Abstract
The trinions TN are a class of 5d \( \mathcal{N} \) = 1 superconformal field theories (SCFTs) realized as M-theory on ℂ3/ℤN × ℤN . We apply to TN , as well as closely-related SCFTs that are obtained by mass deformations, a multitude of recently developed approaches to studying 5d SCFTs and their IR gauge theory descriptions. Thereby we provide a complete picture of the theories both on the Coulomb branch and Higgs branch, from various geometric points of view — toric and gluing of compact surfaces as well as combined fiber diagrams — to magnetic quivers and Hasse diagrams.
Article PDF
Similar content being viewed by others
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].
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].
O. DeWolfe, A. Hanany, A. Iqbal and E. Katz, Five-branes, seven-branes and five-dimensional En 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].
O. Bergman and G. Zafrir, Lifting 4d dualities to 5d, JHEP 04 (2015) 141 [arXiv:1410.2806] [INSPIRE].
G. Zafrir, Duality and enhancement of symmetry in 5d gauge theories, JHEP 12 (2014) 116 [arXiv:1408.4040] [INSPIRE].
O. Bergman and G. Zafrir, 5d fixed points from brane webs and O7-planes, JHEP 12 (2015) 163 [arXiv:1507.03860] [INSPIRE].
K. Ohmori and H. Shimizu, S1 /T 2 compactifications of 6d \( \mathcal{N} \) = (1, 0) theories and brane webs, JHEP 03 (2016) 024 [arXiv:1509.03195] [INSPIRE].
G. Zafrir, Brane webs and O5-planes, JHEP 03 (2016) 109 [arXiv:1512.08114] [INSPIRE].
G. Zafrir, Brane webs, 5d gauge theories and 6d \( \mathcal{N} \) = (1, 0) SCFT’s, JHEP 12 (2015) 157 [arXiv:1509.02016] [INSPIRE].
G. Zafrir, Instanton operators and symmetry enhancement in 5d supersymmetric USp, SO and exceptional gauge theories, JHEP 07 (2015) 087 [arXiv:1503.08136] [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].
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].
F. Apruzzi, S. Schäfer-Nameki and Y.-N. Wang, 5d SCFTs from Decoupling and Gluing, arXiv:1912.04264 [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].
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].
J.J. Heckman, D.R. Morrison and C. Vafa, On the Classification of 6D SCFTs and Generalized ADE Orbifolds, JHEP 05 (2014) 028 [Erratum JHEP 06 (2015) 017] [arXiv:1312.5746] [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].
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].
L. Bhardwaj, P. Jefferson, H.-C. Kim, H.-C. Tarazi and C. Vafa, Twisted Circle Compactifications of 6d SCFTs, arXiv:1909.11666 [INSPIRE].
L. Bhardwaj, Do all 5d SCFTs descend from 6d SCFTs?, arXiv:1912.00025 [INSPIRE].
P. Jefferson, H.-C. Kim, C. Vafa and G. Zafrir, Towards Classification of 5d SCFTs: Single Gauge Node, arXiv:1705.05836 [INSPIRE].
L. Bhardwaj and P. Jefferson, Classifying 5d SCFTs via 6d SCFTs: Rank one, JHEP 07 (2019) 178 [Addendum JHEP 01 (2020) 153] [arXiv:1809.01650] [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: Arbitrary rank, JHEP 10 (2019) 282 [arXiv:1811.10616] [INSPIRE].
L. Bhardwaj, Dualities of 5d gauge theories from S-duality, arXiv:1909.05250 [INSPIRE].
L. Bhardwaj and G. Zafrir, Classification of 5d N = 1 gauge theories, arXiv:2003.04333 [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].
A. Bourget et al., The Higgs mechanism — Hasse diagrams for symplectic singularities, JHEP 01 (2020) 157 [arXiv:1908.04245] [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 A. Zajac, Minimally Unbalanced Quivers, JHEP 02 (2019) 180 [arXiv:1810.01495] [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].
A. Bourget, J.F. Grimminger, A. Hanany, M. Sperling and Z. Zhong, Magnetic Quivers from Brane Webs with O5 Planes, arXiv:2004.04082 [INSPIRE].
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
D. Gaiotto and J. Maldacena, The Gravity duals of N = 2 superconformal field theories, JHEP 10 (2012) 189 [arXiv:0904.4466] [INSPIRE].
O. Chacaltana and J. Distler, Tinkertoys for Gaiotto Duality, JHEP 11 (2010) 099 [arXiv:1008.5203] [INSPIRE].
F. Benini, Y. Tachikawa and B. Wecht, Sicilian gauge theories and N = 1 dualities, JHEP 01 (2010) 088 [arXiv:0909.1327] [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].
H. Hayashi, Y. Tachikawa and K. Yonekura, Mass-deformed TN as a linear quiver, JHEP 02 (2015) 089 [arXiv:1410.6868] [INSPIRE].
L. Bao, V. Mitev, E. Pomoni, M. Taki and F. Yagi, Non-Lagrangian Theories from Brane Junctions, JHEP 01 (2014) 175 [arXiv:1310.3841] [INSPIRE].
H. Hayashi, H.-C. Kim and T. Nishinaka, Topological strings and 5d TN partition functions, JHEP 06 (2014) 014 [arXiv:1310.3854] [INSPIRE].
