Abstract
In this paper we study 7D maximally supersymmetric Yang-Mills on a specific 3-Sasakian manifold that is the total space of an SO(3)-bundle over ℂP2. The novelty of this example is that the manifold is not a toric Sasaki-Einstein manifold. The hyperkähler cone of this manifold is a Swann bundle with hypertoric symmetry and this allows us to calculate the perturbative part of the partition function of the theory. The result is also verified by an index calculation. We also discuss a factorisation of this result and compare it with analogous results for S7.
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References
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
V. Pestun et al., Localization techniques in quantum field theories, J. Phys. A 50 (2017) 440301 [arXiv:1608.02952] [INSPIRE].
V. Pestun and M. Zabzine, Introduction to localization in quantum field theory, J. Phys. A 50 (2017) 443001 [arXiv:1608.02953] [INSPIRE].
B. Willett, Localization on three-dimensional manifolds, J. Phys. A 50 (2017) 443006.
G. Festuccia and N. Seiberg, Rigid supersymmetric theories in curved superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [INSPIRE].
J. Qiu and M. Zabzine, Review of localization for 5d supersymmetric gauge theories, J. Phys. A 50 (2017) 443014 [arXiv:1608.02966] [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.A. Minahan and M. Zabzine, Gauge theories with 16 supersymmetries on spheres, JHEP 03 (2015) 155 [arXiv:1502.07154] [INSPIRE].
K. Polydorou, A. Rocén and M. Zabzine, 7D supersymmetric Yang-Mills on curved manifolds, JHEP 12 (2017) 152 [arXiv:1710.09653] [INSPIRE].
J. Kallen, J.A. Minahan, A. Nedelin and M. Zabzine, N 3 -behavior from 5D Yang-Mills theory, JHEP 10 (2012) 184 [arXiv:1207.3763] [INSPIRE].
A. Losev, G.W. Moore and S.L. Shatashvili, M & m’s, Nucl. Phys. B 522 (1998) 105 [hep-th/9707250] [INSPIRE].
W. Nahm, Supersymmetries and their representations, Nucl. Phys. B 135 (1978) 149.
M. Blau, Killing spinors and SYM on curved spaces, JHEP 011 (2000) 023.
J. Schmude, Localisation on Sasaki-Einstein manifolds from holomorphic functions on the cone, JHEP 01 (2015) 119 [arXiv:1401.3266] [INSPIRE].
M.F. Atiyah, Elliptic operators and compact groups, Springer, Berlin Germany (1974).
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].
C.P. Boyer and K. Galicki, 3-Sasakian manifolds, Surveys Diff. Geom. 7 (1999) 123 [hep-th/9810250] [INSPIRE].
A. Swann, Hyperkähler and quaternionic Kähler geometry, Math. Ann. 289 (1991) 421.
N. Hitchin, Kählerian twistor spaces, Proc. London Math. Soc. 3 (1981) 133.
S. Salamon, Quaternionic Kähler manifolds, Inv. Math. 67 (1982) 143.
C.P. Boyer and K. Galicki, The twistor space of a 3-Sasakian manifold, Int. J. Math. 8 (1997) 31.
C.P. Boyer, K. Galicki and B.M. Mann, Quaternionic reduction and Einstein manifolds, Commun. Anal. Geom. 1 (1993) 229.
M. Konishi, On manifolds with sasakian 3-structure over quaternion Kähler manifolds, Kodai Mathematical Seminar Reports volume 26, Tokyo Institute of Technology, Tokyo (1975).
E. Calabi, Métriques kählériennes et fibrés holomorphes, Ann. Sci. École Norm. S. 12 (1979) 269.
R. Bielawski and A.S. Dancer, The geometry and topology of toric hyper-Kähler manifolds, Comm. Anal. Geom. 8 (2000) 727.
N. Proudfoot, A survey of hypertoric geometry and topology, Toric Topol. 460 (2008) 323, arXiv:0705.4236.
H. Konno, The geometry of toric hyper-Kähler varieties, Toric Topol., arXiv:0709.1252.
J. De Boer, K. Hori, H. Ooguri and Y. Oz, Mirror symmetry in three-dimensional gauge theories, quivers and d-branes, Nucl. Phys. B 493 (1997) 101.
T. Hausel and B. Sturmfels, Toric hyper-Kähler varieties, Doc. Math. 7 (2002) 495 [math/0203096].
R. Bielawski, Complete hyper-Kähler 4n-manifolds with a local tri-Hamiltonian ℝn -action, Math. Ann. 314 (1999) 505.
M. Bullimore, T. Dimofte, D. Gaiotto and J. Hilburn, Boundaries, mirror symmetry and symplectic duality in 3D \( \mathcal{N}=4 \) gauge theory, JHEP 10 (2016) 108 [arXiv:1603.08382] [INSPIRE].
N.A. Nekrasov, Instanton partition functions and M-theory, in the proceedings of the 15th International Seminar on High Energy Physics (Quarks 2008), May 23–29, Sergiev Posad, Russia (2008).
J. Winding, Multiple elliptic gamma functions associated to cones, Adv. Math. 325 (2018) 56 [arXiv:1609.02384] [INSPIRE].
A. Narukawa, The modular properties and the integral representations of the multiple elliptic gamma functions, Adv. Math. 189 (2004) 247.
D. Martelli, J. Sparks and S.-T. Yau, Sasaki-Einstein manifolds and volume minimisation, Commun. Math. Phys. 280 (2008) 611 [hep-th/0603021] [INSPIRE].
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Rocén, A. 7D supersymmetric Yang-Mills on a 3-Sasakian manifold. J. High Energ. Phys. 2018, 24 (2018). https://doi.org/10.1007/JHEP11(2018)024
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DOI: https://doi.org/10.1007/JHEP11(2018)024