Abstract
We consider the 6d (2, 0) theory on a fibration by genus g curves, and dimensionally reduce along the fiber to 4d theories with duality defects. This generalizes class S theories, for which the fibration is trivial. The non-trivial fibration in the present setup implies that the gauge couplings of the 4d theory, which are encoded in the complex structures of the curve, vary and can undergo S-duality transformations. These monodromies occur around 2d loci in space-time, the duality defects, above which the fiber is singular. The key role that the fibration plays here motivates refering to this setup as theories of class F. In the simplest instance this gives rise to 4d \( \mathcal{N}=4 \) Super-Yang-Mills with space-time dependent coupling that undergoes SL(2, ℤ) monodromies. We determine the anomaly polynomial for these theories by pushing forward the anomaly polynomial of the 6d (2, 0) theory along the fiber. This gives rise to corrections to the anomaly polynomials of 4d \( \mathcal{N}=4 \) SYM and theories of class S. For the torus case, this analysis is complemented with a field theoretic derivation of a U(1) anomaly in 4d \( \mathcal{N}=4 \) SYM. The corresponding anomaly polynomial is tested against known expressions of anomalies for wrapped D3-branes with varying coupling, which are known field theoretically and from holography. Extensions of the construction to 4d \( \mathcal{N}=0 \) and 1, and 2d theories with varying coupling, are also discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [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].
L. Martucci, Topological duality twist and brane instantons in F-theory, JHEP 06 (2014) 180 [arXiv:1403.2530] [INSPIRE].
B. Haghighat, S. Murthy, C. Vafa and S. Vandoren, F-Theory, Spinning Black Holes and Multi-string Branches, JHEP 01 (2016) 009 [arXiv:1509.00455] [INSPIRE].
B. Assel and S. Schäfer-Nameki, Six-dimensional origin of \( \mathcal{N}=4 \) SYM with duality defects, JHEP 12 (2016) 058 [arXiv:1610.03663] [INSPIRE].
C. Lawrie, S. Schäfer-Nameki and T. Weigand, Chiral 2d theories from N = 4 SYM with varying coupling, JHEP 04 (2017) 111 [arXiv:1612.05640] [INSPIRE].
C. Couzens, C. Lawrie, D. Martelli, S. Schäfer-Nameki and J.-M. Wong, F-theory and AdS3/CF T2, JHEP 08 (2017) 043 [arXiv:1705.04679] [INSPIRE].
J. Choi, J.J. Fernandez-Melgarejo and S. Sugimoto, Supersymmetric Gauge Theory with Space-time-Dependent Couplings, PTEP 2018 (2018) 013B01 [arXiv:1710.09792] [INSPIRE].
C. Couzens, D. Martelli and S. Schäfer-Nameki, F-theory and AdS 3/CF T 2 (2, 0), JHEP 06 (2018) 008 [arXiv:1712.07631] [INSPIRE].
J. Choi, J.J. Fernández-Melgarejo and S. Sugimoto, Deformation of \( \mathcal{N}=4 \) SYM with varying couplings via fluxes and intersecting branes, JHEP 03 (2018) 128 [arXiv:1801.09394] [INSPIRE].
R. Miranda, The basic theory of elliptic surfaces, Dottorato di Ricerca in Matematica, ETS Editrice, Pisa (1989).
C.P. Bachas, P. Bain and M.B. Green, Curvature terms in D-brane actions and their M-theory origin, JHEP 05 (1999) 011 [hep-th/9903210] [INSPIRE].
A. Kapustin and E. Witten, Electric-Magnetic Duality And The Geometric Langlands Program, Commun. Num. Theor. Phys. 1 (2007) 1 [hep-th/0604151] [INSPIRE].
K. Kodaira, On compact analytic surfaces: II, Ann. Math. 77 (1963) 563.
L.F. Alday, F. Benini and Y. Tachikawa, Liouville/Toda central charges from M5-branes, Phys. Rev. Lett. 105 (2010) 141601 [arXiv:0909.4776] [INSPIRE].
Y. Tachikawa, A review of the T N theory and its cousins, PTEP 2015 (2015) 11B102 [arXiv:1504.01481] [INSPIRE].
Y. Tachikawa and K. Yonekura, Anomalies involving the space of couplings and the Zamolodchikov metric, JHEP 12 (2017) 140 [arXiv:1710.03934] [INSPIRE].
