Abstract
We describe how the geometry of the Higgs branch of 5d superconformal field theories is transformed under movement along the extended Coulomb branch. Working directly with the (unitary) magnetic quiver, we demonstrate a correspondence between Fayet-Iliopoulos deformations in 3d and 5d mass deformations. When the Higgs branch has multiple cones, characterised by a collection of magnetic quivers, the mirror map is not globally well-defined, however we are able to utilize the correspondence to establish a local version of mirror symmetry. We give several detailed examples of deformations, including decouplings and weak-coupling limits, in (Dn, Dn) conformal matter theories, TN theory and its parent PN, for which we find new Lagrangian descriptions given by quiver gauge theories with fundamental and anti-symmetric matter.
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References
N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [INSPIRE].
D. R. Morrison and N. Seiberg, Extremal transitions and five-dimensional supersymmetric field theories, Nucl. Phys. B 483 (1997) 229 [hep-th/9609070] [INSPIRE].
K. A. Intriligator, D. R. Morrison and N. Seiberg, Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces, Nucl. Phys. B 497 (1997) 56 [hep-th/9702198] [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].
O. Aharony and A. Hanany, Branes, superpotentials and superconformal fixed points, Nucl. Phys. B 504 (1997) 239 [hep-th/9704170] [INSPIRE].
O. DeWolfe, A. Hanany, A. Iqbal and E. Katz, Five-branes, seven-branes and five-dimensional E(n) field theories, JHEP 03 (1999) 006 [hep-th/9902179] [INSPIRE].
O. Bergman, D. Rodríguez-Gómez and G. Zafrir, 5-brane webs, symmetry enhancement, and duality in 5d supersymmetric gauge theory, JHEP 03 (2014) 112 [arXiv:1311.4199] [INSPIRE].
G. Zafrir, Duality and enhancement of symmetry in 5d gauge theories, JHEP 12 (2014) 116 [arXiv:1408.4040] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, 6d SCFTs, 5d dualities and tao web diagrams, JHEP 05 (2019) 203 [arXiv:1509.03300] [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].
O. Bergman and G. Zafrir, 5d fixed points from brane webs and O7-planes, JHEP 12 (2015) 163 [arXiv:1507.03860] [INSPIRE].
G. Zafrir, Brane webs and O5-planes, JHEP 03 (2016) 109 [arXiv:1512.08114] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, Dualities and 5-brane webs for 5d rank 2 SCFTs, JHEP 12 (2018) 016 [arXiv:1806.10569] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, 5-brane webs for 5d \( \mathcal{N} \) = 1 G2 gauge theories, JHEP 03 (2018) 125 [arXiv:1801.03916] [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, Rank-3 antisymmetric matter on 5-brane webs, JHEP 05 (2019) 133 [arXiv:1902.04754] [INSPIRE].
O. Bergman and D. Rodríguez-Gómez, The cat’s cradle: deforming the higher rank E1 and Ẽ1 theories, JHEP 02 (2021) 122 [arXiv:2011.05125] [INSPIRE].
M. Martone and G. Zafrir, On the compactification of 5d theories to 4d, JHEP 08 (2021) 017 [arXiv:2106.00686] [INSPIRE].
M. Bertolini and F. Mignosa, Supersymmetry breaking deformations and phase transitions in five dimensions, arXiv:2109.02662 [INSPIRE].
D.-E. Diaconescu and R. Entin, Calabi-Yau spaces and five-dimensional field theories with exceptional gauge symmetry, Nucl. Phys. B 538 (1999) 451 [hep-th/9807170] [INSPIRE].
H. Hayashi, C. Lawrie and S. Schäfer-Nameki, Phases, flops and F-theory: SU(5) gauge theories, JHEP 10 (2013) 046 [arXiv:1304.1678] [INSPIRE].
H. Hayashi, C. Lawrie, D. R. Morrison and S. Schäfer-Nameki, Box graphs and singular fibers, JHEP 05 (2014) 048 [arXiv:1402.2653] [INSPIRE].
M. Del Zotto, J. J. Heckman and D. R. Morrison, 6D SCFTs and phases of 5D theories, JHEP 09 (2017) 147 [arXiv:1703.02981] [INSPIRE].
D. Xie and S.-T. Yau, Three dimensional canonical singularity and five dimensional \( \mathcal{N} \) = 1 SCFT, JHEP 06 (2017) 134 [arXiv:1704.00799] [INSPIRE].