H. Hayashi and G. Zoccarato, Exact partition functions of Higgsed 5d TN theories, JHEP 01 (2015) 093 [arXiv:1409.0571] [INSPIRE].
H. Hayashi and G. Zoccarato, Topological vertex for Higgsed 5d TN theories, JHEP 09 (2015) 023 [arXiv:1505.00260] [INSPIRE].
O. Bergman, D. Rodŕıguez-Gómez and C.F. Uhlemann, Testing AdS6 /CFT5 in Type IIB with stringy operators, JHEP 08 (2018) 127 [arXiv:1806.07898] [INSPIRE].
M. Fluder and C.F. Uhlemann, Precision Test of AdS6 /CFT5 in Type IIB String Theory, Phys. Rev. Lett. 121 (2018) 171603 [arXiv:1806.08374] [INSPIRE].
J. Kaidi and C.F. Uhlemann, M-theory curves from warped AdS6 in Type IIB, JHEP 11 (2018) 175 [arXiv:1809.10162] [INSPIRE].
C.F. Uhlemann, Exact results for 5d SCFTs of long quiver type, JHEP 11 (2019) 072 [arXiv:1909.01369] [INSPIRE].
F. Apruzzi, M. Fazzi, A. Passias, D. Rosa and A. Tomasiello, AdS6 solutions of type-II supergravity, JHEP 11 (2014) 099 [Erratum JHEP 05 (2015) 012] [arXiv:1406.0852] [INSPIRE].
E. D’Hoker, M. Gutperle, A. Karch and C.F. Uhlemann, Warped AdS6 × S2 in Type IIB supergravity I: Local solutions, JHEP 08 (2016) 046 [arXiv:1606.01254] [INSPIRE].
E. D’Hoker, M. Gutperle and C.F. Uhlemann, Warped AdS6 × S2 in Type IIB supergravity II: Global solutions and five-brane webs, JHEP 05 (2017) 131 [arXiv:1703.08186] [INSPIRE].
V.I. Danilov, The geometry of toric varieties, Russ. Math. Surv. 33 (1978) 97 [Uspekhi Mat. Nauk 33 (1978) 85].
W. Fulton, Introduction to toric varieties, in Annals of Mathematics Studies 131, Princeton University Press, Princeton New Jersey U.S.A. (1993).
D.A. Cox, J.B. Little and H.K. Schenck, Toric varieties, in Graduate Studies in Mathematics 124, American Mathematical Society, Providence Rhode Island U.S.A. (2011).
D.R. Morrison and W. Taylor, Toric bases for 6D F-theory models, Fortsch. Phys. 60 (2012) 1187 [arXiv:1204.0283] [INSPIRE].
U. Derenthal, Singular del Pezzo surfaces whose universal torsors are hypersurfaces, Proc. Lond. Math. Soc. 108 (2014) 638 [math.AG/0604194].
A.P. Braun and S. Schäfer-Nameki, Box Graphs and Resolutions I, Nucl. Phys. B 905 (2016) 447 [arXiv:1407.3520] [INSPIRE].
A.P. Braun and S. Schäfer-Nameki, Box Graphs and Resolutions II: From Coulomb Phases to Fiber Faces, Nucl. Phys. B 905 (2016) 480 [arXiv:1511.01801] [INSPIRE].
E. Witten, Phase transitions in M-theory and F-theory, Nucl. Phys. B 471 (1996) 195 [hep-th/9603150] [INSPIRE].
S. Kachru and M. Zimet, A comment on 4d and 5d BPS states, JHEP 01 (2020) 060 [arXiv:1808.01529] [INSPIRE].
J. Tian and Y.-N. Wang, E-string spectrum and typical F-theory geometry, arXiv:1811.02837 [INSPIRE].
H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, Complete prepotential for 5d \( \mathcal{N} \) = 1 superconformal field theories, JHEP 02 (2020) 074 [arXiv:1912.10301] [INSPIRE].
M. Esole and M.J. Kang, The Geometry of the SU(2) × G2 -model, JHEP 02 (2019) 091 [arXiv:1805.03214] [INSPIRE].
M. Esole, R. Jagadeesan and M.J. Kang, 48 Crepant Paths to SU(2) × SU(3), arXiv:1905.05174 [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].
K. Hori, H. Ooguri and C. Vafa, NonAbelian conifold transitions and N = 4 dualities in three-dimensions, Nucl. Phys. B 504 (1997) 147 [hep-th/9705220] [INSPIRE].
O. Aharony and A. Hanany, Branes, superpotentials and superconformal fixed points, Nucl. Phys. B 504 (1997) 239 [hep-th/9704170] [INSPIRE].
O. Bergman and A. Fayyazuddin, String junction transitions in the moduli space of N = 2 SYM, Nucl. Phys. B 535 (1998) 139 [hep-th/9806011] [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].
D.R. Morrison, S. Schäfer-Nameki and B. Willett, Higher-Form Symmetries in 5d, arXiv:2005.12296 [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
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2004.15007
Rights and permissions
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.
About this article
Cite this article
Eckhard, J., Schäfer-Nameki, S. & Wang, YN. Trifectas for TN in 5d. J. High Energ. Phys. 2020, 199 (2020). https://doi.org/10.1007/JHEP07(2020)199
Received:
Accepted:
Published:
Version of record:
DOI: https://doi.org/10.1007/JHEP07(2020)199