Abstract
We propose a graph-based approach to 5d superconformal field theories (SCFTs) based on their realization as M-theory compactifications on singular elliptic Calabi-Yau threefolds. Field-theoretically, these 5d SCFTs descend from 6d \( \mathcal{N} \) = (1, 0) SCFTs by circle compactification and mass deformations. We derive a description of these theories in terms of graphs, so-called Combined Fiber Diagrams, which encode salient features of the partially resolved Calabi-Yau geometry, and provides a combinatorial way of characterizing all 5d SCFTs that descend from a given 6d theory. Remarkably, these graphs manifestly capture strongly coupled data of the 5d SCFTs, such as the superconformal flavor symmetry, BPS states, and mass deformations. The capabilities of this approach are demonstrated by deriving all rank one and rank two 5d SCFTs. The full potential, how- ever, becomes apparent when applied to theories with higher rank. Starting with the higher rank conformal matter theories in 6d, we are led to the discovery of previously unknown flavor symmetry enhancements and new 5d SCFTs.
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References
E. Witten, Some comments on string dynamics, in the proceedings of Future perspectives in string theory (Strings’95), March 13–18, Los Angeles, U.S.A. (1995), hep-th/9507121 [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 ibid. 06 (2015) 017] [arXiv:1312.5746] [INSPIRE].
L. Bhardwaj, Classification of 6d \( \mathcal{N} \) = (1, 0) gauge theories, JHEP 11 (2015) 002 [arXiv:1502.06594] [INSPIRE].
J.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa, Atomic classification of 6D SCFTs, Fortsch. Phys. 63 (2015) 468 [arXiv:1502.05405] [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].
P. Jefferson, S. Katz, H.-C. Kim and C. Vafa, On geometric classification of 5d SCFTs, JHEP 04 (2018) 103 [arXiv:1801.04036] [INSPIRE].
P. Argyres, M. Lotito, Y. Lü and M. Martone, Geometric constraints on the space of \( \mathcal{N} \) = 2 SCFTs. Part I. Physical constraints on relevant deformations, JHEP 02 (2018) 001 [arXiv:1505.04814] [INSPIRE].
F. Apruzzi et al., 5d superconformal field theories and graphs, arXiv:1906.11820 [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].
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].
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].
P. Jefferson, H.-C. Kim, C. Vafa and G. Zafrir, Towards classification of 5d SCFTs: single gauge node, arXiv:1705.05836 [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].
O. Aharony and A. Hanany, Branes, superpotentials and superconformal fixed points, Nucl. Phys. B 504 (1997) 239 [hep-th/9704170] [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 E(n) field theories, JHEP 03 (1999) 006 [hep-th/9902179] [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, 5d gauge theories and 6d \( \mathcal{N} \) = (1, 0) SCFT’s, JHEP 12 (2015) 157 [arXiv:1509.02016] [INSPIRE].
G. Zafrir, Brane webs and O5-planes, JHEP 03 (2016) 109 [arXiv:1512.08114] [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].
H. Hayashi, S.-S. Kim, K. Lee and F. Yagi, Discrete theta angle from an O5-plane, JHEP 11 (2017) 041 [arXiv:1707.07181] [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, Dualities and 5-brane webs for 5d rank 2 SCFTs, JHEP 12 (2018) 016 [arXiv:1806.10569] [INSPIRE].
N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [INSPIRE].
A. Brandhuber and Y. Oz, The D4-D8 brane system and five-dimensional fixed points, Phys. Lett. B 460 (1999) 307 [hep-th/9905148] [INSPIRE].
O. Bergman and D. Rodriguez-Gomez, 5d quivers and their AdS6 duals, JHEP 07 (2012) 171 [arXiv:1206.3503] [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].
