Abstract
We construct rigid supersymmetric gauge theories on Riemannian five-manifolds. We follow a holographic approach, realizing the manifold as the conformal boundary of a six-dimensional bulk supergravity solution. This leads to a systematic classification of five-dimensional supersymmetric backgrounds with gravity duals. We show that the background metric is furnished with a conformal Killing vector, which generates a transversely holomorphic foliation with a transverse Hermitian structure. Moreover, we prove that any such metric defines a supersymmetric background. Finally, we construct supersymmetric Lagrangians for gauge theories coupled to arbitrary matter on such backgrounds.
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J. Källén and M. Zabzine, Twisted supersymmetric 5D Yang-Mills theory and contact geometry, JHEP 05 (2012) 125 [arXiv:1202.1956] [INSPIRE].
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].
J. Källén, J. Qiu and M. Zabzine, The perturbative partition function of supersymmetric 5D Yang-Mills theory with matter on the five-sphere, JHEP 08 (2012) 157 [arXiv:1206.6008] [INSPIRE].
H.-C. Kim and S. Kim, M 5-branes from gauge theories on the 5-sphere, JHEP 05 (2013) 144 [arXiv:1206.6339] [INSPIRE].
H.-C. Kim, S.-S. Kim and K. Lee, 5-dim superconformal index with enhanced E n global symmetry, JHEP 10 (2012) 142 [arXiv:1206.6781] [INSPIRE].
S. Terashima, Supersymmetric gauge theories on S 4 × S 1, Phys. Rev. D 89 (2014) 125001 [arXiv:1207.2163] [INSPIRE].
Y. Imamura, Supersymmetric theories on squashed five-sphere, Prog. Theor. Exp. Phys. 2013 (2013) 013B04 [arXiv:1209.0561] [INSPIRE].
Y. Imamura, Perturbative partition function for squashed S 5, arXiv:1210.6308 [INSPIRE].
T. Kawano and N. Matsumiya, 5D SYM on 3D sphere and 2D YM, Phys. Lett. B 716 (2012) 450 [arXiv:1206.5966] [INSPIRE].
Y. Fukuda, T. Kawano and N. Matsumiya, 5D SYM and 2D q-deformed YM, Nucl. Phys. B 869 (2013) 493 [arXiv:1210.2855] [INSPIRE].
J. Yagi, 3d TQFT from 6d SCFT, JHEP 08 (2013) 017 [arXiv:1305.0291] [INSPIRE].
S. Lee and M. Yamazaki, 3d Chern-Simons theory from M 5-branes, JHEP 12 (2013) 035 [arXiv:1305.2429] [INSPIRE].
C. Cordova and D.L. Jafferis, Complex Chern-Simons from M 5-branes on the squashed three-sphere, arXiv:1305.2891 [INSPIRE].
G. Festuccia and N. Seiberg, Rigid supersymmetric theories in curved superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [INSPIRE].
M. Zucker, Minimal off-shell supergravity in five-dimensions, Nucl. Phys. B 570 (2000) 267 [hep-th/9907082] [INSPIRE].
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].
Y. Pan, Rigid supersymmetry on 5-dimensional Riemannian manifolds and contact geometry, JHEP 05 (2014) 041 [arXiv:1308.1567] [INSPIRE].
Y. Imamura and H. Matsuno, Supersymmetric backgrounds from 5d N = 1 supergravity, JHEP 07 (2014) 055 [arXiv:1404.0210] [INSPIRE].
Y. Pan, 5d Higgs branch localization, Seiberg-Witten equations and contact geometry, JHEP 01 (2015) 145 [arXiv:1406.5236] [INSPIRE].
S.M. Kuzenko, J. Novak and G. Tartaglino-Mazzucchelli, Symmetries of curved superspace in five dimensions, JHEP 10 (2014) 175 [arXiv:1406.0727] [INSPIRE].
J. Sparks, Sasaki-Einstein manifolds, Surveys Diff. Geom. 16 (2011) 265 [arXiv:1004.2461] [INSPIRE].
J. Qiu and M. Zabzine, 5D super Yang-Mills on Y p,q Sasaki-Einstein manifolds, Commun. Math. Phys. 333 (2015) 861 [arXiv:1307.3149] [INSPIRE].
D. Martelli, J. Sparks and S.-T. Yau, The geometric dual of a-maximisation for toric Sasaki-Einstein manifolds, Commun. Math. Phys. 268 (2006) 39 [hep-th/0503183] [INSPIRE].
