Abstract
We present a formula for the Hilbert series that counts gauge invariant chiral operators in a large class of 3d \( \mathcal{N}\ge 2 \) Yang-Mills-Chern-Simons theories. The formula counts ’t Hooft monopole operators dressed by gauge invariants of a residual gauge theory of massless fields in the monopole background. We provide a general formula for the case of abelian theories, where nonperturbative corrections are absent, and consider a few examples of nonabelian theories where nonperturbative corrections are well understood. We also analyze in detail nonabelian ABJ(M) theories as well as worldvolume theories of M2-branes probing Calabi-Yau fourfold and hyperKähler twofold singularities with \( \mathcal{N}\ge 2 \) and \( \mathcal{N}\ge 3 \) supersymmetry.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
D. Tong, Dynamics of N = 2 supersymmetric Chern-Simons theories, JHEP 07 (2000) 019 [hep-th/0005186] [INSPIRE].
K. Intriligator and N. Seiberg, Aspects of 3d N = 2 Chern-Simons-matter theories, JHEP 07 (2013) 079 [arXiv:1305.1633] [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].
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].
D. Bashkirov and A. Kapustin, Supersymmetry enhancement by monopole operators, JHEP 05 (2011) 015 [arXiv:1007.4861] [INSPIRE].
D. Bashkirov, Examples of global symmetry enhancement by monopole operators, arXiv:1009.3477 [INSPIRE].
S. Cremonesi, A. Hanany and A. Zaffaroni, Monopole operators and Hilbert series of Coulomb branches of 3d N = 4 gauge theories, JHEP 01 (2014) 005 [arXiv:1309.2657] [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].
S. Cremonesi, The Hilbert series of 3d N = 2 Yang-Mills theories with vectorlike matter, J. Phys. A 48 (2015) 455401 [arXiv:1505.02409] [INSPIRE].
A. Hanany, C. Hwang, H. Kim, J. Park and R.-K. Seong, Hilbert series for theories with Aharony duals, JHEP 11 (2015) 132 [Addendum ibid. 04 (2016) 064] [arXiv:1505.02160] [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].
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.S. Razamat and B. Willett, Down the rabbit hole with theories of class S, JHEP 10 (2014) 099 [arXiv:1403.6107] [INSPIRE].
N. Dorey and D. Tong, Mirror symmetry and toric geometry in three-dimensional gauge theories, JHEP 05 (2000) 018 [hep-th/9911094] [INSPIRE].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
P.B. Kronheimer and H. Nakajima, Yang-Mills instantons on ALE gravitational instantons, Math. Ann. 288 (1990) 263.
B. Assel, Hanany-Witten effect and SL(2, Z) dualities in matrix models, JHEP 10 (2014) 117 [arXiv:1406.5194] [INSPIRE].
F. Benini, C. Closset and S. Cremonesi, Chiral flavors and M2-branes at toric CY 4 singularities, JHEP 02 (2010) 036 [arXiv:0911.4127] [INSPIRE].
S. Cremonesi, Type IIB construction of flavoured ABJ(M) and fractional M2 branes, JHEP 01 (2011) 076 [arXiv:1007.4562] [INSPIRE].
D.L. Jafferis, Quantum corrections to N = 2 Chern-Simons theories with flavor and their AdS 4 duals, JHEP 08 (2013) 046 [arXiv:0911.4324] [INSPIRE].
F. Benini, C. Closset and S. Cremonesi, Quantum moduli space of Chern-Simons quivers, wrapped D6-branes and AdS 4 /CFT 3, JHEP 09 (2011) 005 [arXiv:1105.2299] [INSPIRE].
C. Closset and S. Cremonesi, Toric Fano varieties and Chern-Simons quivers, JHEP 05 (2012) 060 [arXiv:1201.2431] [INSPIRE].
D. Martelli and J. Sparks, AdS 4 /CFT 3 duals from M2-branes at hypersurface singularities and their deformations, JHEP 12 (2009) 017 [arXiv:0909.2036] [INSPIRE].
D.L. Jafferis, Quantum corrections to N = 2 Chern-Simons theories with flavor and their AdS 4 duals, JHEP 08 (2013) 046 [arXiv:0911.4324] [INSPIRE].
Y. Imamura and S. Yokoyama, Index for three dimensional superconformal field theories with general R-charge assignments, JHEP 04 (2011) 007 [arXiv:1101.0557] [INSPIRE].
D.R. Grayson and M.E. Stillman, Macaulay2, a software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2/.