J. Tian and Y.-N. Wang, E-string spectrum and typical F-theory geometry, arXiv:1811.02837 [INSPIRE].
F. Apruzzi et al., Fibers add flavor. Part III. Higher rank,
F. Apruzzi et al., Fibers add flavor. Part II. 5d SCFTs, gauge theories and dualities, arXiv:1909.09128 [INSPIRE].
H.-C. Kim, S.-S. Kim and K. Lee, 5-dim superconformal index with enhanced en global symmetry, JHEP 10 (2012) 142 [arXiv:1206.6781] [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].
V. Mitev, E. Pomoni, M. Taki and F. Yagi, Fiber-base duality and global symmetry enhancement, JHEP 04 (2015) 052 [arXiv:1411.2450] [INSPIRE].
C. Hwang, J. Kim, S. Kim and J. Park, General instanton counting and 5d SCFT, JHEP 07 (2015) 063 [arXiv:1406.6793] [INSPIRE].
D. Gaiotto and H.-C. Kim, Duality walls and defects in 5d \( \mathcal{N} \) = 1 theories, JHEP 01 (2017) 019 [arXiv:1506.03871] [INSPIRE].
Y. Tachikawa, Instanton operators and symmetry enhancement in 5d supersymmetric gauge theories, PTEP 2015 (2015) 043B06 [arXiv:1501.01031] [INSPIRE].
K. Yonekura, Instanton operators and symmetry enhancement in 5d supersymmetric quiver gauge theories, JHEP 07 (2015) 167 [arXiv:1505.04743] [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].
O. Bergman and D. Rodriguez-Gomez, A note on instanton operators, instanton particles and supersymmetry, JHEP 05 (2016) 068 [arXiv:1601.00752] [INSPIRE].
H. Hayashi et al., A new 5d description of 6d D-type minimal conformal matter, JHEP 08 (2015) 097 [arXiv:1505.04439] [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, A. Hanany and F. Yagi, Tropical geometry and five dimensional Higgs branches at infinite coupling, JHEP 01 (2019) 068 [arXiv:1810.01379] [INSPIRE].
M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6d conformal matter, JHEP 02 (2015) 054 [arXiv:1407.6359] [INSPIRE].
C. Lawrie and S. Schäfer-Nameki, The Tate form on steroids: resolution and higher codimension fibers, JHEP 04 (2013) 061 [arXiv:1212.2949] [INSPIRE].
P. Candelas et al., Codimension three bundle singularities in F-theory, JHEP 06 (2002) 014 [hep-th/0009228] [INSPIRE].
V. Braun, Toric elliptic fibrations and F-theory compactifications, JHEP 01 (2013) 016 [arXiv:1110.4883] [INSPIRE].
V. Braun, T.W. Grimm and J. Keitel, Geometric engineering in toric F-theory and GUTs with U(1) gauge factors, JHEP 12 (2013) 069 [arXiv:1306.0577] [INSPIRE].
M. Cvetič, A. Grassi, D. Klevers and H. Piragua, Chiral four-dimensional f-theory compactifications with SU(5) and multiple U(1)-factors, JHEP 04 (2014) 010 [arXiv:1306.3987] [INSPIRE].
F. Baume, E. Palti and S. Schwieger, On E8 and F-theory GUTs, JHEP 06 (2015) 039 [arXiv:1502.03878] [INSPIRE].
L.B. Anderson, X. Gao, J. Gray and S.-J. Lee, Tools for CICYs in F-theory, JHEP 11 (2016) 004 [arXiv:1608.07554] [INSPIRE].
W. Buchmüller, M. Dierigl, P.-K. Oehlmann and F. Ruehle, The toric SO(10) F-theory landscape, JHEP 12 (2017) 035 [arXiv:1709.06609] [INSPIRE].
L.B. Anderson, X. Gao, J. Gray and S.-J. Lee, Fibrations in CICY threefolds, JHEP 10 (2017) 077 [arXiv:1708.07907] [INSPIRE].
Y.-C. Huang and W. Taylor, Comparing elliptic and toric hypersurface Calabi-Yau threefolds at large Hodge numbers, JHEP 02 (2019) 087 [arXiv:1805.05907] [INSPIRE].
