Abstract
We discuss consistency at the quantum level in the rigid \( \mathcal{N} \) = 1 supersymmetric field theories with a U(1)R symmetry in four-dimensional curved space which are formulated via coupling to the new-minimal supergravity background fields. By analyzing correlation functions of the current operators in the \( \mathrm{\mathcal{R}} \)-multiplet, we show that the quantum consistency with the (unbroken) supersymmetry requires the U(1)R anomaly coefficient, which depends only on the field content of the theory, to vanish. This consistency condition is obtained under the assumption that the supercurrent Ward identity is non-anomalous and that the vacuum is supersymmetric.
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References
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [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.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
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. Klare, A. Tomasiello and A. Zaffaroni, Supersymmetry on Curved Spaces and Holography, JHEP 08 (2012) 061 [arXiv:1205.1062] [INSPIRE].
T.T. Dumitrescu, An introduction to supersymmetric field theories in curved space, J. Phys. A 50 (2017) 443005 [arXiv:1608.02957] [INSPIRE].
M.F. Sohnius and P.C. West, An Alternative Minimal Off-Shell Version of N = 1 Supergravity, Phys. Lett. B 105 (1981) 353 [INSPIRE].
M. Sohnius and P.C. West, The Tensor Calculus and Matter Coupling of the Alternative Minimal Auxiliary Field Formulation of N = 1 Supergravity, Nucl. Phys. B 198 (1982) 493 [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].
B. Assel, D. Cassani and D. Martelli, Localization on Hopf surfaces, JHEP 08 (2014) 123 [arXiv:1405.5144] [INSPIRE].
D. Cassani and D. Martelli, Supersymmetry on curved spaces and superconformal anomalies, JHEP 10 (2013) 025 [arXiv:1307.6567] [INSPIRE].
B. Assel, D. Cassani, L. Di Pietro, Z. Komargodski, J. Lorenzen and D. Martelli, The Casimir Energy in Curved Space and its Supersymmetric Counterpart, JHEP 07 (2015) 043 [arXiv:1503.05537] [INSPIRE].
D. Cassani and D. Martelli, The gravity dual of supersymmetric gauge theories on a squashed S 1 × S 3, JHEP 08 (2014) 044 [arXiv:1402.2278] [INSPIRE].
S.J. Gates Jr., M.T. Grisaru and W. Siegel, Auxiliary Field Anomalies, Nucl. Phys. B 203 (1982) 189 [INSPIRE].
S.J. Gates Jr., M.T. Grisaru, M. Roček and W. Siegel, Superspace Or One Thousand and One Lessons in Supersymmetry, Front. Phys. 58 (1983) 1 [hep-th/0108200] [INSPIRE].
P. Benetti Genolini, D. Cassani, D. Martelli and J. Sparks, The holographic supersymmetric Casimir energy, Phys. Rev. D 95 (2017) 021902 [arXiv:1606.02724] [INSPIRE].
P. Benetti Genolini, D. Cassani, D. Martelli and J. Sparks, Holographic renormalization and supersymmetry, JHEP 02 (2017) 132 [arXiv:1612.06761] [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].
H. Osborn, Weyl consistency conditions and a local renormalization group equation for general renormalizable field theories, Nucl. Phys. B 363 (1991) 486 [INSPIRE].
I. Papadimitriou, Lectures on Holographic Renormalization, Springer Proc. Phys. 176 (2016) 131 [INSPIRE].
K. Jensen, R. Loganayagam and A. Yarom, Thermodynamics, gravitational anomalies and cones, JHEP 02 (2013) 088 [arXiv:1207.5824] [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York U.S.A. (1997).
I. Papadimitriou, Holographic renormalization as a canonical transformation, JHEP 11 (2010) 014 [arXiv:1007.4592] [INSPIRE].
S.M. Christensen and M.J. Duff, Axial and Conformal Anomalies for Arbitrary Spin in Gravity and Supergravity, Phys. Lett. B 76 (1978) 571 [INSPIRE].
D. Anselmi, J. Erlich, D.Z. Freedman and A.A. Johansen, Positivity constraints on anomalies in supersymmetric gauge theories, Phys. Rev. D 57 (1998) 7570 [hep-th/9711035] [INSPIRE].
I. Papadimitriou, Supercurrent anomalies in 4d SCFTs, JHEP 07 (2017) 038 [arXiv:1703.04299] [INSPIRE].
O.S. An, Anomaly-corrected supersymmetry algebra and supersymmetric holographic renormalization, JHEP 12 (2017) 107 [arXiv:1703.09607] [INSPIRE].
