Abstract
Abelian 6d (2,0) theory has SO(5) R symmetry. We twist this theory by identifying the R symmetry group with the SO(5) subgroup of the SO(1,5) Lorentz group. This twisted theory can be put on any five-manifold M, times R, while preserving one scalar supercharge. We subsequently assume the existence of one unit normalized Killing vector field on M, and we find a corresponding SO(4) twist that preserves two supercharges and is a generalization of the geometric Langlands twist of 4d SYM. We generalize the story to non-Abelian gauge group for the corresponding 5d SYM theories on M. We derive a vanishing theorem for BPS contact instantons by identifying the 6d potential energy and its BPS bound, in the 5d theory. To this end we need to perform a Wick rotation that complexifies the gauge field.
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References
K. Hosomichi, R.-K. Seong and S. Terashima, Supersymmetric gauge theories on the five-sphere, Nucl. Phys. B 865 (2012) 376 [arXiv:1203.0371] [INSPIRE].
H.-C. Kim, J. Kim and S. Kim, Instantons on the 5-sphere and M5-branes, arXiv:1211.0144 [INSPIRE].
G. Lockhart and C. Vafa, Superconformal partition functions and non-perturbative topological strings, arXiv:1210.5909 [INSPIRE].
J.A. Minahan, A. Nedelin and M. Zabzine, 5D super Yang-Mills theory and the correspondence to AdS7/CFT6, J. Phys. A 46 (2013) 355401 [arXiv:1304.1016] [INSPIRE].
A. Gustavsson, Euclidean quantum M5 brane theory on S1 × S5, J. Phys. A 48 (2015) 265402 [arXiv:1501.06977] [INSPIRE].
M. Bershadsky, C. Vafa and V. Sadov, D-branes and topological field theories, Nucl. Phys. B 463 (1996) 420 [hep-th/9511222] [INSPIRE].
C. Imbimbo and D. Rosa, Topological anomalies for Seifert 3-manifolds, arXiv:1411.6635 [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].
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].
E. Witten, Fivebranes and knots, arXiv:1101.3216 [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].
L. Anderson, Five-dimensional topologically twisted maximally supersymmetric Yang-Mills theory, JHEP 02 (2013) 131 [arXiv:1212.5019] [INSPIRE].
J. Qiu and M. Zabzine, On twisted N = 2 5D super Yang-Mills theory, arXiv:1409.1058 [INSPIRE].
J. Källén and M. Zabzine, Twisted supersymmetric 5D Yang-Mills theory and contact geometry, JHEP 05 (2012) 125 [arXiv:1202.1956] [INSPIRE].
C.M. Hull and N. Lambert, Emergent time and the M5-brane, JHEP 06 (2014) 016 [arXiv:1403.4532] [INSPIRE].
E. Witten, Quantization of Chern-Simons gauge theory with complex gauge group, Commun. Math. Phys. 137 (1991) 29 [INSPIRE].
E. Witten, A new look at the path integral of quantum mechanics, arXiv:1009.6032 [INSPIRE].
E. Witten, Geometric langlands from six dimensions, arXiv:0905.2720 [INSPIRE].
H. Linander and F. Ohlsson, (2,0) theory on circle fibrations, JHEP 01 (2012) 159 [arXiv:1111.6045] [INSPIRE].
C. Cordova and D.L. Jafferis, Complex Chern-Simons from M5-branes on the squashed three-sphere, arXiv:1305.2891 [INSPIRE].
S. Lee and M. Yamazaki, 3D Chern-Simons theory from M5-branes, JHEP 12 (2013) 035 [arXiv:1305.2429] [INSPIRE].
J. Yagi, 3D TQFT from 6D SCFT, JHEP 08 (2013) 017 [arXiv:1305.0291] [INSPIRE].
K.-M. Lee and H.-U. Yee, BPS string webs in the 6-dim (2, 0) theories, JHEP 03 (2007) 057 [hep-th/0606150] [INSPIRE].
M. Ünsal, Twisted supersymmetric gauge theories and orbifold lattices, JHEP 10 (2006) 089 [hep-th/0603046] [INSPIRE].
H.-C. Kim, S. Kim, S.-S. Kim and K. Lee, The general M5-brane superconformal index, arXiv:1307.7660 [INSPIRE].
H.-C. Kim and K. Lee, Supersymmetric M5 brane theories on R × CP2, JHEP 07 (2013) 072 [arXiv:1210.0853] [INSPIRE].
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ArXiv ePrint: 1504.00099
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Bak, D., Gustavsson, A. The geometric Langlands twist in five and six dimensions. J. High Energ. Phys. 2015, 13 (2015). https://doi.org/10.1007/JHEP07(2015)013
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DOI: https://doi.org/10.1007/JHEP07(2015)013