P. Jefferson, H.-C. Kim, C. Vafa and G. Zafrir, Towards classification of 5d SCFTs: single gauge node, arXiv:1705.05836 [INSPIRE].
P. Jefferson, S. Katz, H.-C. Kim and C. Vafa, On geometric classification of 5d SCFTs, JHEP 04 (2018) 103 [arXiv:1801.04036] [INSPIRE].
L. Bhardwaj and P. Jefferson, Classifying 5d SCFTs via 6d SCFTs: rank one, JHEP 07 (2019) 178 [Addendum ibid. 01 (2020) 153] [arXiv:1809.01650] [INSPIRE].
L. Bhardwaj and P. Jefferson, Classifying 5d SCFTs via 6d SCFTs: arbitrary rank, JHEP 10 (2019) 282 [arXiv:1811.10616] [INSPIRE].
F. Apruzzi, L. Lin and C. Mayrhofer, Phases of 5d SCFTs from M-/F-theory on non-flat fibrations, JHEP 05 (2019) 187 [arXiv:1811.12400] [INSPIRE].
C. Closset, M. Del Zotto and V. Saxena, Five-dimensional SCFTs and gauge theory phases: an M-theory/type IIA perspective, SciPost Phys. 6 (2019) 052 [arXiv:1812.10451] [INSPIRE].
F. Apruzzi, C. Lawrie, L. Lin, S. Schäfer-Nameki and Y.-N. Wang, 5d superconformal field theories and graphs, Phys. Lett. B 800 (2020) 135077 [arXiv:1906.11820] [INSPIRE].
F. Apruzzi, C. Lawrie, L. Lin, S. Schäfer-Nameki and Y.-N. Wang, Fibers add flavor. Part I. Classification of 5d SCFTs, flavor symmetries and BPS states, JHEP 11 (2019) 068 [arXiv:1907.05404] [INSPIRE].
F. Apruzzi, C. Lawrie, L. Lin, S. Schäfer-Nameki and Y.-N. Wang, Fibers add flavor. Part II. 5d SCFTs, gauge theories, and dualities, JHEP 03 (2020) 052 [arXiv:1909.09128] [INSPIRE].
L. Bhardwaj, On the classification of 5d SCFTs, JHEP 09 (2020) 007 [arXiv:1909.09635] [INSPIRE].
L. Bhardwaj, P. Jefferson, H.-C. Kim, H.-C. Tarazi and C. Vafa, Twisted circle compactifications of 6d SCFTs, JHEP 12 (2020) 151 [arXiv:1909.11666] [INSPIRE].
L. Bhardwaj, Dualities of 5d gauge theories from S-duality, JHEP 07 (2020) 012 [arXiv:1909.05250] [INSPIRE].
V. Saxena, Rank-two 5d SCFTs from M-theory at isolated toric singularities: a systematic study, JHEP 04 (2020) 198 [arXiv:1911.09574] [INSPIRE].
F. Apruzzi, S. Schäfer-Nameki and Y.-N. Wang, 5d SCFTs from decoupling and gluing, JHEP 08 (2020) 153 [arXiv:1912.04264] [INSPIRE].
C. Closset and M. Del Zotto, On 5d SCFTs and their BPS quivers. Part I: B-branes and brane tilings, arXiv:1912.13502 [INSPIRE].
L. Bhardwaj, Do all 5d SCFTs descend from 6d SCFTs?, JHEP 04 (2021) 085 [arXiv:1912.00025] [INSPIRE].
L. Bhardwaj and G. Zafrir, Classification of 5d \( \mathcal{N} \) = 1 gauge theories, JHEP 12 (2020) 099 [arXiv:2003.04333] [INSPIRE].
J. Eckhard, S. Schäfer-Nameki and Y.-N. Wang, Trifectas for TN in 5d, JHEP 07 (2020) 199 [arXiv:2004.15007] [INSPIRE].
L. Bhardwaj, More 5d KK theories, arXiv:2005.01722 [INSPIRE].
M. Hubner, 5d SCFTs from (En, Em) conformal matter, JHEP 12 (2020) 014 [arXiv:2006.01694] [INSPIRE].
L. Bhardwaj, Flavor symmetry of 5d SCFTs, Part 1: general setup, arXiv:2010.13230 [INSPIRE].
L. Bhardwaj, Flavor symmetry of 5d SCFTs. Part II. Applications, JHEP 04 (2021) 221 [arXiv:2010.13235] [INSPIRE].
L. Bhardwaj and S. Schäfer-Nameki, Higher-form symmetries of 6d and 5d theories, JHEP 02 (2021) 159 [arXiv:2008.09600] [INSPIRE].