M. Dierigl, P.-K. Oehlmann and F. Ruehle, Global tensor-matter transitions in F-theory, Fortsch. Phys. 66 (2018) 1800037 [arXiv:1804.07386] [INSPIRE].
I. Achmed-Zade, I. García-Etxebarria and C. Mayrhofer, A note on non-flat points in the SU(5) × U(1)PQ F-theory model, JHEP 05 (2019) 013 [arXiv:1806.05612] [INSPIRE].
R. Miranda, Smooth models for elliptic threefolds, in The birational geometry of degenerations, R. Friedman ed., Progress in Mathematics volume 29, Birkhäuser, Boston U.S.A. (1983).
M. Esole, M.J. Kang and S.-T. Yau, Mordell-Weil torsion, anomalies and phase transitions, arXiv:1712.02337 [INSPIRE].
M. Esole, R. Jagadeesan and M.J. Kang, 48 Crepant paths to SU(2) × SU(3), arXiv:1905.05174 [INSPIRE].
M. Esole and M.J. Kang, Flopping and slicing: SO(4) and Spin(4)-models, arXiv:1802.04802 [INSPIRE].
M. Esole and M.J. Kang, The geometry of the SU(2) × G2 -model, JHEP 02 (2019) 091 [arXiv:1805.03214] [INSPIRE].
M. Bertolini, P.R. Merkx and D.R. Morrison, On the global symmetries of 6D superconformal field theories, JHEP 07 (2016) 005 [arXiv:1510.08056] [INSPIRE].
P.R. Merkx, Classifying global symmetries of 6D SCFTs, JHEP 03 (2018) 163 [arXiv:1711.05155] [INSPIRE].
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].
C. Lawrie, S. Schäfer-Nameki and J.-M. Wong, F-theory and all things rational: surveying U(1) symmetries with rational sections, JHEP 09 (2015) 144 [arXiv:1504.05593] [INSPIRE].
O.J. Ganor, D.R. Morrison and N. Seiberg, Branes, Calabi-Yau spaces and toroidal compactification of the N = 1 six-dimensional E8 theory, Nucl. Phys. B 487 (1997) 93 [hep-th/9610251] [INSPIRE].
A.C. Cadavid, A. Ceresole, R. D’Auria and S. Ferrara, Eleven-dimensional supergravity compactified on Calabi-Yau threefolds, Phys. Lett. B 357 (1995) 76 [hep-th/9506144] [INSPIRE].
S. Ferrara, R.R. Khuri and R. Minasian, M theory on a Calabi-Yau manifold, Phys. Lett. B 375 (1996) 81 [hep-th/9602102] [INSPIRE].
E. Witten, Phase transitions in M-theory and F-theory, Nucl. Phys. B 471 (1996) 195 [hep-th/9603150] [INSPIRE].
S. Ferrara, R. Minasian and A. Sagnotti, Low-energy analysis of M and F theories on Calabi-Yau threefolds, Nucl. Phys. B 474 (1996) 323 [hep-th/9604097] [INSPIRE].
L. Bhardwaj and P. Jefferson, Classifying 5d SCFTs via 6d SCFTs: rank one, JHEP 07 (2019) 178 [arXiv:1809.01650] [INSPIRE].
L. Bhardwaj and P. Jefferson, Classifying 5d SCFTs via 6d SCFTs: arbitrary rank, arXiv:1811.10616 [INSPIRE].
C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [INSPIRE].
D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. 1, Nucl. Phys. B 473 (1996) 74 [hep-th/9602114] [INSPIRE].
D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. 2, Nucl. Phys. B 476 (1996) 437 [hep-th/9603161] [INSPIRE].
S.H. Katz and C. Vafa, Matter from geometry, Nucl. Phys. B 497 (1997) 146 [hep-th/9606086] [INSPIRE].
