Abstract
We study the topologically twisted compactification of the 6d (2, 0) M5-brane theory on an elliptically fibered Kähler three-fold preserving two supercharges. We show that upon reducing on the elliptic fiber, the 4d theory is \( \mathcal{N} \) = 4 Super-Yang Mills, with varying complexified coupling τ , in the presence of defects. For abelian gauge group this agrees with the so-called duality twisted theory, and we determine a non-abelian generalization to U(N). When the elliptic fibration is singular, the 4d theory contains 3d walls (along the branch-cuts of τ ) and 2d surface defects, around which the 4d theory undergoes \( \mathrm{S}\mathrm{L}\left(2,\mathbb{Z}\right) \) duality transformations. Such duality defects carry chiral fields, which from the 6d point of view arise as modes of the two-form B in the tensor multiplet. Each duality defect has a flavor symmetry associated to it, which is encoded in the structure of the singular elliptic fiber above the defect. Generically 2d surface defects will intersect in points in 4d, where there is an enhanced flavor symmetry. The 6d point of view provides a complete characterization of this 4d-3d-2d-0d ‘Matroshka’-defect configuration.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Montonen and D.I. Olive, Magnetic Monopoles as Gauge Particles?, Phys. Lett. B 72 (1977) 117 [INSPIRE].
P. Goddard, J. Nuyts and D.I. Olive, Gauge Theories and Magnetic Charge, Nucl. Phys. B 125 (1977) 1 [INSPIRE].
C. Vafa and E. Witten, A strong coupling test of S duality, Nucl. Phys. B 431 (1994) 3 [hep-th/9408074] [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].
E. Witten, Geometric Langlands From Six Dimensions, arXiv:0905.2720 [INSPIRE].
C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [INSPIRE].
L. Martucci, Topological duality twist and brane instantons in F-theory, JHEP 06 (2014) 180 [arXiv:1403.2530] [INSPIRE].
K.A. Intriligator, Bonus symmetries of N = 4 super Yang-Mills correlation functions via AdS duality, Nucl. Phys. B 551 (1999) 575 [hep-th/9811047] [INSPIRE].
E.I. Buchbinder, J. Gomis and F. Passerini, Holographic gauge theories in background fields and surface operators, JHEP 12 (2007) 101 [arXiv:0710.5170] [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, G 4 flux, chiral matter and singularity resolution in F-theory compactifications, Nucl. Phys. B 858 (2012) 1 [arXiv:1109.3454] [INSPIRE].
T.W. Grimm and H. Hayashi, F-theory fluxes, Chirality and Chern-Simons theories, JHEP 03 (2012) 027 [arXiv:1111.1232] [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].
H. Linander and F. Ohlsson, (2,0) theory on circle fibrations, JHEP 01 (2012) 159 [arXiv:1111.6045] [INSPIRE].
F. Ohlsson, (2, 0) theory on Taub-NUT: A note on WZW models on singular fibrations, arXiv:1205.0694 [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].
C. Lawrie, S. Schäfer-Nameki and T. Weigand, Strings from Wrapped D3-branes in F-theory, to appear.
T. Maxfield, Supergravity Backgrounds for Four-Dimensional Maximally Supersymmetric Yang-Mills, arXiv:1609.05905 [INSPIRE].
A. Gadde, S. Gukov and P. Putrov, Duality Defects, arXiv:1404.2929 [INSPIRE].
E. Witten, Nonperturbative superpotentials in string theory, Nucl. Phys. B 474 (1996) 343 [hep-th/9604030] [INSPIRE].
O.J. Ganor, A Note on zeros of superpotentials in F-theory, Nucl. Phys. B 499 (1997) 55 [hep-th/9612077] [INSPIRE].
J.J. Heckman, J. Marsano, N. Saulina, S. Schäfer-Nameki and C. Vafa, Instantons and SUSY breaking in F-theory, arXiv:0808.1286 [INSPIRE].
J. Marsano, N. Saulina and S. Schäfer-Nameki, An Instanton Toolbox for F-theory Model Building, JHEP 01 (2010) 128 [arXiv:0808.2450] [INSPIRE].
M. Cvetič, I. Garcia Etxebarria and J. Halverson, Three Looks at Instantons in F-theory — New Insights from Anomaly Inflow, String Junctions and Heterotic Duality, JHEP 11 (2011) 101 [arXiv:1107.2388] [INSPIRE].