C. Closset, S. Giacomelli, S. Schäfer-Nameki and Y.-N. Wang, 5d and 4d SCFTs: canonical singularities, trinions and S-dualities, JHEP 05 (2021) 274 [arXiv:2012.12827] [INSPIRE].
A. P. Braun, J. Chen, B. Haghighat, M. Sperling and S. Yang, Fibre-base duality of 5d KK theories, JHEP 05 (2021) 200 [arXiv:2103.06066] [INSPIRE].
F. Apruzzi, L. Bhardwaj, J. Oh and S. Schäfer-Nameki, The global form of flavor symmetries and 2-group symmetries in 5d SCFTs, arXiv:2105.08724 [INSPIRE].
C. Closset and H. Magureanu, The U -plane of rank-one 4d \( \mathcal{N} \) = 2 KK theories, arXiv:2107.03509 [INSPIRE].
A. Collinucci, A. Sangiovanni and R. Valandro, Genus zero Gopakumar-Vafa invariants from open strings, JHEP 09 (2021) 059 [arXiv:2104.14493] [INSPIRE].
A. Collinucci, M. De Marco, A. Sangiovanni and R. Valandro, Higgs branches of 5d rank-zero theories from geometry, arXiv:2105.12177 [INSPIRE].
O. Aharony, A. Hanany and B. Kol, Webs of (p, q) five-branes, five-dimensional field theories and grid diagrams, JHEP 01 (1998) 002 [hep-th/9710116] [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].
L. Bao, E. Pomoni, M. Taki and F. Yagi, M 5-branes, toric diagrams and gauge theory duality, JHEP 04 (2012) 105 [arXiv:1112.5228] [INSPIRE].
M. Taki, Seiberg duality, 5d SCFTs and Nekrasov partition functions, arXiv:1401.7200 [INSPIRE].
M. van Beest, A. Bourget, J. Eckhard and S. Schäfer-Nameki, (Symplectic) leaves and (5d Higgs) branches in the poly(go)nesian tropical rain forest, JHEP 11 (2020) 124 [arXiv:2008.05577] [INSPIRE].
M. Van Beest, A. Bourget, J. Eckhard and S. Schäfer-Nameki, (5d RG-flow) trees in the tropical rain forest, JHEP 03 (2021) 241 [arXiv:2011.07033] [INSPIRE].
C. Closset, S. Schäfer-Nameki and Y.-N. Wang, Coulomb and Higgs branches from canonical singularities: part 0, JHEP 02 (2021) 003 [arXiv:2007.15600] [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].
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, Coulomb branches of 3d \( \mathcal{N} \) = 4 quiver gauge theories and slices in the affine Grassmannian, Adv. Theor. Math. Phys. 23 (2019) 75 [arXiv:1604.03625] [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].
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].
S. Cabrera and A. Hanany, Quiver subtractions, JHEP 09 (2018) 008 [arXiv:1803.11205] [INSPIRE].
S. Cabrera, A. Hanany and F. Yagi, Tropical geometry and five dimensional Higgs branches at infinite coupling, JHEP 01 (2019) 068 [arXiv:1810.01379] [INSPIRE].
S. Cabrera, A. Hanany and M. Sperling, Magnetic quivers, Higgs branches, and 6d N = (1, 0) theories, JHEP 06 (2019) 071 [Erratum ibid. 07 (2019) 137] [arXiv:1904.12293] [INSPIRE].
A. Bourget et al., The Higgs mechanism — Hasse diagrams for symplectic singularities, JHEP 01 (2020) 157 [arXiv:1908.04245] [INSPIRE].
A. Bourget, S. Cabrera, J. F. Grimminger, A. Hanany and Z. Zhong, Brane webs and magnetic quivers for SQCD, JHEP 03 (2020) 176 [arXiv:1909.00667] [INSPIRE].
S. Cabrera, A. Hanany and M. Sperling, Magnetic quivers, Higgs branches, and 6d \( \mathcal{N} \) = (1, 0) theories — orthogonal and symplectic gauge groups, JHEP 02 (2020) 184 [arXiv:1912.02773] [INSPIRE].
