Abstract
We study 6d superconformal field theories (SCFTs) compactified on a circle with arbitrary twists. The theories obtained after compactification, often referred to as 5d Kaluza-Klein (KK) theories, can be viewed as starting points for RG flows to 5d SCFTs. According to a conjecture, all 5d SCFTs can be obtained in this fashion. We compute the Coulomb branch prepotential for all 5d KK theories obtainable in this manner and associate to these theories a smooth local genus one fibered Calabi-Yau threefold in which is encoded information about all possible RG flows to 5d SCFTs. These Calabi-Yau threefolds provide hitherto unknown M-theory duals of F-theory configurations compactified on a circle with twists. For certain exceptional KK theories that do not admit a standard geometric description we propose an algebraic description that appears to retain the properties of the local Calabi-Yau threefolds necessary to determine RG flows to 5d SCFTs, along with other relevant physical data.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
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].
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].
M. Del Zotto and G. Lockhart, Universal features of BPS strings in six-dimensional SCFTs, JHEP 08 (2018) 173 [arXiv:1804.09694] [INSPIRE].
N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [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. Aharony and A. Hanany, Branes, superpotentials and superconformal fixed points, Nucl. Phys. B 504 (1997) 239 [hep-th/9704170] [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, Rank-3 antisymmetric matter on 5-brane webs, JHEP 05 (2019) 133 [arXiv:1902.04754] [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 N = 1 G2 gauge theories, JHEP 03 (2018) 125 [arXiv:1801.03916] [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].
M.R. Douglas, S.H. Katz and C. Vafa, Small instantons, del Pezzo surfaces and type-I-prime theory, Nucl. Phys. B 497 (1997) 155 [hep-th/9609071] [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].
H. Hayashi, S.-S. Kim, K. Lee, M. Taki and F. Yagi, More on 5d descriptions of 6d SCFTs, JHEP 10 (2016) 126 [arXiv:1512.08239] [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].
G. Zafrir, Brane webs, 5d gauge theories and 6d N = (1, 0) SCFT’s, JHEP 12 (2015) 157 [arXiv:1509.02016] [INSPIRE].
S.-S. Kim, M. Taki and F. Yagi, Tao probing the end of the world, PTEP 2015 (2015) 083B02 [arXiv:1504.03672] [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].
E. Witten, Toroidal compactification without vector structure, JHEP 02 (1998) 006 [hep-th/9712028] [INSPIRE].
J. de Boer et al., Triples, fluxes, and strings, Adv. Theor. Math. Phys. 4 (2002) 995 [hep-th/0103170] [INSPIRE].
Y. Tachikawa, Frozen singularities in M and F-theory, JHEP 06 (2016) 128 [arXiv:1508.06679] [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].
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].
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 N = (1, 0) gauge theories, JHEP 11 (2015) 002 [arXiv:1502.06594] [INSPIRE].
L. Bhardwaj, Revisiting the classifications of 6d SCFTs and LSTs, JHEP 03 (2020) 171 [arXiv:1903.10503] [INSPIRE].
D.R. Morrison and W. Taylor, Sections, multisections, and U(1) fields in F-theory, arXiv:1404.1527 [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].
D.D. Frey and T. Rudelius, 6d SCFTs and the classification of homomorphisms ΓADE → E8, Adv. Theor. Math. Phys. 24 (2020) 709 [arXiv:1811.04921] [INSPIRE].
F. Bonetti and T.W. Grimm, Six-dimensional (1, 0) effective action of F-theory via M-theory on Calabi-Yau threefolds, JHEP 05 (2012) 019 [arXiv:1112.1082] [INSPIRE].
F. Bonetti, T.W. Grimm and S. Hohenegger, Exploring 6d origins of 5d supergravities with Chern-Simons terms, JHEP 05 (2013) 124 [arXiv:1303.2661] [INSPIRE].
T.W. Grimm and A. Kapfer, Anomaly cancelation in field theory and F-theory on a circle, JHEP 05 (2016) 102 [arXiv:1502.05398] [INSPIRE].
T.W. Grimm, The N = 1 effective action of F-theory compactifications, Nucl. Phys. B 845 (2011) 48 [arXiv:1008.4133] [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].
M. Esole and S.-H. Shao, M-theory on elliptic Calabi-Yau threefolds and 6d anomalies, arXiv:1504.01387 [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, The geometry of G2, Spin(7), and Spin(8)-models, arXiv:1709.04913 [INSPIRE].
M. Esole, P. Jefferson and M.J. Kang, The geometry of F4-models, arXiv:1704.08251 [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].
M. Esole, S.-H. Shao and S.-T. Yau, Singularities and gauge theory phases, Adv. Theor. Math. Phys. 19 (2015) 1183 [arXiv:1402.6331] [INSPIRE].
M. Esole, S.-H. Shao and S.-T. Yau, Singularities and gauge theory phases II, Adv. Theor. Math. Phys. 20 (2016) 683 [arXiv:1407.1867] [INSPIRE].
K. Oguiso, On algebraic fiber space structures on a Calabi-Yau 3-fold, Int. J. Math. 04 (1993) 439.
P.M.H. Wilson, The existence of elliptic fibre space structures on Calabi-Yau threefolds, Math. Annalen 300 (1994) 693.
L. Bhardwaj, Dualities of 5d gauge theories from S-duality, JHEP 07 (2020) 012 [arXiv:1909.05250] [INSPIRE].
Y. Tachikawa, On S-duality of 5d super Yang-Mills on S1, JHEP 11 (2011) 123 [arXiv:1110.0531] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 1909.11666
Supplementary Information
ESM 1
(ZIP 40 kb)
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
Bhardwaj, L., Jefferson, P., Kim, HC. et al. Twisted circle compactifications of 6d SCFTs. J. High Energ. Phys. 2020, 151 (2020). https://doi.org/10.1007/JHEP12(2020)151
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2020)151