- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Editors' Suggestion
  • Open Access

From quivers to geometry: 5D conformal matter

Antoine Bourget1,*, Mario De Marco6,†, Michele Del Zotto4,5,2,‡, Julius F. Grimminger3,§, and Andrea Sangiovanni4,5,2,∥

  • *Contact author: antoine.bourget@ipht.fr
  • †Contact author: mario.de.marco@ulb.be
  • ‡Contact author: michele.delzotto@math.uu.se
  • §Contact author: julius.grimminger@maths.ox.ac.uk
  • ∥Contact author: andrea.sangiovanni@math.uu.se
PDF

Phys. Rev. D 114, 086009 – Published 9 October, 2026

DOI: https://doi.org/10.1103/8qw1-xyj6

Abstract

We show that all 5D balanced (ADE-shaped) special unitary quivers with no Chern-Simons level admit a UV completion which is a 5D conformal matter superconformal field theory (SCFT). We give explicit local Calabi-Yau threefolds realizing each of these models in M-theory. This unified description enables a systematic exploration of their physical properties, such as their Higgs branch, as well as connections to class-S constructions and the affine Grassmannian.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (89)

  1. E. Witten, String theory dynamics in various dimensions, Nucl. Phys. B443, 85 (1995).
  2. A. Strominger, Open p-branes, Phys. Lett. B 383, 44 (1996).
  3. E. Witten, Five-branes and M theory on an orbifold, Nucl. Phys. B463, 383 (1996).
  4. O. J. Ganor and A. Hanany, Small E(8) instantons and tensionless noncritical strings, Nucl. Phys. B474, 122 (1996).
  5. N. Seiberg, Nontrivial fixed points of the renormalization group in six-dimensions, Phys. Lett. B 390, 169 (1997).
  6. N. Seiberg, Five dimensional SUSY field theories, non-trivial fixed points and string dynamics, Phys. Lett. B 388, 753 (1996).
  7. D. R. Morrison and N. Seiberg, Extremal transitions and five-dimensional supersymmetric field theories, Nucl. Phys. B483, 229 (1997).
  8. M. R. Douglas, S. H. Katz, and C. Vafa, Small instantons, Del Pezzo surfaces and type I-prime theory, Nucl. Phys. B497, 155 (1997).
  9. J. J. Heckman, D. R. Morrison, and C. Vafa, On the classification of 6D SCFTs and generalized ADE orbifolds, J. High Energy Phys. 05 (2014) 028; 06 (2015) 017(E).
  10. M. Del Zotto, J. J. Heckman, A. Tomasiello, and C. Vafa, 6d conformal matter, J. High Energy Phys. 02 (2015) 054.
  11. J. J. Heckman, D. R. Morrison, T. Rudelius, and C. Vafa, Atomic classification of 6D SCFTs, Fortschr. Phys. 63, 468 (2015).
  12. L. Bhardwaj, D. R. Morrison, Y. Tachikawa, and A. Tomasiello, The frozen phase of F-theory, J. High Energy Phys. 08 (2018) 138.
  13. P. Jefferson, S. Katz, H.-C. Kim, and C. Vafa, On geometric classification of 5D SCFTs, J. High Energy Phys. 04 (2018) 103.
  14. M. Del Zotto, J. J. Heckman, and D. R. Morrison, 6D SCFTs and phases of 5D theories, J. High Energy Phys. 09 (2017) 147.
  15. L. Bhardwaj and P. Jefferson, Classifying 5d SCFTs via 6d SCFTs: Rank one, J. High Energy Phys. 07 (2019) 178; 01 (2020) 153(E).
  16. L. Bhardwaj and P. Jefferson, Classifying 5d SCFTs via 6d SCFTs: Arbitrary rank, J. High Energy Phys. 10 (2019) 282.
  17. L. Bhardwaj, Do all 5d SCFTs descend from 6d SCFTs?, J. High Energy Phys. 04 (2021) 085.
  18. L. Bhardwaj and G. Zafrir, Classification of 5d N=1 gauge theories, J. High Energy Phys. 12 (2020) 099.