L. Martucci and T. Weigand, Hidden Selection Rules, M5-instantons and Fluxes in F-theory, JHEP 10 (2015) 131 [arXiv:1507.06999] [INSPIRE].
R. Blumenhagen, M. Cvetič, S. Kachru and T. Weigand, D-Brane Instantons in Type II Orientifolds, Ann. Rev. Nucl. Part. Sci. 59 (2009) 269 [arXiv:0902.3251] [INSPIRE].
L.B. Anderson, I. García-Etxebarria, T.W. Grimm and J. Keitel, Physics of F-theory compactifications without section, JHEP 12 (2014) 156 [arXiv:1406.5180] [INSPIRE].
A. Grassi, On minimal models of elliptic threefolds, Math. Ann. 290 (1991) 287.
M. Green, J. Schwarz and E. Witten, Superstring Theory: Volume 2, Loop Amplitudes, Anomalies and Phenomenology, Cambridge Monographs on Mathematical Physics, Cambridge University Press (1987).
E. Witten, Some comments on string dynamics, hep-th/9507121 [INSPIRE].
E. Bergshoeff, E. Sezgin and A. Van Proeyen, (2, 0) tensor multiplets and conformal supergravity in D = 6, Class. Quant. Grav. 16 (1999) 3193 [hep-th/9904085] [INSPIRE].
A. Kapustin and M. Tikhonov, Abelian duality, walls and boundary conditions in diverse dimensions, JHEP 11 (2009) 006 [arXiv:0904.0840] [INSPIRE].
E. Witten, Five-brane effective action in M-theory, J. Geom. Phys. 22 (1997) 103 [hep-th/9610234] [INSPIRE].
K. Kodaira, On compact analytic surfaces: II, Ann. Math. 77 (1963) 563.
T.W. Grimm, The N = 1 effective action of F-theory compactifications, Nucl. Phys. B 845 (2011) 48 [arXiv:1008.4133] [INSPIRE].
H. Hayashi, R. Tatar, Y. Toda, T. Watari and M. Yamazaki, New Aspects of Heterotic-F Theory Duality, Nucl. Phys. B 806 (2009) 224 [arXiv:0805.1057] [INSPIRE].
R. Donagi and M. Wijnholt, Higgs Bundles and UV Completion in F-theory, Commun. Math. Phys. 326 (2014) 287 [arXiv:0904.1218] [INSPIRE].
J. Marsano, N. Saulina and S. Schäfer-Nameki, Monodromies, Fluxes and Compact Three-Generation F-theory GUTs, JHEP 08 (2009) 046 [arXiv:0906.4672] [INSPIRE].
G.W. Gibbons and S.W. Hawking, Classification of Gravitational Instanton Symmetries, Commun. Math. Phys. 66 (1979) 291 [INSPIRE].
P.J. Ruback, The Motion of Kaluza-Klein Monopoles, Commun. Math. Phys. 107 (1986) 93 [INSPIRE].
A. Sen, Dynamics of multiple Kaluza-Klein monopoles in M and string theory, Adv. Theor. Math. Phys. 1 (1998) 115 [hep-th/9707042] [INSPIRE].
N. Lambert and H. Liu, Charged States in M-theory, unpublished.
E. Witten, Branes, Instantons, And Taub-NUT Spaces, JHEP 06 (2009) 067 [arXiv:0902.0948] [INSPIRE].
C. Lawrie, Coulomb Branch and non-Kodaira Fibers, to appear.
B. Assel, S. Schäfer-Nameki and J.-M. Wong, M5-branes on S 2 × M 4 : Nahm’s equations and 4d topological σ-models, JHEP 09 (2016) 120 [arXiv:1604.03606] [INSPIRE].
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
A. Van Proeyen, Tools for supersymmetry, Ann. U. Craiova Phys. 9 (1999) 1 [hep-th/9910030] [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
Additional information
ArXiv ePrint: 1610.03663
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Assel, B., Schäfer-Nameki, S. Six-dimensional origin of \( \mathcal{N} \) = 4 SYM with duality defects. J. High Energ. Phys. 2016, 58 (2016). https://doi.org/10.1007/JHEP12(2016)058
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2016)058