N. Seiberg and W. Taylor, Charge lattices and consistency of 6D supergravity, JHEP 06 (2011) 001 [arXiv:1103.0019] [INSPIRE].
D. Forcella, A. Hanany and A. Zaffaroni, Baryonic generating functions, JHEP 12 (2007) 022 [hep-th/0701236] [INSPIRE].
A. Kapustin and M.J. Strassler, On mirror symmetry in three-dimensional Abelian gauge theories, JHEP 04 (1999) 021 [hep-th/9902033] [INSPIRE].
E. Witten, SL(2, Z) action on three-dimensional conformal field theories with Abelian symmetry, hep-th/0307041 [INSPIRE].
K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].
O. Aharony and A. Hanany, Branes, superpotentials and superconformal fixed points, Nucl. Phys. B 504 (1997) 239 [hep-th/9704170] [INSPIRE].
M. Aganagic, K. Hori, A. Karch and D. Tong, Mirror symmetry in (2 + 1)-dimensions and (1 + 1)-dimensions, JHEP 07 (2001) 022 [hep-th/0105075] [INSPIRE].
P. Goddard, J. Nuyts and D.I. Olive, Gauge theories and magnetic charge, Nucl. Phys. B 125 (1977) 1 [INSPIRE].
D. Jafferis and X. Yin, A duality appetizer, arXiv:1103.5700 [INSPIRE].
O. Bergman, A. Hanany, A. Karch and B. Kol, Branes and supersymmetry breaking in three-dimensional gauge theories, JHEP 10 (1999) 036 [hep-th/9908075] [INSPIRE].
F. Benini, C. Closset and S. Cremonesi, Comments on 3d Seiberg-like dualities, JHEP 10 (2011) 075 [arXiv:1108.5373] [INSPIRE].
O. Aharony, O. Bergman and D.L. Jafferis, Fractional M2-branes, JHEP 11 (2008) 043 [arXiv:0807.4924] [INSPIRE].
D. Gaiotto and A. Tomasiello, The gauge dual of Romans mass, JHEP 01 (2010) 015 [arXiv:0901.0969] [INSPIRE].
I.R. Klebanov and E. Witten, Superconformal field theory on three-branes at a Calabi-Yau singularity, Nucl. Phys. B 536 (1998) 199 [hep-th/9807080] [INSPIRE].
D. Forcella, A. Hanany, Y.-H. He and A. Zaffaroni, The master space of N = 1 gauge theories, JHEP 08 (2008) 012 [arXiv:0801.1585] [INSPIRE].
A. Hanany and A. Zaffaroni, Tilings, Chern-Simons theories and M2 branes, JHEP 10 (2008) 111 [arXiv:0808.1244] [INSPIRE].
A. Hanany, D. Vegh and A. Zaffaroni, Brane tilings and M2 branes, JHEP 03 (2009) 012 [arXiv:0809.1440] [INSPIRE].
O. Aharony, A note on the holographic interpretation of string theory backgrounds with varying flux, JHEP 03 (2001) 012 [hep-th/0101013] [INSPIRE].
A. Dymarsky, I.R. Klebanov and N. Seiberg, On the moduli space of the cascading SU(M + p) × SU(p) gauge theory, JHEP 01 (2006) 155 [hep-th/0511254] [INSPIRE].
K. Furuuchi and K. Okuyama, D-branes wrapped on fuzzy del Pezzo surfaces, JHEP 01 (2011) 043 [arXiv:1008.5012] [INSPIRE].
D.L. Jafferis and A. Tomasiello, A simple class of N = 3 gauge/gravity duals, JHEP 10 (2008) 101 [arXiv:0808.0864] [INSPIRE].
K. Hosomichi, K.-M. Lee, S. Lee, S. Lee and J. Park, N = 4 superconformal Chern-Simons theories with hyper and twisted hyper multiplets, JHEP 07 (2008) 091 [arXiv:0805.3662] [INSPIRE].
Y. Imamura and K. Kimura, On the moduli space of elliptic Maxwell-Chern-Simons theories, Prog. Theor. Phys. 120 (2008) 509 [arXiv:0806.3727] [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].
M. Porrati and A. Zaffaroni, M theory origin of mirror symmetry in three-dimensional gauge theories, Nucl. Phys. B 490 (1997) 107 [hep-th/9611201] [INSPIRE].
A. Dey and J. Distler, Three dimensional mirror symmetry and partition function on S 3, JHEP 10 (2013) 086 [arXiv:1301.1731] [INSPIRE].
