Abstract
We construct a supergeometry based on S 2×S 2 on which four dimensional \( \mathcal{N} \) = 2 gauge theories can be placed supersymmetrically while preserving all supersymmetries. By embedding the supergeometry in four dimensional \( \mathcal{N} \) = 2 supergravity we are able to construct an arbitrary \( \mathcal{N} \) = 2 gauge theory on S 2 × S 2. We show that \( \mathcal{N} \) = 2 gauge theories are invariant under the exceptional superalgebra D(2, 1, α), where α is the ratio of the radii of the two S 2’s. We solve the supersymmetry fixed points equations for a choice of supercharge in D(2, 1, α). The solution of these BPS equations, which we find, would serve as the exact saddle point configurations of a localization computation of the partition function of \( \mathcal{N} \) = 2 gauge theories on S 2 × S 2.
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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].
E. Gerchkovitz, J. Gomis and Z. Komargodski, Sphere partition functions and the Zamolodchikov metric, JHEP 11 (2014) 001 [arXiv:1405.7271] [INSPIRE].
J. Gomis and N. Ishtiaque, Kähler potential and ambiguities in 4D \( \mathcal{N} \) = 2 SCFTs, JHEP 04 (2015) 169 [arXiv:1409.5325] [INSPIRE].
N. Hama and K. Hosomichi, Seiberg-Witten theories on ellipsoids, JHEP 09 (2012) 033 [arXiv:1206.6359] [INSPIRE].
C. Klare and A. Zaffaroni, Extended supersymmetry on curved spaces, JHEP 10 (2013) 218 [arXiv:1308.1102] [INSPIRE].
A. Bawane, G. Bonelli, M. Ronzani and A. Tanzini, \( \mathcal{N} \) = 2 supersymmetric gauge theories on S 2 × S 2 and Liouville gravity, JHEP 07 (2015) 054 [arXiv:1411.2762] [INSPIRE].
F. Benini and S. Cremonesi, Partition functions of \( \mathcal{N} \) = (2, 2) gauge theories on S 2 and vortices, Commun. Math. Phys. 334 (2015) 1483 [arXiv:1206.2356] [INSPIRE].
N. Doroud, J. Gomis, B. Le Floch and S. Lee, Exact results in D = 2 supersymmetric gauge theories, JHEP 05 (2013) 093 [arXiv:1206.2606] [INSPIRE].
G. Festuccia and N. Seiberg, Rigid supersymmetric theories in curved superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [INSPIRE].
B. de Wit, J.W. van Holten and A. Van Proeyen, Structure of N = 2 supergravity, Nucl. Phys. B 184 (1981) 77 [Erratum ibid. B 222 (1983) 516] [INSPIRE].
D.Z. Freedman and A.V. Proeyen, Supergravity, Cambridge University Press, Cambridge U.K. (2012).
B. de Wit, J.W. van Holten and A. Van Proeyen, Transformation rules of N = 2 supergravity multiplets, Nucl. Phys. B 167 (1980) 186 [INSPIRE].
B. de Wit and J.W. van Holten, Multiplets of linearized SO(2) supergravity, Nucl. Phys. B 155 (1979) 530 [INSPIRE].
B. de Wit, R. Philippe and A. Van Proeyen, The Improved Tensor Multiplet in N = 2 Supergravity, Nucl. Phys. B 219 (1983) 143 [INSPIRE].
A.V. Proeyen, N = 2 supergravity in d = 4, 5, 6 and its matter couplings, http://itf.fys.kuleuven.be/∼toine/LectParis.pdf.
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
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ArXiv ePrint: 1411.4918
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Sinamuli, M. On \( \mathcal{N} \) = 2 supersymmetric gauge theories on S 2 × S 2 . J. High Energ. Phys. 2016, 62 (2016). https://doi.org/10.1007/JHEP05(2016)062
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DOI: https://doi.org/10.1007/JHEP05(2016)062