  19. A. Collinucci, F. Del Monte, M. De Marco, M. Moleti, and R. Valandro, Non-toric 5d SCFTs from Reid’s Pagoda, J. High Energy Phys. 06 (2026) 128.
  20. O. Aharony and A. Hanany, Branes, superpotentials and superconformal fixed points, Nucl. Phys. B504, 239 (1997).
  21. O. Aharony, A. Hanany, and B. Kol, Webs of (p,q) five-branes, five-dimensional field theories and grid diagrams, J. High Energy Phys. 01 (1998) 002.
  22. O. DeWolfe, A. Hanany, A. Iqbal, and E. Katz, Five-branes, seven-branes and five-dimensional E(n) field theories, J. High Energy Phys. 03 (1999) 006.
  23. F. Benini, S. Benvenuti, and Y. Tachikawa, Webs of five-branes and N=2 superconformal field theories, J. High Energy Phys. 09 (2009) 052.
  24. O. Bergman, D. Rodríguez-Gómez, and G. Zafrir, 5-brane webs, symmetry enhancement, and duality in 5d supersymmetric gauge theory, J. High Energy Phys. 03 (2014) 112.
  25. G. Zafrir, Duality and enhancement of symmetry in 5d gauge theories, J. High Energy Phys. 12 (2014) 116.
  26. H. Hayashi, S.-S. Kim, K. Lee, and F. Yagi, 6d SCFTs, 5d dualities and Tao web diagrams, J. High Energy Phys. 05 (2019) 203.
  27. H. Hayashi, S.-S. Kim, K. Lee, M. Taki, and F. Yagi, A new 5d description of 6d D-type minimal conformal matter, J. High Energy Phys. 08 (2015) 097.
  28. O. Bergman and G. Zafrir, 5d fixed points from brane webs and O7-planes, J. High Energy Phys. 12 (2015) 163.
  29. H. Hayashi, S.-S. Kim, K. Lee, and F. Yagi, Dualities and 5-brane webs for 5d rank 2 SCFTs, J. High Energy Phys. 12 (2018) 016.
  30. H. Hayashi, S.-S. Kim, K. Lee, and F. Yagi, 5-brane webs for 5d N= 1  G2 gauge theories, J. High Energy Phys. 03 (2018) 125.
  31. H. Hayashi, S.-S. Kim, K. Lee, and F. Yagi, Rank-3 antisymmetric matter on 5-brane webs, J. High Energy Phys. 05 (2019) 133.
  32. H. Hayashi, S.-S. Kim, K. Lee, and F. Yagi, Complete prepotential for 5d N=1 superconformal field theories, J. High Energy Phys. 02 (2020) 074.
  33. O. Bergman and D. Rodríguez-Gómez, The Cat’s cradle: Deforming the higher rank E1 and E˜1 theories, J. High Energy Phys. 02 (2021) 122.
  34. N. C. Leung and C. Vafa, Branes and toric geometry, arXiv:hep-th/9711013
  35. A. Bourget, A. Collinucci, and S. Schafer-Nameki, Generalized toric polygons, T-branes, and 5d SCFTs, SciPost Phys. 18, 079 (2025).
  36. V. Alexeev, H. Argüz, and P. Bousseau, Non-toric brane webs, Calabi-Yau 3-folds, and 5d SCFTs, Commun. Math. Phys. 406, 287 (2025).
  37. D. Xie and S.-T. Yau, Three dimensional canonical singularity and five dimensional n N=1 SCFT, J. High Energy Phys. 06 (2017) 134.
  38. S. S. T. Yau and Y. Yu, Classification of 3-dimensional isolated rational hypersurface singularities with c*-action (2003).
  39. J. Tian and Y.-N. Wang, 5D and 6D SCFTs from C3 orbifolds, SciPost Phys. 12, 127 (2022).
  40. C. Closset, S. Schafer-Nameki, and Y.-N. Wang, Coulomb and Higgs branches from canonical singularities: Part 0, J. High Energy Phys. 02 (2021) 003.
  41. C. Closset, S. Giacomelli, S. Schafer-Nameki, and Y.-N. Wang, 5d and 4d SCFTs: Canonical singularities, trinions and s-dualities, J. High Energy Phys. 05 (2021) 274.