D. Martelli and J. Sparks, Moduli spaces of Chern-Simons quiver gauge theories and AdS 4 /CFT 3, Phys. Rev. D 78 (2008) 126005 [arXiv:0808.0912] [INSPIRE].
S. Hohenegger and I. Kirsch, A note on the holography of Chern-Simons matter theories with flavour, JHEP 04 (2009) 129 [arXiv:0903.1730] [INSPIRE].
D. Gaiotto and D.L. Jafferis, Notes on adding D6 branes wrapping Rp 3 in AdS 4 ×CP 3, JHEP 11 (2012) 015 [arXiv:0903.2175] [INSPIRE].
K.-M. Lee and H.-U. Yee, New AdS 4 × X 7 geometries with N = 6 in M-theory, JHEP 03 (2007) 012 [hep-th/0605214] [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].
D. Martelli, J. Sparks and S.-T. Yau, The geometric dual of a-maximisation for toric Sasaki-Einstein manifolds, Commun. Math. Phys. 268 (2006) 39 [hep-th/0503183] [INSPIRE].
J.P. Gauntlett, D. Martelli, J.F. Sparks and D. Waldram, A new infinite class of Sasaki-Einstein manifolds, Adv. Theor. Math. Phys. 8 (2004) 987 [hep-th/0403038] [INSPIRE].
J.P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, Supersymmetric AdS backgrounds in string and M-theory, IRMA Lect. Math. Theor. Phys. 8 (2005) 217 [hep-th/0411194] [INSPIRE].
D. Martelli and J. Sparks, Notes on toric Sasaki-Einstein seven-manifolds and AdS 4 /CFT 3, JHEP 11 (2008) 016 [arXiv:0808.0904] [INSPIRE].
A. Hanany and R. Kalveks, Highest weight generating functions for Hilbert series, JHEP 10 (2014) 152 [arXiv:1408.4690] [INSPIRE].
D. Fabbri et al., 3D superconformal theories from Sasakian seven manifolds: new nontrivial evidences for AdS 4 /CFT 3, Nucl. Phys. B 577 (2000) 547 [hep-th/9907219] [INSPIRE].
S.M. Chester, L.V. Iliesiu, S.S. Pufu and R. Yacoby, Bootstrapping O(N) vector models with four supercharges in 3 ≤ d ≤ 4, JHEP 05 (2016) 103 [arXiv:1511.07552] [INSPIRE].
H. Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories, I, arXiv:1503.03676 [INSPIRE].
D.R. Morrison and M.R. Plesser, Nonspherical horizons. 1, Adv. Theor. Math. Phys. 3 (1999) 1 [hep-th/9810201] [INSPIRE].
J. Gray, A. Hanany, Y.-H. He, V. Jejjala and N. Mekareeya, SQCD: a geometric aperçu, JHEP 05 (2008) 099 [arXiv:0803.4257] [INSPIRE].
A. Hanany and N. Mekareeya, Counting gauge invariant operators in SQCD with classical gauge groups, JHEP 10 (2008) 012 [arXiv:0805.3728] [INSPIRE].
E. Witten, Branes, instantons, and Taub-NUT spaces, JHEP 06 (2009) 067 [arXiv:0902.0948] [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.B. Kronheimer, Instantons and the geometry of the nilpotent variety, J. Diff. Geom. 32 (1990) 473 [INSPIRE].
R. Brylinski, Instantons and Kähler geometry of nilpotent orbits, NATO Adv. Sci. Inst. Ser. C 514 (1998) 85 [math/9811032] [INSPIRE].
P. Kobak and A. Swann, The hyper-Kähler geometry associated to Wolf spaces, Boll. Unione Mat. Ital. B 4 (2001) 587 [math/0001025].
D. Gaiotto, A. Neitzke and Y. Tachikawa, Argyres-Seiberg duality and the Higgs branch, Commun. Math. Phys. 294 (2010) 389 [arXiv:0810.4541] [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].
A. Butti, D. Forcella and A. Zaffaroni, Counting BPS baryonic operators in CFTs with Sasaki-Einstein duals, JHEP 06 (2007) 069 [hep-th/0611229] [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 rint: 1607.05728
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
Cremonesi, S., Mekareeya, N. & Zaffaroni, A. The moduli spaces of 3d \( \mathcal{N}\ge 2 \) Chern-Simons gauge theories and their Hilbert series. J. High Energ. Phys. 2016, 46 (2016). https://doi.org/10.1007/JHEP10(2016)046
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2016)046