J. F. Grimminger and A. Hanany, Hasse diagrams for 3d \( \mathcal{N} \) = 4 quiver gauge theories — Inversion and the full moduli space, JHEP 09 (2020) 159 [arXiv:2004.01675] [INSPIRE].
A. Bourget, J. F. Grimminger, A. Hanany, M. Sperling and Z. Zhong, Magnetic quivers from brane webs with O5 planes, JHEP 07 (2020) 204 [arXiv:2004.04082] [INSPIRE].
A. Bourget, J. F. Grimminger, A. Hanany, M. Sperling, G. Zafrir and Z. Zhong, Magnetic quivers for rank 1 theories, JHEP 09 (2020) 189 [arXiv:2006.16994] [INSPIRE].
M. Akhond, F. Carta, S. Dwivedi, H. Hayashi, S.-S. Kim and F. Yagi, Five-brane webs, Higgs branches and unitary/orthosymplectic magnetic quivers, JHEP 12 (2020) 164 [arXiv:2008.01027] [INSPIRE].
A. Bourget, S. Giacomelli, J. F. Grimminger, A. Hanany, M. Sperling and Z. Zhong, S-fold magnetic quivers, JHEP 02 (2021) 054 [arXiv:2010.05889] [INSPIRE].
A. Dancer, A. Hanany and F. Kirwan, Symplectic duality and implosions, arXiv:2004.09620 [INSPIRE].
A. Bourget, A. Hanany and D. Miketa, Quiver origami: discrete gauging and folding, JHEP 01 (2021) 086 [arXiv:2005.05273] [INSPIRE].
A. Bourget, J. F. Grimminger, A. Hanany, R. Kalveks, M. Sperling and Z. Zhong, Magnetic lattices for orthosymplectic quivers, JHEP 12 (2020) 092 [arXiv:2007.04667] [INSPIRE].
M. Akhond, F. Carta, S. Dwivedi, H. Hayashi, S.-S. Kim and F. Yagi, Factorised 3d \( \mathcal{N} \) = 4 orthosymplectic quivers, JHEP 05 (2021) 269 [arXiv:2101.12235] [INSPIRE].
A. Bourget, J. F. Grimminger, A. Hanany, M. Sperling and Z. Zhong, Branes, quivers, and the affine Grassmannian, arXiv:2102.06190 [INSPIRE].
A. Bourget, A. Dancer, J. F. Grimminger, A. Hanany, F. Kirwan and Z. Zhong, Orthosymplectic implosions, JHEP 08 (2021) 012 [arXiv:2103.05458] [INSPIRE].
M. Akhond and F. Carta, Magnetic quivers from brane webs with O7+-planes, arXiv:2107.09077 [INSPIRE].
J. Bao, A. Hanany, Y.-H. He and E. Hirst, Some open questions in quiver gauge theory, arXiv:2108.05167 [INSPIRE].
K. Gledhill and A. Hanany, Coulomb branch global symmetry and quiver addition, arXiv:2109.07237 [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].
A. Hanany and R. Kalveks, Highest weight generating functions for Hilbert series, JHEP 10 (2014) 152 [arXiv:1408.4690] [INSPIRE].
N. Mekareeya, K. Ohmori, Y. Tachikawa and G. Zafrir, E8 instantons on type-A ALE spaces and supersymmetric field theories, JHEP 09 (2017) 144 [arXiv:1707.04370] [INSPIRE].
K. Altmann, The versal deformation of an isolated toric Gorenstein singularity, Invent. Math. 128 (1997) 443.
K. Altmann, Infinitesimal deformations and obstructions for toric singularities, alg-geom/9405008.
A. Dey, Higgs branches of Argyres-Douglas theories as quiver varieties, arXiv:2109.07493 [INSPIRE].
M. Del Zotto, J. J. Heckman, A. Tomasiello and C. Vafa, 6d conformal matter, JHEP 02 (2015) 054 [arXiv:1407.6359] [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].
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van Beest, M., Giacomelli, S. Connecting 5d Higgs branches via Fayet-Iliopoulos deformations. J. High Energ. Phys. 2021, 202 (2021). https://doi.org/10.1007/JHEP12(2021)202
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DOI: https://doi.org/10.1007/JHEP12(2021)202