  42. C. Closset, S. Schäfer-Nameki, and Y.-N. Wang, Coulomb and Higgs branches from canonical singularities. Part I. Hypersurfaces with smooth Calabi-Yau resolutions, J. High Energy Phys. 04 (2022) 061.
  43. A. Collinucci, M. De Marco, A. Sangiovanni, and R. Valandro, Higgs branches of 5d rank-zero theories from geometry, J. High Energy Phys. 10 (2021) 018.
  44. J. Mu, Y.-N. Wang, and H. N. Zhang, 5d SCFTs from isolated complete intersection singularities, J. High Energy Phys. 10 (2021) 018.
  45. M. De Marco, M. del Zotto, M. Graffeo, and A. Sangiovanni, 5d trinions and tetraons, arXiv:2605.16497.
  46. D. Dramburg, S. N. Meynet, and A. Sangiovanni, On the orbifold origin of higher form symmetries in geometric engineering, Phys. Rev. D 114, 066007 (2026).
  47. O. Chacaltana and J. Distler, Tinkertoys for gaiotto duality, J. High Energy Phys. 11 (2010) 099.
  48. O. Chacaltana and J. Distler, Tinkertoys for the DN series, J. High Energy Phys. 02 (2013) 110.
  49. 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, 1340006 (2013).
  50. O. Chacaltana, J. Distler, and Y. Tachikawa, Gaiotto duality for the twisted A2N−1 series, J. High Energy Phys. 05 (2015) 075.
  51. O. Chacaltana, J. Distler, and A. Trimm, Tinkertoys for the twisted D-series, J. High Energy Phys. 04 (2015) 173.
  52. O. Chacaltana, J. Distler, and A. Trimm, Tinkertoys for the E6 theory, J. High Energy Phys. 09 (2015) 007.
  53. O. Chacaltana, J. Distler, and A. Trimm, Tinkertoys for the twisted E6 theory, arXiv:1501.00357.
  54. O. Chacaltana, J. Distler, and A. Trimm, Tinkertoys for the Z3-twisted D4 theory, arXiv:1601.02077.
  55. O. Chacaltana, J. Distler, A. Trimm, and Y. Zhu, Tinkertoys for the E7 theory, J. High Energy Phys. 05 (2018) 031.
  56. O. Chacaltana, J. Distler, A. Trimm, and Y. Zhu, Tinkertoys for the E8 Theory, arXiv:1802.09626.
  57. M. De Marco, M. Del Zotto, M. Graffeo, and A. Sangiovanni, 5d conformal matter, J. High Energy Phys. 05 (2024) 306; 08 (2024) 067(E).
  58. M. De Marco, M. Del Zotto, J. F. Grimminger, and A. Sangiovanni, Remarks on the Higgs branch of 5d conformal matter, J. High Energy Phys. 10 (2025) 144.
  59. K. Yonekura, Instanton operators and symmetry enhancement in 5d supersymmetric quiver gauge theories, J. High Energy Phys. 07 (2015) 167.
  60. See Supplemental Materials at http://link.aps.org/supplemental/10.1103/8qw1-xyj6 for macaulay2 code attached to the publication.
  61. A. Bourget, S. Cabrera, J. F. Grimminger, A. Hanany, M. Sperling, A. Zajac, and Z. Zhong, The Higgs mechanism—Hasse diagrams for symplectic singularities, J. High Energy Phys. 01 (2020) 157.
  62. A. Collinucci and R. Valandro, The role of U(1)’s in 5d theories, Higgs branches, and geometry, J. High Energy Phys. 10 (2020) 178.
  63. A. Braverman, M. Finkelberg, and H. Nakajima, Coulomb branches of 3d N=4 quiver gauge theories and slices in the affine Grassmannian, Adv. Theor. Math. Phys. 23, 75 (2019).
  64. M. Bullimore, T. Dimofte, and D. Gaiotto, The Coulomb branch of 3d N=4 theories, Commun. Math. Phys. 354, 671 (2017).
  65. A. Bourget, J. F. Grimminger, A. Hanany, M. Sperling, and Z. Zhong, Branes, quivers, and the affine Grassmannian, Adv. Stud. Pure Math. 88, 331 (2023).
  66. P. N. Achar and A. Henderson, Geometric satake springer correspondence and small representations, Sel. Math. Sov. 19, 949 (2013).
