Abstract
For \( \mathcal{N} \)-extended superconformal field theories in three spacetime dimensions (3D), with 1 ≤ \( \mathcal{N} \) ≤ 3, we compute the two- and three-point correlation functions of the supercurrent and the flavour current multiplets. We demonstrate that supersymmetry imposes additional restrictions on the correlators of conserved currents as compared with the non-supersymmetric case studied by Osborn and Petkou in hep-th/9307010. It is shown that the three-point function of the supercurrent is determined by a single functional form consistent with the conservation equation and all the symmetry properties. Similarly, the three-point function of the flavour current multiplets is also determined by a single functional form in the \( \mathcal{N}=1 \) and \( \mathcal{N}=3 \) cases. The specific feature of the \( \mathcal{N}=2 \) case is that two independent structures are allowed for the three-point function of flavour current multiplets, but only one of them contributes to the three-point function of the conserved currents contained in these multiplets. Since the supergravity and super-Yang-Mills Ward identities are expected to relate the coefficients of the two- and three-point functions under consideration, the results obtained for 3D superconformal field theory are analogous to those in 2D conformal field theory.
In addition, we present a new supertwistor construction for compactified Minkowski superspace. It is suitable for developing superconformal field theory on 3D spacetimes other than Minkowski space, such as S 1 × S 2 and its universal covering space \( \mathrm{\mathbb{R}}\times {S}^2 \).
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References
A.M. Polyakov, Conformal symmetry of critical fluctuations, JETP Lett. 12 (1970) 381 [Pisma Zh. Eksp. Teor. Fiz. 12 (1970) 538] [INSPIRE].
E.J. Schreier, Conformal symmetry and three-point functions, Phys. Rev. D 3 (1971) 980 [INSPIRE].
A.A. Migdal, On hadronic interactions at small distances, Phys. Lett. B 37 (1971) 98 [INSPIRE].
A.A. Migdal, Conformal invariance and bootstrap, Phys. Lett. B 37 (1971) 386 [INSPIRE].
S. Ferrara, A.F. Grillo and R. Gatto, Manifestly conformal-covariant expansion on the light cone, Phys. Rev. D 5 (1972) 3102 [INSPIRE].
S. Ferrara, A.F. Grillo and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion, Annals Phys. 76 (1973) 161 [INSPIRE].
K. Koller, The significance of conformal inversion in quantum field theory, Commun. Math. Phys. 40 (1975) 15.
G. Mack, Convergence of operator product expansions on the vacuum in conformal invariant quantum field theory, Commun. Math. Phys. 53 (1977) 155 [INSPIRE].
I.T. Todorov, M.C. Mintchev and V.P. Petkova, Conformal Invariance in Quantum Field Theory, Scuola Normale Superiore, Pisa Italy (1978).
E.S. Fradkin and M.Y. Palchik, Recent developments in conformal invariant quantum field theory, Phys. Rept. 44 (1978) 249 [INSPIRE].
H. Osborn and A.C. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311 [hep-th/9307010] [INSPIRE].
Y.S. Stanev, Stress-energy tensor and U(1) current operator product expansions in conformal QFT, Bulg. J. Phys. 15 (1988) 93.
S. Giombi, S. Prakash and X. Yin, A note on CFT correlators in three dimensions, JHEP 07 (2013) 105 [arXiv:1104.4317] [INSPIRE].
S. Giombi et al., Chern-Simons Theory with Vector Fermion Matter, Eur. Phys. J. C 72 (2012) 2112 [arXiv:1110.4386] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
J. Erdmenger and H. Osborn, Conserved currents and the energy momentum tensor in conformally invariant theories for general dimensions, Nucl. Phys. B 483 (1997) 431 [hep-th/9605009] [INSPIRE].
S. Ferrara and B. Zumino, Transformation properties of the supercurrent, Nucl. Phys. B 87 (1975) 207 [INSPIRE].
V. Ogievetsky and E. Sokatchev, On vector superfield generated by supercurrent, Nucl. Phys. B 124 (1977) 309 [INSPIRE].
S. Ferrara and B. Zumino, Structure of conformal supergravity, Nucl. Phys. B 134 (1978) 301 [INSPIRE].
W. Siegel, A derivation of the supercurrent superfield, Harvard preprint HUTP-77/A089 (1977).
I.L. Buchbinder and S.M. Kuzenko, Ideas and Methods of Supersymmetry and Supergravity or a Walk Through Superspace, IOP, Bristol U.K. (1995).
