dimensional N = 4 SYM and mirror symmetry, JHEP 03 (2004) 008 [hep-th/0310254] [INSPIRE].
V.A. Borokhov, Monopole operators and mirror symmetry in three-dimensional gauge theories, Ph.D. thesis, Caltech, U.S.A. (2004).
D. Bashkirov, Examples of global symmetry enhancement by monopole operators, arXiv:1009.3477 [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.K. Benna, I.R. Klebanov and T. Klose, Charges of monopole operators in Chern-Simons Yang-Mills theory, JHEP 01 (2010) 110 [arXiv:0906.3008] [INSPIRE].
D. Bashkirov and A. Kapustin, Supersymmetry enhancement by monopole operators, JHEP 05 (2011) 015 [arXiv:1007.4861] [INSPIRE].
A. Dey, A. Hanany, P. Koroteev and N. Mekareeya, Mirror symmetry in three dimensions via gauged linear quivers, JHEP 06 (2014) 059 [arXiv:1402.0016] [INSPIRE].
S. Cremonesi, A. Hanany, N. Mekareeya and A. Zaffaroni, Coulomb branch Hilbert series and Hall-Littlewood polynomials, JHEP 09 (2014) 178 [arXiv:1403.0585] [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].
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].
S. Cremonesi, A. Hanany, N. Mekareeya and A. Zaffaroni, T σ ρ (G) theories and their Hilbert series, JHEP 01 (2015) 150 [arXiv:1410.1548] [INSPIRE].
N. Mekareeya, The moduli space of instantons on an ALE space from 3D \( \mathcal{N} \) = 4 field theories, JHEP 12 (2015) 174 [arXiv:1508.06813] [INSPIRE].
A. Hanany and R. Kalveks, Construction and deconstruction of single instanton Hilbert series, JHEP 12 (2015) 118 [arXiv:1509.01294] [INSPIRE].
A. Hanany and M. Sperling, Coulomb branches for rank 2 gauge groups in 3d \( \mathcal{N} \) = 4 gauge theories, JHEP 08 (2016) 016 [arXiv:1605.00010] [INSPIRE].
A. Hanany, C. Hwang, H. Kim, J. Park and R.-K. Seong, Hilbert series for theories with aharony duals, JHEP 11 (2015) 132 [arXiv:1505.02160] [INSPIRE].
S. Cremonesi, The Hilbert series of 3D \( \mathcal{N} \) = 2 Yang-Mills theories with vectorlike matter, J. Phys. A 48 (2015) 455401 [arXiv:1505.02409] [INSPIRE].
S. Cremonesi, N. Mekareeya and A. Zaffaroni, The moduli spaces of 3D \( \mathcal{N} \) ≥ 2 Chern-Simons gauge theories and their Hilbert series, JHEP 10 (2016) 046 [arXiv:1607.05728] [INSPIRE].
I. Affleck, J.A. Harvey and E. Witten, Instantons and (super)symmetry breaking in (2 + 1)-dimensions, Nucl. Phys. B 206 (1982) 413 [INSPIRE].
J. de Boer, K. Hori and Y. Oz, Dynamics of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 500 (1997) 163 [hep-th/9703100] [INSPIRE].
O. Aharony, A. Hanany, K.A. Intriligator, N. Seiberg and M.J. Strassler, Aspects of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 499 (1997) 67 [hep-th/9703110] [INSPIRE].
O. Chacaltana, J. Distler and Y. Tachikawa, Nilpotent orbits and codimension-two defects of 6d N = (2, 0) theories, Int. J. Mod. Phys. A 28 (2013) 1340006 [arXiv:1203.2930] [INSPIRE].
K. Yonekura, Supersymmetric gauge theory, (2, 0) theory and twisted 5D super-Yang-Mills, JHEP 01 (2014) 142 [arXiv:1310.7943] [INSPIRE].
D. Xie and K. Yonekura, The moduli space of vacua of \( \mathcal{N} \) = 2 class \( \mathcal{S} \) theories, JHEP 10 (2014) 134 [arXiv:1404.7521] [INSPIRE].
A. Hanany and N. Mekareeya, Complete intersection moduli spaces in N = 4 gauge theories in three dimensions, JHEP 01 (2012) 079 [arXiv:1110.6203] [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].
J. Gray, A. Hanany, Y.-H. He, V. Jejjala and N. Mekareeya, SQCD: a geometric Apercu, JHEP 05 (2008) 099 [arXiv:0803.4257] [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, N. Mekareeya and S.S. Razamat, Hilbert series for moduli spaces of two instantons, JHEP 01 (2013) 070 [arXiv:1205.4741] [INSPIRE].
A. Dey, A. Hanany, N. Mekareeya, D. Rodríguez-Gómez and R.-K. Seong, Hilbert series for moduli spaces of instantons on C 2 /Z n , JHEP 01 (2014) 182 [arXiv:1309.0812] [INSPIRE].
P. Goddard, J. Nuyts and D.I. Olive, Gauge theories and magnetic charge, Nucl. Phys. B 125 (1977) 1 [INSPIRE].
F. Englert and P. Windey, Quantization condition for ’t Hooft monopoles in compact simple Lie groups, Phys. Rev. D 14 (1976) 2728 [INSPIRE].
A. Kapustin, Wilson-’t Hooft operators in four-dimensional gauge theories and S-duality, Phys. Rev. D 74 (2006) 025005 [hep-th/0501015] [INSPIRE].
A.M. Polyakov, Quark confinement and topology of gauge groups, Nucl. Phys. B 120 (1977) 429 [INSPIRE].
W. Fulton and J. Harris, Representation theory: a first course, 5th printing, Graduate texts in mathematics, Springer-Verlag, Germany (1999).
H. Derksen and G. Kemper, Computational invariant theory, Encyclopaedia of mathematical sciences, Springer, Berlin, Germany (2002).
A. Hanany and C. Romelsberger, Counting BPS operators in the chiral ring of N = 2 supersymmetric gauge theories or N = 2 braine surgery, Adv. Theor. Math. Phys. 11 (2007) 1091 [hep-th/0611346] [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: 1609.08034
La Caixa-Severo Ochoa Scholar (Federico Carta).
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
Carta, F., Hayashi, H. Hilbert Series and Mixed Branches of T [SU(N )] theories. J. High Energ. Phys. 2017, 37 (2017). https://doi.org/10.1007/JHEP02(2017)037
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2017)037