Abstract
We use brane techniques to study the space of vacua of abelian 3d \( \mathcal{N}=3 \) gauge theories. The coordinates on these spaces are the vevs of chiral monopole and meson operators, which are realized in the type IIB brane configuration of the theory by adding semi-infinite (1, k) strings or F1 strings. The study of various brane setups allows us to determine a basis of chiral operators and chiral ring relations relevant to each branch of vacua, leading to the algebraic description of these branches. The method is mostly graphical and does not require actual computations. We apply it and provide explicit results in various examples. For linear quivers we find that the space of vacua has in general a collection of Coulomb-like branches, a Higgs branch and mixed branches. For circular quivers we find an extra branch, the geometric branch, parametrized by monopoles with equal magnetic charges in all U(1) nodes and meson operators. We explain how to include FI and mass deformations. We also study \( \mathcal{N}=3 \) theories realized with (p, q) 5-branes.
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References
N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, hep-th/9607163 [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, 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].
V. Borokhov, A. Kapustin and X.-k. Wu, Monopole operators and mirror symmetry in three-dimensions, JHEP 12 (2002) 044 [hep-th/0207074] [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].
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].
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].
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].
B. Assel, Ring relations and mirror map from branes, JHEP 03 (2017) 152 [arXiv:1701.08766] [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].
F. Benini, C. Closset and S. Cremonesi, Chiral flavors and M 2-branes at toric CY 4 singularities, JHEP 02 (2010) 036 [arXiv:0911.4127] [INSPIRE].
D.L. Jafferis, Quantum corrections to \( \mathcal{N} = 2 \) 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 AdS4/CFT3, JHEP 09 (2011) 005 [arXiv:1105.2299] [INSPIRE].
S. Cremonesi, N. Mekareeya and A. Zaffaroni, The moduli spaces of 3d \( \mathcal{N}\ge 2 \) Chern-Simons gauge theories and their Hilbert series, JHEP 10 (2016) 046 [arXiv:1607.05728] [INSPIRE].
J.P. Gauntlett, G.W. Gibbons, G. Papadopoulos and P.K. Townsend, Hyper-Kähler manifolds and multiply intersecting branes, Nucl. Phys. B 500 (1997) 133 [hep-th/9702202] [INSPIRE].
T. Kitao, K. Ohta and N. Ohta, Three-dimensional gauge dynamics from brane configurations with (p, q)-five-brane, Nucl. Phys. B 539 (1999) 79 [hep-th/9808111] [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].
D. Gaiotto and E. Witten, Supersymmetric boundary conditions in N = 4 super Yang-Mills theory, J. Statist. Phys. 135 (2009) 789 [arXiv:0804.2902] [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].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M 2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [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].
D. Gaiotto and X. Yin, Notes on superconformal Chern-Simons-matter theories, JHEP 08 (2007) 056 [arXiv:0704.3740] [INSPIRE].
A. Kapustin and M.J. Strassler, On mirror symmetry in three-dimensional Abelian gauge theories, JHEP 04 (1999) 021 [hep-th/9902033] [INSPIRE].
H.-C. Kao, K.-M. Lee and T. Lee, The Chern-Simons coefficient in supersymmetric Yang-Mills Chern-Simons theories, Phys. Lett. B 373 (1996) 94 [hep-th/9506170] [INSPIRE].
D.L. Jafferis and X. Yin, Chern-Simons-matter theory and mirror symmetry, arXiv:0810.1243 [INSPIRE].
J.P. Gauntlett, G.W. Gibbons, G. Papadopoulos and P.K. Townsend, Hyper-Kähler manifolds and multiply intersecting branes, Nucl. Phys. B 500 (1997) 133 [hep-th/9702202] [INSPIRE].
J.P. Gauntlett, Intersecting branes, in the proceedings of the APCTP Winter School: Dualities in gauge and string theories, February 17-28, Sokcho, Korea (1997), hep-th/9705011 [INSPIRE].
B. Assel, Hanany-Witten effect and SL(2, ℤ) dualities in matrix models, JHEP 10 (2014) 117 [arXiv:1406.5194] [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].
D. Gaiotto and E. Witten, Janus configurations, Chern-Simons couplings, and the θ-angle in N = 4 super Yang-Mills theory, JHEP 06 (2010) 097 [arXiv:0804.2907] [INSPIRE].
D. Bashkirov, Examples of global symmetry enhancement by monopole operators, arXiv:1009.3477 [INSPIRE].
C. Cordova, T.T. Dumitrescu and K. Intriligator, Deformations of superconformal theories, JHEP 11 (2016) 135 [arXiv:1602.01217] [INSPIRE].
O. Aharony, IR duality in D = 3 N = 2 supersymmetric U Sp(2N c ) and U (N c ) gauge theories, Phys. Lett. B 404 (1997) 71 [hep-th/9703215] [INSPIRE].
F. Benini, S. Benvenuti and S. Pasquetti, SUSY monopole potentials in 2+1 dimensions, arXiv:1703.08460 [INSPIRE].
A. Amariti, D. Orlando and S. Reffert, Monopole Quivers and new 3D N = 2 dualities, arXiv:1705.09297 [INSPIRE].
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Assel, B. The space of vacua of 3d \( \mathcal{N}=3 \) abelian theories. J. High Energ. Phys. 2017, 11 (2017). https://doi.org/10.1007/JHEP08(2017)011
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DOI: https://doi.org/10.1007/JHEP08(2017)011