M. Magro, I. Sachs and S. Wolf, Superfield Noether procedure, Annals Phys. 298 (2002) 123 [hep-th/0110131] [INSPIRE].
Z. Komargodski and N. Seiberg, Comments on supercurrent multiplets, supersymmetric field theories and supergravity, JHEP 07 (2010) 017 [arXiv:1002.2228] [INSPIRE].
S.M. Kuzenko, Variant supercurrent multiplets, JHEP 04 (2010) 022 [arXiv:1002.4932] [INSPIRE].
S.M. Kuzenko, Variant supercurrents and Noether procedure, Eur. Phys. J. C 71 (2011) 1513 [arXiv:1008.1877] [INSPIRE].
M.F. Sohnius, The multiplet of currents for N = 2 extended supersymmetry, Phys. Lett. B 81 (1979) 8 [INSPIRE].
P.S. Howe, K.S. Stelle and P.K. Townsend, Supercurrents, Nucl. Phys. B 192 (1981) 332 [INSPIRE].
S.M. Kuzenko and S. Theisen, Correlation functions of conserved currents in N = 2 superconformal theory, Class. Quant. Grav. 17 (2000) 665 [hep-th/9907107] [INSPIRE].
D. Butter and S.M. Kuzenko, N = 2 supergravity and supercurrents, JHEP 12 (2010) 080 [arXiv:1011.0339] [INSPIRE].
H. Osborn, N = 1 superconformal symmetry in four-dimensional quantum field theory, Annals Phys. 272 (1999) 243 [hep-th/9808041] [INSPIRE].
L. Bonora, P. Pasti and M. Tonin, Cohomologies and anomalies in supersymmetric theories, Nucl. Phys. B 252 (1985) 458 [INSPIRE].
I.L. Buchbinder and S.M. Kuzenko, Matter superfields in external supergravity: Green functions, effective action and superconformal anomalies, Nucl. Phys. B 274 (1986) 653 [INSPIRE].
M.F. Sohnius, The conformal group in superspace, in Proceedings of Feldafing 1976, in Quantum Theory and the Structure of Time and Space. Vol. 2, L. Castell, M. Drieschner and C.F. von Weizsäcker eds., Carl Hanser Verlag, München Germany (1977), pg. 241.
W. Lang, Currents in Supersymmetric Gauge Theories, Nucl. Phys. B 150 (1979) 201 [INSPIRE].
W. Lang, Construction of the minimal superspace translation tensor and the derivation of the supercurrent, Nucl. Phys. B 179 (1981) 106 [INSPIRE].
K.-i. Shizuya, Supercurrents and superconformal symmetry, Phys. Rev. D 35 (1987) 1848 [INSPIRE].
S.M. Kuzenko, J. Novak and G. Tartaglino-Mazzucchelli, N = 6 superconformal gravity in three dimensions from superspace, JHEP 01 (2014) 121 [arXiv:1308.5552] [INSPIRE].
D. Butter, S.M. Kuzenko, J. Novak and G. Tartaglino-Mazzucchelli, Conformal supergravity in three dimensions: new off-shell formulation, JHEP 09 (2013) 072 [arXiv:1305.3132] [INSPIRE].
T.T. Dumitrescu and N. Seiberg, Supercurrents and brane currents in diverse dimensions, JHEP 07 (2011) 095 [arXiv:1106.0031] [INSPIRE].
S.M. Kuzenko and G. Tartaglino-Mazzucchelli, Three-dimensional N = 2 (AdS) supergravity and associated supercurrents, JHEP 12 (2011) 052 [arXiv:1109.0496] [INSPIRE].
S.J. Gates, 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].
B.M. Zupnik and D.G. Pak, Superfield formulation of the simplest three-dimensional gauge theories and conformal supergravities, Theor. Math. Phys. 77 (1988) 1070 [INSPIRE].
S.M. Kuzenko, Prepotentials for N = 2 conformal supergravity in three dimensions, JHEP 12 (2012) 021 [arXiv:1209.3894] [INSPIRE].
A.S. Galperin, E.A. Ivanov, V.I. Ogievetsky and E.S. Sokatchev, Harmonic Superspace, Cambridge University Press, Cambridge U.K. (2001).