  67. M. De Marco, M. Del Zotto, J. F. Grimminger, and A. Sangiovanni, 5d Higgs branches: Stratifications from geometry, arXiv:2607.06851.
  68. H. Hayashi and K. Ohmori, 5d/6d DE instantons from trivalent gluing of web diagrams, J. High Energy Phys. 06 (2017) 078.
  69. H. Hayashi, P. Jefferson, H.-C. Kim, K. Ohmori, and C. Vafa, SCFTs, holography, and topological strings, Surv. Diff. Geom. 23, 105 (2018).
  70. R. Feger, T. W. Kephart, and R. J. Saskowski, LieART 2.0—A Mathematica application for Lie algebras and representation theory, Comput. Phys. Commun. 257, 107490 (2020).
  71. A. Broer, The sum of generalized exponents and Chevalley’s restriction theorem for modules of covariants, Indagat. Math 6, 385 (1995).
  72. OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences, http://oeis.org/.
  73. Y. Tachikawa, Instanton operators and symmetry enhancement in 5d supersymmetric gauge theories, Prog. Theor. Exp. Phys. 2015, 043B06 (2015).
  74. D. Xie and S.-T. Yau, Three dimensional canonical singularity and five dimensional N=1 SCFT, J. High Energy Phys. 06 (2017) 134.
  75. M. Reid, Canonical 3-folds, J. Géom. Algébrique d’Angers Sijthoff Nordhoff 273 (1980), https://cir.nii.ac.jp/crid/1570009749342399360.
  76. A. Malkin, V. Ostrik, and M. Vybornov, The minimal degeneration singularities in the affine Grassmannians, arXiv:math/0305095.
  77. B. Fu, D. Juteau, P. Levy, and E. Sommers, Generic singularities of nilpotent orbit closures, Adv. Math. 305, 1 (2017).
  78. D. Xie and S.-T. Yau, 4d N=2 SCFT and singularity theory part I: Classification, arXiv:1510.01324.
  79. B. S. Acharya, Confinement in five dimensions, arXiv:2407.03171.
  80. C. Closset, M. Del Zotto, and V. Saxena, Five-dimensional SCFTs and gauge theory phases: An m-theory/type IIa perspective, SciPost Phys. 6, 052 (2019).
  81. P. Jefferson, H.-C. Kim, C. Vafa, and G. Zafrir, Towards classification of 5d SCFTs: Single gauge node, SciPost Phys. 14, 122 (2023).
  82. K. Ohmori, H. Shimizu, Y. Tachikawa, and K. Yonekura, 6d N=(1,0) theories on T2 and class S theories: Part I, J. High Energy Phys. 07 (2015) 014.
  83. K. Ohmori, H. Shimizu, Y. Tachikawa, and K. Yonekura, 6d N=(1,0) theories on S1/T2 and class S theories: Part II, J. High Energy Phys. 12 (2015) 131.
  84. S. S. Razamat, C. Vafa, and G. Zafrir, 4d N=1 from 6d (1, 0), J. High Energy Phys. 04 (2017) 064.
  85. H.-C. Kim, S. S. Razamat, C. Vafa, and G. Zafrir, E-string theory on Riemann surfaces, Fortschr. Phys. 66, 1700074 (2018).
  86. H.-C. Kim, S. S. Razamat, C. Vafa, and G. Zafrir, D-type conformal matter and SU/USp quivers, J. High Energy Phys. 06 (2018) 058.
  87. H.-C. Kim, S. S. Razamat, C. Vafa, and G. Zafrir, Compactifications of ADE conformal matter on a torus, J. High Energy Phys. 09 (2018) 110.
  88. L. Bhardwaj, P. Jefferson, H.-C. Kim, H.-C. Tarazi, and C. Vafa, Twisted circle compactifications of 6d SCFTs, J. High Energy Phys. 12 (2020) 151.
  89. R. Feger and T. W. Kephart, Lieart—a Mathematica application for Lie algebras and representation theory, Comput. Phys. Commun. 192, 166 (2015).

Outline

Information

Phys. Rev. D 114, 086009– Published 9 October, 2026
  • Received 24 July 2026
  • Accepted 1 September 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation