Abstract
We study supersymmetry of \( \mathcal{N}=4 \) super Yang-Mills theory in four dimensions deformed in the Ω-background. We take the Nekrasov-Shatashvili limit of the background so that two-dimensional super Poincaré symmetry is recovered. We compute the deformed central charge of the superalgebra and study the 1/2 and 1/4 BPS states. We obtain the Ω-deformed 1/2 and 1/4 BPS dyon equations from the deformed supersymmetry transformation and the Bogomol’nyi completion of the energy.
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References
G.W. Moore, N. Nekrasov and S. Shatashvili, Integrating over Higgs branches, Commun. Math. Phys. 209 (2000) 97 [hep-th/9712241] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [INSPIRE].
A.S. Losev, A. Marshakov and N.A. Nekrasov, Small instantons, little strings and free fermions, in From fields to strings, volume 1, M. Shifman et al. eds., pg. 581–621, [hep-th/0302191] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, hep-th/0306238 [INSPIRE].
K. Ito, H. Nakajima, T. Saka and S. Sasaki, N = 2 Instanton Effective Action in Ω-background and D3/D(-1)-brane System in RR Background, JHEP 11 (2010) 093 [arXiv:1009.1212] [INSPIRE].
K. Ito, H. Nakajima, T. Saka and S. Sasaki, N = 4 Instanton Calculus in Omega and RR Backgrounds, Nucl. Phys. B 860 (2012) 267 [arXiv:1111.6709] [INSPIRE].
K. Ito, H. Nakajima and S. Sasaki, Torsion and Supersymmetry in Omega-background, JHEP 11 (2012) 132 [arXiv:1209.2561] [INSPIRE].
S. Hellerman, D. Orlando and S. Reffert, String theory of the Omega deformation, JHEP 01 (2012) 148 [arXiv:1106.0279] [INSPIRE].
S. Hellerman, D. Orlando and S. Reffert, The Omega Deformation From String and M-theory, JHEP 07 (2012) 061 [arXiv:1204.4192] [INSPIRE].
S. Hellerman, D. Orlando and S. Reffert, BPS States in the Duality Web of the Omega deformation, JHEP 06 (2013) 047 [arXiv:1210.7805] [INSPIRE].
S. Shadchin, On F-term contribution to effective action, JHEP 08 (2007) 052 [hep-th/0611278] [INSPIRE].
N. Nekrasov, Instanton partition functions and M-theory, Japan J. Math. 4 (2009) 63.
T. Dimofte, S. Gukov and L. Hollands, Vortex Counting and Lagrangian 3-manifolds, Lett. Math. Phys. 98 (2011) 225 [arXiv:1006.0977] [INSPIRE].
S. Reffert, General Omega Deformations from Closed String Backgrounds, JHEP 04 (2012) 059 [arXiv:1108.0644] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, arXiv:0908.4052 [INSPIRE].
N. Nekrasov and E. Witten, The Omega Deformation, Branes, Integrability and Liouville Theory, JHEP 09 (2010) 092 [arXiv:1002.0888] [INSPIRE].
K. Bulycheva, H.-Y. Chen, A. Gorsky and P. Koroteev, BPS States in Omega Background and Integrability, JHEP 10 (2012) 116 [arXiv:1207.0460] [INSPIRE].
K. Ito, S. Kamoshita and S. Sasaki, BPS Monopole Equation in Omega-background, JHEP 04 (2011) 023 [arXiv:1103.2589] [INSPIRE].
K. Ito, S. Kamoshita and S. Sasaki, Deformed BPS Monopole in Ω-background, Phys. Lett. B 710 (2012) 240 [arXiv:1110.1455] [INSPIRE].
P. Forgacs, Z. Horvath and L. Palla, Generating the BPS One Monopole by a Backlund Transformation, Phys. Rev. Lett. 45 (1980) 505 [INSPIRE].
D. Tong and C. Turner, Vortices and Monopoles in a Harmonic Trap, arXiv:1508.01207 [INSPIRE].
K. Ito, H. Nakajima and S. Sasaki, Twisted \( \mathcal{N}=4 \) Super Yang-Mills Theory in Ω-background, JHEP 10 (2013) 080 [arXiv:1307.7565] [INSPIRE].
J.P. Yamron, Topological Actions From Twisted Supersymmetric Theories, Phys. Lett. B 213 (1988) 325 [INSPIRE].
E. Witten, Topological Quantum Field Theory, Commun. Math. Phys. 117 (1988) 353 [INSPIRE].
C. Vafa and E. Witten, A strong coupling test of S duality, Nucl. Phys. B 431 (1994) 3 [hep-th/9408074] [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].
O. Bergman, Three pronged strings and 1/4 BPS states in N = 4 super Yang-Mills theory, Nucl. Phys. B 525 (1998) 104 [hep-th/9712211] [INSPIRE].
K. Hashimoto, H. Hata and N. Sasakura, 3-String junction and BPS saturated solutions in SU(3) supersymmetric Yang-Mills theory, Phys. Lett. B 431 (1998) 303 [hep-th/9803127] [INSPIRE].
T. Kawano and K. Okuyama, String network and 1/4 BPS states in N = 4 SU(N) supersymmetric Yang-Mills theory, Phys. Lett. B 432 (1998) 338 [hep-th/9804139] [INSPIRE].
O. Bergman and B. Kol, String webs and 1/4 BPS monopoles, Nucl. Phys. B 536 (1998) 149 [hep-th/9804160] [INSPIRE].
K. Hashimoto, H. Hata and N. Sasakura, Multipronged strings and BPS saturated solutions in SU(N) supersymmetric Yang-Mills theory, Nucl. Phys. B 535 (1998) 83 [hep-th/9804164] [INSPIRE].
K.-M. Lee and P. Yi, Dyons in N = 4 supersymmetric theories and three pronged strings, Phys. Rev. D 58 (1998) 066005 [hep-th/9804174] [INSPIRE].
C.J. Houghton and K.-M. Lee, Nahm data and the mass of 1/4 BPS states, Phys. Rev. D 61 (2000) 106001 [hep-th/9909218] [INSPIRE].
H. Osborn, Topological Charges for N = 4 Supersymmetric Gauge Theories and Monopoles of Spin 1, Phys. Lett. B 83 (1979) 321 [INSPIRE].
S. Yokoyama, Supersymmetry Algebra in Super Yang-Mills Theories, JHEP 09 (2015) 211 [arXiv:1506.03522] [INSPIRE].
C. Fraser and T.J. Hollowood, Semiclassical quantization in N = 4 supersymmetric Yang-Mills theory and duality, Phys. Lett. B 402 (1997) 106 [hep-th/9704011] [INSPIRE].
E.J. Weinberg and P. Yi, Magnetic Monopole Dynamics, Supersymmetry and Duality, Phys. Rept. 438 (2007) 65 [hep-th/0609055] [INSPIRE].
W. Nahm, A Simple Formalism for the BPS Monopole, Phys. Lett. B 90 (1980) 413 [INSPIRE].
N. Lambert, D. Orlando and S. Reffert, Omega-Deformed Seiberg-Witten Effective Action from the M5-brane, Phys. Lett. B 723 (2013) 229 [arXiv:1304.3488] [INSPIRE].
N. Lambert, D. Orlando and S. Reffert, Alpha- and Omega-Deformations from fluxes in M-theory, JHEP 11 (2014) 162 [arXiv:1409.1219] [INSPIRE].
M. Billó, M. Frau, F. Fucito, A. Lerda and J.F. Morales, S-duality and the prepotential in \( \mathcal{N}={2}^{\star } \) theories (I): the ADE algebras, JHEP 11 (2015) 024 [arXiv:1507.07709] [INSPIRE].
M. Billó, M. Frau, F. Fucito, A. Lerda and J.F. Morales, S-duality and the prepotential of \( \mathcal{N}={2}^{\star } \) theories (II): the non-simply laced algebras, JHEP 11 (2015) 026 [arXiv:1507.08027] [INSPIRE].
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Ito, K., Kanayama, Y., Nakajima, H. et al. BPS equations in Ω-deformed \( \mathcal{N}=4 \) super Yang-Mills theory. J. High Energ. Phys. 2015, 152 (2015). https://doi.org/10.1007/JHEP11(2015)152
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DOI: https://doi.org/10.1007/JHEP11(2015)152