W. Siegel, Unextended superfields in extended supersymmetry, Nucl. Phys. B 156 (1979) 135 [INSPIRE].
N.J. Hitchin, A. Karlhede, U. Lindström and M. Roček, HyperKähler metrics and supersymmetry, Commun. Math. Phys. 108 (1987) 535 [INSPIRE].
B.M. Zupnik and D.V. Khetselius, Three-dimensional extended supersymmetry in the harmonic superspace (in Russian), Sov. J. Nucl. Phys. 47 (1988) 730 [INSPIRE].
B. Zupnik, Harmonic superpotentials and symmetries in gauge theories with eight supercharges, Nucl. Phys. B 554 (1999) 365 [Erratum ibid. B 644 (2002) 405] [hep-th/9902038] [INSPIRE].
B.M. Zupnik, Three-dimensional N = 4 superconformal superfield theories, Theor. Math. Phys. 162 (2010) 74 [arXiv:0905.1179] [INSPIRE].
J.-H. Park, Superconformal symmetry in three-dimensions, J. Math. Phys. 41 (2000) 7129 [hep-th/9910199] [INSPIRE].
J.-H. Park, Superconformal symmetry and correlation functions, Nucl. Phys. B 559 (1999) 455 [hep-th/9903230] [INSPIRE].
J.-H. Park, Superconformal symmetry in six-dimensions and its reduction to four-dimensions, Nucl. Phys. B 539 (1999) 599 [hep-th/9807186] [INSPIRE].
S.M. Kuzenko, J.-H. Park, G. Tartaglino-Mazzucchelli and R. Unge, Off-shell superconformal nonlinear σ-models in three dimensions, JHEP 01 (2011) 146 [arXiv:1011.5727] [INSPIRE].
C. Closset, T.T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, Contact terms, unitarity and F-maximization in three-dimensional superconformal theories, JHEP 10 (2012) 053 [arXiv:1205.4142] [INSPIRE].
C. Closset, T.T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, Comments on Chern-Simons contact terms in three dimensions, JHEP 09 (2012) 091 [arXiv:1206.5218] [INSPIRE].
J.-H. Park, N = 1 superconformal symmetry in four-dimensions, Int. J. Mod. Phys. A 13 (1998) 1743 [hep-th/9703191] [INSPIRE].
S.M. Kuzenko, On compactified harmonic/projective superspace, 5D superconformal theories and all that, Nucl. Phys. B 745 (2006) 176 [hep-th/0601177] [INSPIRE].
S.M. Kuzenko, Conformally compactified Minkowski superspaces revisited, JHEP 10 (2012) 135 [arXiv:1206.3940] [INSPIRE].
S.M. Kuzenko and D. Sorokin, Superconformal structures on the three-sphere, JHEP 1410 (2014) 80 [arXiv:1406.7090] [INSPIRE].
A.A. Rosly, Gauge fields in superspace and twistors, Class. Quant. Grav. 2 (1985) 693 [INSPIRE].
J. Lukierski and A. Nowicki, General superspaces from supertwistors, Phys. Lett. B 211 (1988) 276 [INSPIRE].
P.S. Howe and G.G. Hartwell, A superspace survey, Class. Quant. Grav. 12 (1995) 1823 [INSPIRE].
S.M. Kuzenko, Variant supercurrents and Noether procedure, Eur. Phys. J. C 71 (2011) 1513 [arXiv:1008.1877] [INSPIRE].
D.V. Volkov and V.P. Akulov, Possible universal neutrino interaction, JETP Lett. 16 (1972) 438 [Pisma Zh. Eksp. Teor. Fiz. 16 (1972) 621] [INSPIRE].
D.V. Volkov and V.P. Akulov, Is the neutrino a Goldstone particle?, Phys. Lett. B 46 (1973) 109 [INSPIRE].
V.P. Akulov and D.V. Volkov, Goldstone fields with spin 1/2, Theor. Math. Phys. 18 (1974) 28 [INSPIRE].
S.M. Kuzenko and G. Tartaglino-Mazzucchelli, Conformal supergravities as Chern-Simons theories revisited, JHEP 03 (2013) 113 [arXiv:1212.6852] [INSPIRE].
H. Nicolai, E. Sezgin and Y. Tanii, Conformally invariant supersymmetric field theories on S p × S 1 and super p-branes, Nucl. Phys. B 305 (1988) 483 [INSPIRE].
E. Sezgin and Y. Tanii, Superconformal σ-models in higher than two-dimensions, Nucl. Phys. B 443 (1995) 70 [hep-th/9412163] [INSPIRE].
