Abstract
We revisit the relation between the anomalies in four and six dimensions and the Chern-Simons couplings one dimension below. While the dimensional reduction of chiral theories is well-understood, the question which three and five-dimensional theories can come from a general circle reduction, and are hence liftable, is more subtle. We argue that existence of an anomaly cancellation mechanism is a necessary condition for liftability. In addition, the anomaly cancellation and the CS couplings in six and five dimensions respectively determine the central charges of string-like BPS objects that cannot be consistently decoupled from gravity, a.k.a. supergravity strings. Following the completeness conjecture and requiring that their worldsheet theory is unitary imposes bounds on the admissible theories. We argue that for the anomaly-free six-dimensional theories it is more advantageous to study the unitarity constraints obtained after reduction to five dimensions. In general these are slightly more stringent and can be cast in a more geometric form, highly reminiscent of the Kodaira positivity condition (KPC). Indeed, for the F-theoretic models which have an underlying Calabi-Yau threefold these can be directly compared. The unitarity constraints (UC) are in general weaker than KPC, and maybe useful in understanding the consistent models without F-theoretic realisation. We catalogue the cases when UC is more restrictive than KPC, hinting at more refined hidden structure in elliptic Calabi-Yau threefolds with certain singularity structure.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
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].
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].
A. Grassi and D.R. Morrison, Anomalies and the Euler characteristic of elliptic Calabi-Yau threefolds, Commun. Num. Theor. Phys. 6 (2012) 51 [arXiv:1109.0042] [INSPIRE].
V. Kumar, D.R. Morrison and W. Taylor, Global aspects of the space of 6D N = 1 supergravities, JHEP 11 (2010) 118 [arXiv:1008.1062] [INSPIRE].
S. Monnier and G.W. Moore, Remarks on the Green-Schwarz terms of six-dimensional supergravity theories, Commun. Math. Phys. 372 (2019) 963 [arXiv:1808.01334] [INSPIRE].
W. Taylor, TASI lectures on supergravity and string vacua in various dimensions, arXiv:1104.2051 [INSPIRE].
T. Weigand, F-theory, PoS(TASI2017)016 [arXiv:1806.01854] [INSPIRE].
E. Poppitz and M. Ünsal, Index theorem for topological excitations on R3 × S1 and Chern-Simons theory, JHEP 03 (2009) 027 [arXiv:0812.2085] [INSPIRE].
F. Bonetti, T.W. Grimm and S. Hohenegger, One-loop Chern-Simons terms in five dimensions, JHEP 07 (2013) 043 [arXiv:1302.2918] [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].
P. Corvilain, T.W. Grimm and D. Regalado, Chiral anomalies on a circle and their cancellation in F-theory, JHEP 04 (2018) 020 [arXiv:1710.07626] [INSPIRE].
P. Corvilain, 6d \( \mathcal{N} \) = (1, 0) anomalies on S1 and F-theory implications, JHEP 08 (2020) 133 [arXiv:2005.12935] [INSPIRE].
W.A. Bardeen and B. Zumino, Consistent and covariant anomalies in gauge and gravitational theories, Nucl. Phys. B 244 (1984) 421.
H.-C. Kim, G. Shiu and C. Vafa, Branes and the Swampland, Phys. Rev. D 100 (2019) 066006 [arXiv:1905.08261] [INSPIRE].
S. Katz, H.-C. Kim, H.-C. Tarazi and C. Vafa, Swampland constraints on 5d \( \mathcal{N} \) = 1 supergravity, JHEP 07 (2020) 080 [arXiv:2004.14401] [INSPIRE].
H.-C. Tarazi and C. Vafa, On the finiteness of 6d supergravity Landscape, arXiv:2106.10839 [INSPIRE].
C. Closset, T.T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, Contact terms, unitarity, and F-maximization in three-dimensional superconformal theories, JHEP 10 (2012) 053 [arXiv:1205.4142] [INSPIRE].
C. Closset, T.T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, Comments on Chern-Simons contact terms in three dimensions, JHEP 09 (2012) 091 [arXiv:1206.5218] [INSPIRE].
D.L. Jafferis, The exact superconformal R-symmetry extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [INSPIRE].
S.S. Pufu, The F-Theorem and F-Maximization, J. Phys. A 50 (2017) 443008 [arXiv:1608.02960] [INSPIRE].
D. Freed, J.A. Harvey, R. Minasian and G.W. Moore, Gravitational anomaly cancellation for M-theory five-branes, Adv. Theor. Math. Phys. 2 (1998) 601 [hep-th/9803205] [INSPIRE].
J.A. Harvey, R. Minasian and G.W. Moore, Non-Abelian tensor multiplet anomalies, JHEP 09 (1998) 004 [hep-th/9808060] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Large N = 4 holography, JHEP 09 (2013) 036 [arXiv:1305.4181] [INSPIRE].
J.M. Maldacena, A. Strominger and E. Witten, Black hole entropy in M-theory, JHEP 12 (1997) 002 [hep-th/9711053] [INSPIRE].
A. Dabholkar and S. Nampuri, Quantum black holes, Lect. Notes Phys. 851 (2012) 165 [arXiv:1208.4814] [INSPIRE].
M. Dunajski and S.A. Hartnoll, Einstein-Maxwell gravitational instantons and five dimensional solitonic strings, Class. Quant. Grav. 24 (2007) 1841 [hep-th/0610261] [INSPIRE].
E. Witten, Phase transitions in M-theory and F-theory, Nucl. Phys. B 471 (1996) 195 [hep-th/9603150] [INSPIRE].
A.-K. Kashani-Poor, R. Minasian and H. Triendl, Enhanced supersymmetry from vanishing Euler number, JHEP 04 (2013) 058 [arXiv:1301.5031] [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].
S.-J. Lee and T. Weigand, Swampland bounds on the Abelian gauge sector, Phys. Rev. D 100 (2019) 026015 [arXiv:1905.13213] [INSPIRE].
A. Grassi and T. Weigand, Elliptic threefolds with high Mordell-Weil rank, arXiv:2105.02863 [INSPIRE].
D.R. Morrison and W. Taylor, Sections, multisections, and U(1) fields in F-theory, arXiv:1404.1527 [INSPIRE].
M. Bershadsky, K.A. Intriligator, S. Kachru, D.R. Morrison, V. Sadov and C. Vafa, Geometric singularities and enhanced gauge symmetries, Nucl. Phys. B 481 (1996) 215 [hep-th/9605200] [INSPIRE].
S. Donaldson and S. Sun, Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry, I, Acta Math. 213 (2014) 63 [arXiv:1206.2609].
S. Donaldson and S. Sun, Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry, II, J. Diff. Geom. 107 (2017) 327 [arXiv:1507.05082] [INSPIRE].
L. Bhardwaj, M. Del Zotto, J.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa, F-theory and the Classification of Little Strings, Phys. Rev. D 93 (2016) 086002 [Erratum ibid. 100 (2019) 029901] [arXiv:1511.05565] [INSPIRE].
R.A. Bertlmann, Anomalies in quantum field theory, Oxford University Press, Oxford U.K. (1996).
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: 2106.14912
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
Cheng, P., Minasian, R. & Theisen, S. Anomalies as obstructions: from dimensional lifts to swampland. J. High Energ. Phys. 2022, 68 (2022). https://doi.org/10.1007/JHEP01(2022)068
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2022)068