N. Seiberg, Y. Tachikawa and K. Yonekura, Anomalies of Duality Groups and Extended Conformal Manifolds, PTEP 2018 (2018) 073B04 [arXiv:1803.07366] [INSPIRE].
I. Bah and E. Nardoni, Structure of Anomalies of 4d SCFTs from M5-branes and Anomaly Inflow, arXiv:1803.00136 [INSPIRE].
A. Gadde, S. Gukov and P. Putrov, Duality Defects, arXiv:1404.2929 [INSPIRE].
D. Morrison, What is F-theory?, to appear.
A.P. Ogg, On pencils of curves of genus two, Topology 5 (1966) 355.
Y. Namikawa and K. Ueno, The complete classification of fibres in pencils of curves of genus two, Manuscripta Math. 9 (1973) 143.
M.R. Gaberdiel and M.B. Green, An SL(2, Z) anomaly in IIB supergravity and its F-theory interpretation, JHEP 11 (1998) 026 [hep-th/9810153] [INSPIRE].
R. Minasian, S. Sasmal and R. Savelli, Discrete anomalies in supergravity and consistency of string backgrounds, JHEP 02 (2017) 025 [arXiv:1611.09575] [INSPIRE].
G. Festuccia and N. Seiberg, Rigid Supersymmetric Theories in Curved Superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [INSPIRE].
E. Bergshoeff, M. de Roo and B. de Wit, Extended Conformal Supergravity, Nucl. Phys. B 182 (1981) 173 [INSPIRE].
T. Pantev and E. Sharpe, Duality group actions on fermions, JHEP 11 (2016) 171 [arXiv:1609.00011] [INSPIRE].
K.A. Intriligator, Bonus symmetries of N = 4 superYang-Mills correlation functions via AdS duality, Nucl. Phys. B 551 (1999) 575 [hep-th/9811047] [INSPIRE].
K.A. Intriligator and W. Skiba, Bonus symmetry and the operator product expansion of N = 4 SuperYang-Mills, Nucl. Phys. B 559 (1999) 165 [hep-th/9905020] [INSPIRE].
A. Bilal, Lectures on Anomalies, arXiv:0802.0634 [INSPIRE].
T. Maxfield, Supergravity Backgrounds for Four-Dimensional Maximally Supersymmetric Yang-Mills, JHEP 02 (2017) 065 [arXiv:1609.05905] [INSPIRE].
K.A. Intriligator, Anomaly matching and a Hopf-Wess-Zumino term in 6d, \( \mathcal{N}=\left(2,\ 0\right) \) field theories, Nucl. Phys. B 581 (2000) 257 [hep-th/0001205] [INSPIRE].
K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, Anomaly polynomial of general 6d SCFTs, PTEP 2014 (2014) 103B07 [arXiv:1408.5572] [INSPIRE].
K. Intriligator, 6d, \( \mathcal{N}=\left(1,\ 0\right) \) Coulomb branch anomaly matching, JHEP 10 (2014) 162 [arXiv:1408.6745] [INSPIRE].
C. Montonen and D.I. Olive, Magnetic Monopoles as Gauge Particles?, Phys. Lett. B 72 (1977) 117 [INSPIRE].
H. Osborn, Topological Charges for N = 4 Supersymmetric Gauge Theories and Monopoles of Spin 1, Phys. Lett. B 83 (1979) 321 [INSPIRE].
E. Witten, Dyons of Charge eθ/2π, Phys. Lett. B 86 (1979) 283 [INSPIRE].
C. Vafa and E. Witten, A Strong coupling test of S duality, Nucl. Phys. B 431 (1994) 3 [hep-th/9408074] [INSPIRE].
E. Witten, On S duality in Abelian gauge theory, Selecta Math. 1 (1995) 383 [hep-th/9505186] [INSPIRE].
Y. Tachikawa and K. Yonekura, Why are fractional charges of orientifolds compatible with Dirac quantization?, arXiv:1805.02772 [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, NonAbelian tensor multiplet anomalies, JHEP 09 (1998) 004 [hep-th/9808060] [INSPIRE].
P. Yi, Anomaly of (2, 0) theories, Phys. Rev. D 64 (2001) 106006 [hep-th/0106165] [INSPIRE].
J. Fullwood, On elliptic fibrations and F-theory compactifications of string vacua, ProQuest LLC, Ann Arbor, MI (2012) [http://purl.flvc.org/fsu/fd/FSU_migr_etd-4848].