E. Bergshoeff, S. Cecotti, H. Samtleben and E. Sezgin, Superconformal σ-models in three dimensions, Nucl. Phys. B 838 (2010) 266 [arXiv:1002.4411] [INSPIRE].
I.L. Buchbinder, N.G. Pletnev and I.B. Samsonov, Effective action of three-dimensional extended supersymmetric matter on gauge superfield background, JHEP 04 (2010) 124 [arXiv:1003.4806] [INSPIRE].
A. Karlhede, U. Lindström and M. Roček, Selfinteracting tensor multiplets in N = 2 superspace, Phys. Lett. B 147 (1984) 297 [INSPIRE].
U. Lindström and M. Roček, New HyperKähler metrics and new supermultiplets, Commun. Math. Phys. 115 (1988) 21 [INSPIRE].
U. Lindström and M. Roček, N = 2 super Yang-Mills theory in projective superspace, Commun. Math. Phys. 128 (1990) 191 [INSPIRE].
S.M. Kuzenko, Lectures on nonlinear σ-models in projective superspace, J. Phys. A 43 (2010) 443001 [arXiv:1004.0880] [INSPIRE].
B.M. Zupnik, Harmonic superspaces for three-dimensional theories, Lect. Notes Phys. 524 (1999) 116 [hep-th/9804167] [INSPIRE].
A. Galperin, E. Ivanov, S. Kalitsyn, V. Ogievetsky and E. Sokatchev, Unconstrained N = 2 matter, Yang-Mills and supergravity theories in harmonic superspace, Class. Quant. Grav. 1 (1984) 469 [INSPIRE].
I.L. Buchbinder, E.A. Ivanov, O. Lechtenfeld, N.G. Pletnev, I.B. Samsonov B.M. Zupnik, ABJM models in N = 3 harmonic superspace, JHEP 03 (2009) 096 [arXiv:0811.4774] [INSPIRE].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
M.F. Sohnius, Supersymmetry and central charges, Nucl. Phys. B 138 (1978) 109 [INSPIRE].
A.A. Nizami, T. Sharma and V. Umesh, Superspace formulation and correlation functions of 3d superconformal field theories, JHEP 07 (2014) 022 [arXiv:1308.4778] [INSPIRE].
I. Florakis, D. Sorokin and M. Tsulaia, Higher spins in hyper-superspace, Nucl. Phys. B 890 (2014) 279 [arXiv:1408.6675] [INSPIRE].
W.D. Goldberger, W. Skiba and M. Son, Superembedding methods for 4d N = 1 SCFTs, Phys. Rev. D 86 (2012) 025019 [arXiv:1112.0325] [INSPIRE].
M. Maio, Superembedding methods for 4d N-extended SCFTs, Nucl. Phys. B 864 (2012) 141 [arXiv:1205.0389] [INSPIRE].
W.D. Goldberger, Z.U. Khandker, D. Li and W. Skiba, Superembedding methods for current superfields, Phys. Rev. D 88 (2013) 125010 [arXiv:1211.3713] [INSPIRE].
W. Siegel, Green-Schwarz formulation of selfdual superstring, Phys. Rev. D 47 (1993) 2512 [hep-th/9210008] [INSPIRE].
W. Siegel, Super multi-instantons in conformal chiral superspace, Phys. Rev. D 52 (1995) 1042 [hep-th/9412011] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, Z.U. Khandker, D. Li, D. Poland and D. Simmons-Duffin, Covariant Approaches to Superconformal Blocks, JHEP 08 (2014) 129 [arXiv:1402.1167] [INSPIRE].
Z.U. Khandker, D. Li, D. Poland and D. Simmons-Duffin, \( \mathcal{N}=1 \) superconformal blocks for general scalar operators, JHEP 08 (2014) 049 [arXiv:1404.5300] [INSPIRE].
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ArXiv ePrint: 1503.04961
On leave from Tomsk Polytechnic University, 634050 Tomsk, Russia. (Igor B. Samsonov)
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Buchbinder, E.I., Kuzenko, S.M. & Samsonov, I.B. Superconformal field theory in three dimensions: correlation functions of conserved currents. J. High Energ. Phys. 2015, 138 (2015). https://doi.org/10.1007/JHEP06(2015)138
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DOI: https://doi.org/10.1007/JHEP06(2015)138