Abstract
We study the 6d N = (0, 2) superconformal field theory, which describes multiple M5-branes, on the product space S 2 × M 4, and suggest a correspondence between a 2d N = (0, 2) half-twisted gauge theory on S 2 and a topological sigma-model on the four-manifold M 4. To set up this correspondence, we determine in this paper the dimensional reduction of the 6d N = (0, 2) theory on a two-sphere and derive that the four-dimensional theory is a sigma-model into the moduli space of solutions to Nahm’s equations, or equivalently the moduli space of k-centered SU(2) monopoles, where k is the number of M5-branes. We proceed in three steps: we reduce the 6d abelian theory to a 5d Super-Yang-Mills theory on I × M 4, with I an interval, then non-abelianize the 5d theory and finally reduce this to 4d. In the special case, when M 4 is a Hyper-Kähler manifold, we show that the dimensional reduction gives rise to a topological sigma-model based on tri-holomorphic maps. Deriving the theory on a general M 4 requires knowledge of the metric of the target space. For k = 2 the target space is the Atiyah-Hitchin manifold and we twist the theory to obtain a topological sigma-model, which has both scalar fields and self-dual two-forms.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P.S. Howe and E. Sezgin, D = 11, p = 5, Phys. Lett. B 394 (1997) 62 [hep-th/9611008] [INSPIRE].
P.S. Howe, E. Sezgin and P.C. West, Covariant field equations of the M-theory five-brane, Phys. Lett. B 399 (1997) 49 [hep-th/9702008] [INSPIRE].
W. Nahm, A simple formalism for the BPS monopole, Phys. Lett. B 90 (1980) 413 [INSPIRE].
L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, Gauge theories labelled by three-manifolds, Commun. Math. Phys. 325 (2014) 367 [arXiv:1108.4389] [INSPIRE].
Y. Terashima and M. Yamazaki, SL(2, R) Chern-Simons, Liouville and gauge theory on duality walls, JHEP 08 (2011) 135 [arXiv:1103.5748] [INSPIRE].
C. Cordova and D.L. Jafferis, Five-dimensional maximally supersymmetric Yang-Mills in supergravity backgrounds, arXiv:1305.2886 [INSPIRE].
C. Cordova and D.L. Jafferis, Complex Chern-Simons from M 5-branes on the squashed three-sphere, arXiv:1305.2891 [INSPIRE].
N. Seiberg, Notes on theories with 16 supercharges, Nucl. Phys. Proc. Suppl. 67 (1998) 158 [hep-th/9705117] [INSPIRE].
C. Vafa and E. Witten, A strong coupling test of S duality, Nucl. Phys. B 431 (1994) 3 [hep-th/9408074] [INSPIRE].
A. Gadde, S. Gukov and P. Putrov, Fivebranes and 4-manifolds, arXiv:1306.4320 [INSPIRE].
U. Gran, H. Linander and B.E.W. Nilsson, Off-shell structure of twisted (2, 0) theory, JHEP 11 (2014) 032 [arXiv:1406.4499] [INSPIRE].
E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].
J. Gomis, P.-S. Hsin, Z. Komargodski, A. Schwimmer, N. Seiberg and S. Theisen, Anomalies, conformal manifolds and spheres, JHEP 03 (2016) 022 [arXiv:1509.08511] [INSPIRE].
C. Closset, W. Gu, B. Jia and E. Sharpe, Localization of twisted N = (0, 2) gauged linear σ-models in two dimensions, JHEP 03 (2016) 070 [arXiv:1512.08058] [INSPIRE].
M. Bershadsky, C. Vafa and V. Sadov, D-branes and topological field theories, Nucl. Phys. B 463 (1996) 420 [hep-th/9511222] [INSPIRE].
J.P. Gauntlett, N. Kim and D. Waldram, M five-branes wrapped on supersymmetric cycles, Phys. Rev. D 63 (2001) 126001 [hep-th/0012195] [INSPIRE].