M. Bershadsky et al., Geometric singularities and enhanced gauge symmetries, Nucl. Phys. B 481 (1996) 215 [hep-th/9605200] [INSPIRE].
S. Katz, D.R. Morrison, S. Schäfer-Nameki and J. Sully, Tate’s algorithm and F-theory, JHEP 08 (2011) 094 [arXiv:1106.3854] [INSPIRE].
D.S. Park, Anomaly equations and intersection theory, JHEP 01 (2012) 093 [arXiv:1111.2351] [INSPIRE].
M. Esole and S.-T. Yau, Small resolutions of SU(5)-models in F-theory, Adv. Theor. Math. Phys. 17 (2013) 1195 [arXiv:1107.0733] [INSPIRE].
J. Marsano and S. Schäfer-Nameki, Yukawas, G-flux and spectral covers from resolved Calabi-Yau’s, JHEP 11 (2011) 098 [arXiv:1108.1794] [INSPIRE].
S. Krause, C. Mayrhofer and T. Weigand, G4 flux, chiral matter and singularity resolution in F-theory compactifications, Nucl. Phys. B 858 (2012) 1 [arXiv:1109.3454] [INSPIRE].
M. Esole, P. Jefferson and M.J. Kang, Euler characteristics of crepant resolutions of Weierstrass models, Commun. Math. Phys. 371 (2019) 99 [arXiv:1703.00905] [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].
Y. Tachikawa, On S-duality of 5d super Yang-Mills on S1 , JHEP 11 (2011) 123 [arXiv:1110.0531] [INSPIRE].
L. Bhardwaj, D.R. Morrison, Y. Tachikawa and A. Tomasiello, The frozen phase of F-theory, JHEP 08 (2018) 138 [arXiv:1805.09070] [INSPIRE].
F. Apruzzi et al., Supplementary material: CFD-trees, https://people.maths.ox.ac.uk/schafernamek/CFD/.
R. Gopakumar and C. Vafa, M theory and topological strings. 2., hep-th/9812127 [INSPIRE].
S. Kachru and M. Zimet, A comment on 4d and 5d BPS states, arXiv:1808.01529 [INSPIRE].
M. Taki, Seiberg duality, 5D SCFTs and Nekrasov partition functions, arXiv:1401.7200 [INSPIRE].
N. Yamatsu, Finite-dimensional Lie algebras and their representations for unified model building, arXiv:1511.08771 [INSPIRE].
M.-X. Huang, A. Klemm and M. Poretschkin, Refined stable pair invariants for E-, M- and [p, q]-strings, JHEP 11 (2013) 112 [arXiv:1308.0619] [INSPIRE].
S. Cecotti, D. Gaiotto and C. Vafa, tt∗ geometry in 3 and 4 dimensions, JHEP 05 (2014) 055 [arXiv:1312.1008] [INSPIRE].
S. Banerjee, P. Longhi and M. Romo, Exploring 5d BPS spectra with exponential networks, Annales Henri Poincaré 20 (2019) 4055 [arXiv:1811.02875] [INSPIRE].
U. Derenthal, Geometry of universal torsors, Ph.D. thesis, Universit¨at G¨ottingen, G¨ottingen, Germany (2006).
U. Derenthal, Singular del pezzo surfaces whose universal torsors are hypersurfaces, Proc. London Math. Soc. 108 (2014) 638.
W. Taylor and Y.-N. Wang, Non-toric bases for elliptic Calabi-Yau threefolds and 6D F-theory vacua, Adv. Theor. Math. Phys. 21 (2017) 1063 [arXiv:1504.07689] [INSPIRE].
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Apruzzi, F., Lawrie, C., Lin, L. et al. Fibers add flavor. Part I. Classification of 5d SCFTs, flavor symmetries and BPS states. J. High Energ. Phys. 2019, 68 (2019). https://doi.org/10.1007/JHEP11(2019)068
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DOI: https://doi.org/10.1007/JHEP11(2019)068