M. Henningson and K. Skenderis, The Holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].
V. Balasubramanian and P. Kraus, A Stress tensor for Anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [INSPIRE].
J. de Boer, E.P. Verlinde and H.L. Verlinde, On the holographic renormalization group, JHEP 08 (2000) 003 [hep-th/9912012] [INSPIRE].
P. Kraus, F. Larsen and R. Siebelink, The gravitational action in asymptotically AdS and flat space-times, Nucl. Phys. B 563 (1999) 259 [hep-th/9906127] [INSPIRE].
S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].
M. Bianchi, D.Z. Freedman and K. Skenderis, How to go with an RG flow, JHEP 08 (2001) 041 [hep-th/0105276] [INSPIRE].
M. Bianchi, D.Z. Freedman and K. Skenderis, Holographic renormalization, Nucl. Phys. B 631 (2002) 159 [hep-th/0112119] [INSPIRE].
D. Martelli and W. Mueck, Holographic renormalization and Ward identities with the Hamilton-Jacobi method, Nucl. Phys. B 654 (2003) 248 [hep-th/0205061] [INSPIRE].
K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].
I. Papadimitriou and K. Skenderis, AdS/CFT correspondence and geometry, IRMA Lect. Math. Theor. Phys. 8 (2005) 73 [hep-th/0404176] [INSPIRE].
O.S. An, Y.H. Ko and S.-H. Won, Super-Weyl Anomaly from Holography and Rigid Supersymmetry Algebra on Two-Sphere, arXiv:1812.10209 [INSPIRE].
G. Katsianis, I. Papadimitriou, K. Skenderis and M. Taylor, Anomalous Supersymmetry, arXiv:1902.06715 [INSPIRE].
I. Papadimitriou, Supersymmetry anomalies in \( \mathcal{N} \) = 1 conformal supergravity, JHEP 04 (2019) 040 [arXiv:1902.06717] [INSPIRE].
D.Z. Freedman and A. Van Proeyen, Supergravity, Cambridge University Press, Cambridge U.K. (2012).
S.P. Martin, A Supersymmetry primer, Adv. Ser. Direct. High Energy Phys. 21 (2010) 1 [Adv. Ser. Direct. High Energy Phys. 18 (1998) 1] [hep-ph/9709356] [INSPIRE].
J. Wess and J. Bagger, Supersymmetry and supergravity, Princeton University Press, Princeton U.S.A. (1992).
E. Gerchkovitz, Constraints on the R-charges of free bound states from the Römelsberger index, JHEP 07 (2014) 071 [arXiv:1311.0487] [INSPIRE].
L. Álvarez-Gaumé and E. Witten, Gravitational Anomalies, Nucl. Phys. B 234 (1984) 269 [INSPIRE].
A. Cappelli and A. Coste, On the Stress Tensor of Conformal Field Theories in Higher Dimensions, Nucl. Phys. B 314 (1989) 707 [INSPIRE].
L.S. Brown and J.P. Cassidy, Stress Tensors and their Trace Anomalies in Conformally Flat Space-Times, Phys. Rev. D 16 (1977) 1712 [INSPIRE].
E.S. Fradkin and A.A. Tseytlin, Conformal Anomaly in Weyl Theory and Anomaly Free Superconformal Theories, Phys. Lett. B 134 (1984) 187 [INSPIRE].
B. Assel, D. Cassani and D. Martelli, Supersymmetric counterterms from new minimal supergravity, JHEP 11 (2014) 135 [arXiv:1410.6487] [INSPIRE].
D. Anselmi, D.Z. Freedman, M.T. Grisaru and A.A. Johansen, Nonperturbative formulas for central functions of supersymmetric gauge theories, Nucl. Phys. B 526 (1998) 543 [hep-th/9708042] [INSPIRE].
I. Papadimitriou, Holographic Renormalization of general dilaton-axion gravity, JHEP 08 (2011) 119 [arXiv:1106.4826] [INSPIRE].
R. Blumenhagen, D. Lüst and S. Theisen, Basic concepts of string theory, Theoretical and Mathematical Physics, Springer (2013).
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An, O.S., Kang, J.U., Kim, J.C. et al. Quantum consistency in supersymmetric theories with R-symmetry in curved space. J. High Energ. Phys. 2019, 146 (2019). https://doi.org/10.1007/JHEP05(2019)146
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DOI: https://doi.org/10.1007/JHEP05(2019)146