J.M. Maldacena and C. Núñez, Supergravity description of field theories on curved manifolds and a no go theorem, Int. J. Mod. Phys. A 16 (2001) 822 [hep-th/0007018] [INSPIRE].
E. Witten, Fivebranes and knots, arXiv:1101.3216 [INSPIRE].
D. Gaiotto, G.W. Moore and Y. Tachikawa, On 6d N = (2, 0) theory compactified on a Riemann surface with finite area, Prog. Theor. Exp. Phys. 2013 (2013) 013B03 [arXiv:1110.2657] [INSPIRE].
S.K. Donaldson, Nahm’s equations and the classification of monopoles, Commun. Math. Phys. 96 (1984) 387 [INSPIRE].
M. Atiyah and N. Hitchin, The geometry and dynamics of magnetic monopoles, M.B. Porter Lectures, Princeton University Press, Princeton U.S.A. (1988).
P.B. Kronheimer, A hyper-Kählerian structure on coadjoint orbits of a semisimple complex group, J. London Math. Soc. 42 (1990) 193.
R. Bielawski, Lie groups, Nahm’s equations and hyper-Kähler manifolds, in Algebraic groups, Universitätsverlag Göttingen, Göttingen Germany (2007), pg. 1.
R. Bielawski, Hyper-Kähler structures and group actions, J. London Math. Soc. 55 (1997) 400.
L. Álvarez-Gaumé and D.Z. Freedman, Geometrical structure and ultraviolet finiteness in the supersymmetric σ-model, Commun. Math. Phys. 80 (1981) 443 [INSPIRE].
J. Bagger and E. Witten, Matter couplings in N = 2 supergravity, Nucl. Phys. B 222 (1983) 1 [INSPIRE].
A. Kapustin and K. Vyas, A-models in three and four dimensions, arXiv:1002.4241 [INSPIRE].
D. Anselmi and P. Fré, Topological σ-models in four-dimensions and triholomorphic maps, Nucl. Phys. B 416 (1994) 255 [hep-th/9306080] [INSPIRE].
D. Joyce, Compact manifolds with special holonomy, Oxford mathematical monographs, Oxford University Press, Oxford U.K. (2000).
N. Marcus, The other topological twisting of N = 4 Yang-Mills, Nucl. Phys. B 452 (1995) 331 [hep-th/9506002] [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].
D. Bak and A. Gustavsson, Partially twisted superconformal M 5 brane in R-symmetry gauge field backgrounds, JHEP 12 (2015) 093 [arXiv:1508.04496] [INSPIRE].
A.S. Galperin, E.A. Ivanov, V.I. Ogievetsky and E.S. Sokatchev, Harmonic superspace, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge U.K. (2007).
T. Kugo and K. Ohashi, Supergravity tensor calculus in 5D from 6D, Prog. Theor. Phys. 104 (2000) 835 [hep-ph/0006231] [INSPIRE].
T. Kugo and K. Ohashi, Off-shell D = 5 supergravity coupled to matter Yang-Mills system, Prog. Theor. Phys. 105 (2001) 323 [hep-ph/0010288] [INSPIRE].
E. Bergshoeff, E. Sezgin and A. Van Proeyen, Superconformal tensor calculus and matter couplings in six-dimensions, Nucl. Phys. B 264 (1986) 653 [Erratum ibid. B 598 (2001) 667] [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].
F. Riccioni, Tensor multiplets in six-dimensional (2, 0) supergravity, Phys. Lett. B 422 (1998) 126 [hep-th/9712176] [INSPIRE].
N.J. Hitchin, Monopoles and geodesics, Commun. Math. Phys. 83 (1982) 579 [INSPIRE].
N.J. Hitchin, On the construction of monopoles, Commun. Math. Phys. 89 (1983) 145 [INSPIRE].
N.J. Hitchin, A. Karlhede, U. Lindström and M. Roček, Hyper-Kähler metrics and supersymmetry, Commun. Math. Phys. 108 (1987) 535 [INSPIRE].