P. Aluffi and M. Esole, Chern class identities from tadpole matching in type IIB and F-theory, JHEP 03 (2009) 032 [arXiv:0710.2544] [INSPIRE].
P. Aluffi and M. Esole, New Orientifold Weak Coupling Limits in F-theory, JHEP 02 (2010) 020 [arXiv:0908.1572] [INSPIRE].
M. Esole, J. Fullwood and S.-T. Yau, D 5 elliptic fibrations: non-Kodaira fibers and new orientifold limits of F-theory, Commun. Num. Theor. Phys. 09 (2015) 583 [arXiv:1110.6177] [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].
C. Lawrie and S. Schäfer-Nameki, The Tate Form on Steroids: Resolution and Higher Codimension Fibers, JHEP 04 (2013) 061 [arXiv:1212.2949] [INSPIRE].
M. Kuntzler and C. Lawrie, Smooth: A Mathematica package for studying resolutions of singular fibrations, Version 0.4.
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. Katz, D.R. Morrison, S. Schäfer-Nameki and J. Sully, Tate’s algorithm and F-theory, JHEP 08 (2011) 094 [arXiv:1106.3854] [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].
S. Krause, C. Mayrhofer and T. Weigand, G 4 flux, chiral matter and singularity resolution in F-theory compactifications, Nucl. Phys. B 858 (2012) 1 [arXiv:1109.3454] [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.J. Heckman and T. Rudelius, Top Down Approach to 6D SCFTs, arXiv:1805.06467 [INSPIRE].
T. Weigand, TASI Lectures on F-theory, arXiv:1806.01854 [INSPIRE].
M. Bershadsky, A. Johansen, V. Sadov and C. Vafa, Topological reduction of 4d SYM to 2d σ-models, Nucl. Phys. B 448 (1995) 166 [hep-th/9501096] [INSPIRE].
F. Benini and N. Bobev, Exact two-dimensional superconformal R-symmetry and c-extremization, Phys. Rev. Lett. 110 (2013) 061601 [arXiv:1211.4030] [INSPIRE].
E. Witten, On the conformal field theory of the Higgs branch, JHEP 07 (1997) 003 [hep-th/9707093] [INSPIRE].
L. Álvarez-Gaumé and E. Witten, Gravitational Anomalies, Nucl. Phys. B 234 (1984) 269 [INSPIRE].
M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6d Conformal Matter, JHEP 02 (2015) 054 [arXiv:1407.6359] [INSPIRE].
K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, 6d \( \mathcal{N}=\left(1,\ 0\right) \) theories on T 2 and class S theories: Part I, JHEP 07 (2015) 014 [arXiv:1503.06217] [INSPIRE].
F. Apruzzi, F. Hassler, J.J. Heckman and I.V. Melnikov, From 6D SCFTs to Dynamic GLSMs, Phys. Rev. D 96 (2017) 066015 [arXiv:1610.00718] [INSPIRE].
Y. Imamura, H. Isono, K. Kimura and M. Yamazaki, Exactly marginal deformations of quiver gauge theories as seen from brane tilings, Prog. Theor. Phys. 117 (2007) 923 [hep-th/0702049] [INSPIRE].
B. Feng, A. Hanany and Y.-H. He, D-brane gauge theories from toric singularities and toric duality, Nucl. Phys. B 595 (2001) 165 [hep-th/0003085] [INSPIRE].
H. Kim and P. Yi, D-brane anomaly inflow revisited, JHEP 02 (2012) 012 [arXiv:1201.0762] [INSPIRE].
J.W. Milnor and J.D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo (1974).
P. Deligne, Courbes elliptiques: formulaire d’après J. Tate, in Modular functions of one variable, Springer, Berlin, Lect. Notes Math. 476 (1975) 53.
P. Aluffi, Chern classes of blow-ups, Math. Proc. Cambridge Philos. Soc. 148 (2010) 227.
C. Vafa, Black holes and Calabi-Yau threefolds, Adv. Theor. Math. Phys. 2 (1998) 207 [hep-th/9711067] [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
Corresponding author
Additional information
ArXiv ePrint: 1806.06066
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Lawrie, C., Martelli, D. & Schäfer-Nameki, S. Theories of class F and anomalies. J. High Energ. Phys. 2018, 90 (2018). https://doi.org/10.1007/JHEP10(2018)090
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2018)090