J. Qiu and M. Zabzine, Factorization of 5D super Yang-Mills theory on Y p,q spaces, Phys. Rev. D 89 (2014) 065040 [arXiv:1312.3475] [INSPIRE].
J. Schmude, Localisation on Sasaki-Einstein manifolds from holomorphic functions on the cone, JHEP 01 (2015) 119 [arXiv:1401.3266] [INSPIRE].
J. Qiu, L. Tizzano, J. Winding and M. Zabzine, Gluing Nekrasov partition functions, Commun. Math. Phys. 337 (2015) 785 [arXiv:1403.2945] [INSPIRE].
J. Qiu and M. Zabzine, On twisted N = 2 5D super Yang-Mills theory, arXiv:1409.1058 [INSPIRE].
C. Klare, A. Tomasiello and A. Zaffaroni, Supersymmetry on curved spaces and holography, JHEP 08 (2012) 061 [arXiv:1205.1062] [INSPIRE].
L.J. Romans, The F (4) gauged supergravity in six-dimensions, Nucl. Phys. B 269 (1986) 691 [INSPIRE].
L.F. Alday, M. Fluder, P. Richmond and J. Sparks, Gravity dual of supersymmetric gauge theories on a squashed five-sphere, Phys. Rev. Lett. 113 (2014) 141601 [arXiv:1404.1925] [INSPIRE].
L.F. Alday, M. Fluder, C.M. Gregory, P. Richmond and J. Sparks, Supersymmetric gauge theories on squashed five-spheres and their gravity duals, JHEP 09 (2014) 067 [arXiv:1405.7194] [INSPIRE].
L.F. Alday, P. Richmond and J. Sparks, The holographic supersymmetric Rényi entropy in five dimensions, JHEP 02 (2015) 102 [arXiv:1410.0899] [INSPIRE].
N. Hama, T. Nishioka and T. Ugajin, Supersymmetric Rényi entropy in five dimensions, JHEP 12 (2014) 048 [arXiv:1410.2206] [INSPIRE].
C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, Supersymmetric field theories on three-manifolds, JHEP 05 (2013) 017 [arXiv:1212.3388] [INSPIRE].
L.F. Alday, D. Martelli, P. Richmond and J. Sparks, Localization on three-manifolds, JHEP 10 (2013) 095 [arXiv:1307.6848] [INSPIRE].
T.T. Dumitrescu, G. Festuccia and N. Seiberg, Exploring curved superspace, JHEP 08 (2012) 141 [arXiv:1205.1115] [INSPIRE].
C. Klare and A. Zaffaroni, Extended supersymmetry on curved spaces, JHEP 10 (2013) 218 [arXiv:1308.1102] [INSPIRE].
K. Hristov, A. Tomasiello and A. Zaffaroni, Supersymmetry on three-dimensional Lorentzian curved spaces and black hole holography, JHEP 05 (2013) 057 [arXiv:1302.5228] [INSPIRE].
D. Cassani, C. Klare, D. Martelli, A. Tomasiello and A. Zaffaroni, Supersymmetry in Lorentzian curved spaces and holography, Commun. Math. Phys. 327 (2014) 577 [arXiv:1207.2181] [INSPIRE].
A. Lischewski, Charged conformal Killing spinors, J. Math. Phys. 56 (2015) 013510 [arXiv:1403.2311] [INSPIRE].
C.P. Boyer and K. Galicki, Sasakian geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford U.K. (2007).
A. Haefliger and D. Sundararaman, Complexifications of transversely holomorphic foliations, Math. Ann. 272 (1985) 23.
A. Pini, D. Rodriguez-Gomez and J. Schmude, Rigid supersymmetry from conformal supergravity in five dimensions, arXiv:1504.04340 [INSPIRE].
J.P. Gauntlett, D. Martelli and D. Waldram, Superstrings with intrinsic torsion, Phys. Rev. D 69 (2004) 086002 [hep-th/0302158] [INSPIRE].
N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [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].
C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, The geometry of supersymmetric partition functions, JHEP 01 (2014) 124 [arXiv:1309.5876] [INSPIRE].
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Alday, L.F., Genolini, P.B., Fluder, M. et al. Supersymmetric gauge theories on five-manifolds. J. High Energ. Phys. 2015, 7 (2015). https://doi.org/10.1007/JHEP08(2015)007
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DOI: https://doi.org/10.1007/JHEP08(2015)007