A.S. Dancer, Nahm’s equations and hyper-Kähler geometry, Commun. Math. Phys. 158 (1993) 545 [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].
J.A. Harvey and A. Strominger, String theory and the Donaldson polynomial, Commun. Math. Phys. 151 (1993) 221 [hep-th/9108020] [INSPIRE].
J.P. Gauntlett, Low-energy dynamics of N = 2 supersymmetric monopoles, Nucl. Phys. B 411 (1994) 443 [hep-th/9305068] [INSPIRE].
D. Gaiotto and E. Witten, Supersymmetric boundary conditions in N = 4 super Yang-Mills theory, J. Statist. Phys. 135 (2009) 789 [arXiv:0804.2902] [INSPIRE].
D.Z. Freedman and A. Van Proeyen, Supergravity, Cambridge University Press, Cambridge U.K. (2012).
L. Anderson, Five-dimensional topologically twisted maximally supersymmetric Yang-Mills theory, JHEP 02 (2013) 131 [arXiv:1212.5019] [INSPIRE].
S. Salamon, Riemannian geometry and holonomy groups, in Pitman Research Notes in Mathematics Series 201, Longman Scientific & Technical, Harlow U.K. and John Wiley & Sons Inc., New York U.S.A. (1989).
G.W. Gibbons and N.S. Manton, Classical and quantum dynamics of BPS monopoles, Nucl. Phys. B 274 (1986) 183 [INSPIRE].
I.T. Ivanov and M. Roček, Supersymmetric σ-models, twistors and the Atiyah-Hitchin metric, Commun. Math. Phys. 182 (1996) 291 [hep-th/9512075] [INSPIRE].
N. Dorey, V.V. Khoze, M.P. Mattis, D. Tong and S. Vandoren, Instantons, three-dimensional gauge theory and the Atiyah-Hitchin manifold, Nucl. Phys. B 502 (1997) 59 [hep-th/9703228] [INSPIRE].
A. Hanany and B. Pioline, (Anti-)instantons and the Atiyah-Hitchin manifold, JHEP 07 (2000) 001 [hep-th/0005160] [INSPIRE].
S. Franco, D. Ghim, S. Lee, R.-K. Seong and D. Yokoyama, 2d (0, 2) quiver gauge theories and D-branes, JHEP 09 (2015) 072 [arXiv:1506.03818] [INSPIRE].
S. Schäfer-Nameki and T. Weigand, F-theory and 2d (0, 2) theories, JHEP 05 (2016) 059 [arXiv:1601.02015] [INSPIRE].
F. Apruzzi, F. Hassler, J.J. Heckman and I.V. Melnikov, UV completions for non-critical strings, JHEP 07 (2016) 045 [arXiv:1602.04221] [INSPIRE].
F. Benini and N. Bobev, Two-dimensional SCFTs from wrapped branes and c-extremization, JHEP 06 (2013) 005 [arXiv:1302.4451] [INSPIRE].
C. Closset, S. Cremonesi and D.S. Park, The equivariant A-twist and gauged linear σ-models on the two-sphere, JHEP 06 (2015) 076 [arXiv:1504.06308] [INSPIRE].
F. Benini and A. Zaffaroni, A topologically twisted index for three-dimensional supersymmetric theories, JHEP 07 (2015) 127 [arXiv:1504.03698] [INSPIRE].
K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].
S. Lee and M. Yamazaki, 3d Chern-Simons theory from M 5-branes, JHEP 12 (2013) 035 [arXiv:1305.2429] [INSPIRE].
J. Yagi, 3d TQFT from 6d SCFT, JHEP 08 (2013) 017 [arXiv:1305.0291] [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, 3-manifolds and 3d indices, Adv. Theor. Math. Phys. 17 (2013) 975 [arXiv:1112.5179] [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: 1604.03606
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. & Wong, JM. M5-branes on S 2 × M 4: Nahm’s equations and 4d topological sigma-models. J. High Energ. Phys. 2016, 120 (2016). https://doi.org/10.1007/JHEP09(2016)120
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2016)120