Abstract
We study the equations governing rigid \( \mathcal{N}=1 \) supersymmetry in five dimensions. If the supersymmetry spinor satisfies a reality condition, these are foliations admitting families of almost complex structures on the leaves. In other words, all these manifolds have families of almost Cauchy-Riemann (CR) structures. After deriving integrability conditions under which circumstances the almost CR structure defines a CR manifold or a transversally holomorphic foliation (THF), we discuss implications on localization. We also discuss potential global obstructions to the existence of solutions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. Festuccia and N. Seiberg, Rigid supersymmetric theories in curved superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [INSPIRE].
T.T. Dumitrescu, G. Festuccia and N. Seiberg, Exploring curved superspace, JHEP 08 (2012) 141 [arXiv:1205.1115] [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].
C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, The geometry of supersymmetric partition functions, JHEP 01 (2014) 124 [arXiv:1309.5876] [INSPIRE].
C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, From rigid supersymmetry to twisted holomorphic theories, Phys. Rev. D 90 (2014) 085006 [arXiv:1407.2598] [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].
C. Klare, A. Tomasiello and A. Zaffaroni, Supersymmetry on curved spaces and holography, JHEP 08 (2012) 061 [arXiv:1205.1062] [INSPIRE].
C. Klare and A. Zaffaroni, Extended supersymmetry on curved spaces, JHEP 10 (2013) 218 [arXiv:1308.1102] [INSPIRE].
J.T. Liu, L.A. Pando Zayas and D. Reichmann, Rigid supersymmetric backgrounds of minimal off-shell supergravity, JHEP 10 (2012) 034 [arXiv:1207.2785] [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 and M. Zabzine, Twisted supersymmetric 5D Yang-Mills theory and contact geometry, JHEP 05 (2012) 125 [arXiv:1202.1956] [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].
J. Qiu and M. Zabzine, On twisted N = 2 5D super Yang-Mills theory, arXiv:1409.1058 [INSPIRE].
Y. Imamura and H. Matsuno, Supersymmetric backgrounds from 5d \( \mathcal{N}=1 \) supergravity, JHEP 07 (2014) 055 [arXiv:1404.0210] [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].
M. Zucker, Minimal off-shell supergravity in five-dimensions, Nucl. Phys. B 570 (2000) 267 [hep-th/9907082] [INSPIRE].
Y. Pan, Rigid supersymmetry on 5-dimensional Riemannian manifolds and contact geometry, JHEP 05 (2014) 041 [arXiv:1308.1567] [INSPIRE].
C. Cordova and D.L. Jafferis, Five-dimensional maximally supersymmetric Yang-Mills in supergravity backgrounds, arXiv:1305.2886 [INSPIRE].
S.M. Kuzenko and G. Tartaglino-Mazzucchelli, Super-Weyl invariance in 5D supergravity, JHEP 04 (2008) 032 [arXiv:0802.3953] [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].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
J. Schmude, Localisation on Sasaki-Einstein manifolds from holomorphic functions on the cone, JHEP 01 (2015) 119 [arXiv:1401.3266] [INSPIRE].
Y. Pan, 5d Higgs branch localization, Seiberg-Witten equations and contact geometry, JHEP 01 (2015) 145 [arXiv:1406.5236] [INSPIRE].
S. Dragomir and G. Tomassini, Differential geometry and analysis on CR manifolds, Progress in Mathematics, Birkhäuser (2007).
I. Biswas, Transversely projective structures on a transversely holomorphic foliation, Conform. Geom. Dyn. 5 (2001) 74.
H. Jacobowitz, Transversely holomorphic foliations and CR structures, in VI Workshop on Partial Differential Equations, Rio de Janeiro Brazil (1999) [Mat. Contemp. 18 (2000) 175].
L.F. Alday, P.B. Genolini, M. Fluder, P. Richmond and J. Sparks, Supersymmetric gauge theories on five-manifolds, JHEP 08 (2015) 007 [arXiv:1503.09090] [INSPIRE].
D. Rodriguez-Gomez and J. Schmude, Partition functions for equivariantly twisted \( \mathcal{N}=2 \) gauge theories on toric Kähler manifolds, JHEP 05 (2015) 111 [arXiv:1412.4407] [INSPIRE].
E. Witten, Topological quantum field theory, Commun. Math. Phys. 117 (1988) 353 [INSPIRE].
A. Karlhede and M. Roček, Topological quantum field theory and N = 2 conformal supergravity, Phys. Lett. B 212 (1988) 51 [INSPIRE].
E. Witten, Fivebranes and knots, arXiv:1101.3216 [INSPIRE].
L. Anderson, Five-dimensional topologically twisted maximally supersymmetric Yang-Mills theory, JHEP 02 (2013) 131 [arXiv:1212.5019] [INSPIRE].
D. Rodriguez-Gomez and J. Schmude, Supersymmetrizing 5d instanton operators, JHEP 03 (2015) 114 [arXiv:1501.00927] [INSPIRE].
Y. Imamura, Perturbative partition function for squashed S 5, arXiv:1210.6308 [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].
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: 1504.00321
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
Pan, Y., Schmude, J. On rigid supersymmetry and notions of holomorphy in five dimensions. J. High Energ. Phys. 2015, 41 (2015). https://doi.org/10.1007/JHEP11(2015)041
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2015)041