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Weyl versus conformal invariance in quantum field theory. (English) Zbl 1383.81209
Summary: We argue that conformal invariance in flat spacetime implies Weyl invariance in a general curved background metric for all unitary theories in spacetime dimensions \(d \leq 10 \). We also study possible curvature corrections to the Weyl transformations of operators, and show that these are absent for operators of sufficiently low dimensionality and spin. We find possible ‘anomalous’ Weyl transformations proportional to the Weyl (Cotton) tensor for \(d > 3\) (\(d=3\)). The arguments are based on algebraic consistency conditions similar to the Wess-Zumino consistency conditions that classify possible local anomalies. The arguments can be straightforwardly extended to larger operator dimensions and higher \(d\) with additional algebraic complexity.

81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
53Z05 Applications of differential geometry to physics
81T50 Anomalies in quantum field theory
Full Text: DOI
[1] Gross, DJ; Wess, J., Scale invariance, conformal invariance and the high-energy behavior of scattering amplitudes, Phys. Rev., D 2, 753, (1970)
[2] Callan, CG; Coleman, SR; Jackiw, R., A new improved energy-momentum tensor, Annals Phys., 59, 42, (1970) · Zbl 1092.83502
[3] Coleman, SR; Jackiw, R., Why dilatation generators do not generate dilatations?, Annals Phys., 67, 552, (1971)
[4] Iorio, A.; O’Raifeartaigh, L.; Sachs, I.; Wiesendanger, C., Weyl gauging and conformal invariance, Nucl. Phys., B 495, 433, (1997) · Zbl 0935.83026
[5] Cosme, C.; Lopes, JMVP; Penedones, J., Conformal symmetry of the critical 3D Ising model inside a sphere, JHEP, 08, 022, (2015)
[6] Kos, F.; Poland, D.; Simmons-Duffin, D.; Vichi, A., Precision islands in the Ising and O(N) models, JHEP, 08, 036, (2016) · Zbl 1390.81227
[7] Polchinski, J., Scale and conformal invariance in quantum field theory, Nucl. Phys., B 303, 226, (1988)
[8] Luty, MA; Polchinski, J.; Rattazzi, R., The a-theorem and the asymptotics of 4D quantum field theory, JHEP, 01, 152, (2013) · Zbl 1342.81347
[9] Dymarsky, A.; Komargodski, Z.; Schwimmer, A.; Theisen, S., On scale and conformal invariance in four dimensions, JHEP, 10, 171, (2015) · Zbl 1388.81372
[10] Dymarsky, A.; Farnsworth, K.; Komargodski, Z.; Luty, MA; Prilepina, V., Scale invariance, conformality and generalized free fields, JHEP, 02, 099, (2016) · Zbl 1388.81807
[11] K. Yonekura, Unitarity, Locality and Scale versus Conformal Invariance in Four Dimensions, arXiv:1403.4939 [INSPIRE].
[12] Jack, I.; Osborn, H., Analogs for the c-theorem for four-dimensional renormalizable field theories, Nucl. Phys., B 343, 647, (1990)
[13] Osborn, H., Weyl consistency conditions and a local renormalization group equation for general renormalizable field theories, Nucl. Phys., B 363, 486, (1991)
[14] Fortin, J-F; Grinstein, B.; Stergiou, A., Limit cycles and conformal invariance, JHEP, 01, 184, (2013) · Zbl 1342.81488
[15] Baume, F.; Keren-Zur, B.; Rattazzi, R.; Vitale, L., The local callan-symanzik equation: structure and applications, JHEP, 08, 152, (2014) · Zbl 1333.81307
[16] Stergiou, A.; Stone, D.; Vitale, LG, Constraints on perturbative RG flows in six dimensions, JHEP, 08, 010, (2016) · Zbl 1390.81543
[17] Elvang, H.; Freedman, DZ; Hung, L-Y; Kiermaier, M.; Myers, RC; Theisen, S., On renormalization group flows and the a-theorem in 6d, JHEP, 10, 011, (2012)
[18] Cordova, C.; Dumitrescu, TT; Intriligator, K., Anomalies, renormalization group flows and the a-theorem in six-dimensional (1, 0) theories, JHEP, 10, 080, (2016) · Zbl 1390.81369
[19] Nakayama, Y., Scale invariance vs conformal invariance, Phys. Rept., 569, 1, (2015)
[20] C. Graham, Conformally Invariant Powers of the Laplacian II: Nonexistence, J. London Math. Soc.s2-46 (1992) 566.
[21] Eastwood, MG, Higher symmetries of the Laplacian, Annals Math., 161, 1645, (2005) · Zbl 1091.53020
[22] A.R. Gover and K. Hirachi, Conformally invariant powers of the Laplacian: A Complete non-existence theorem, J. Am. Math. Soc.17 (2004) 389 [math.DG/0304082] [INSPIRE]. · Zbl 1066.53037
[23] G.K. Karananas and A. Monin, Weyl vs. Conformal, Phys. Lett.B 757 (2016) 257 [arXiv:1510.08042] [INSPIRE]. · Zbl 1360.81267
[24] Brust, C.; Hinterbichler, K., Free □\^{}{k} scalar conformal field theory, JHEP, 02, 066, (2017) · Zbl 1377.81158
[25] Jackiw, R., Weyl symmetry and the Liouville theory, Theor. Math. Phys., 148, 941, (2006) · Zbl 1177.81088
[26] A. Edery and Y. Nakayama, Restricted Weyl invariance in four-dimensional curved spacetime, Phys. Rev.D 90 (2014) 043007 [arXiv:1406.0060] [INSPIRE].
[27] G.K. Karananas and A. Monin, Weyl and Ricci gauging from the coset construction, Phys. Rev.D 93 (2016) 064013 [arXiv:1510.07589] [INSPIRE]. · Zbl 1360.81267
[28] M. Picco, Critical behavior of the Ising model with long range interactions, arXiv:1207.1018 [INSPIRE].
[29] S. El-Showk, Y. Nakayama and S. Rychkov, What Maxwell Theory in D ≠ 4 teaches us about scale and conformal invariance, Nucl. Phys.B 848 (2011) 578 [arXiv:1101.5385] [INSPIRE]. · Zbl 1215.78006
[30] Wess, J., The conformal invariance in quantum field theory, Nuovo Cim., 18, 1086, (1960) · Zbl 0094.42601
[31] Mack, G.; Salam, A., Finite component field representations of the conformal group, Annals Phys., 53, 174, (1969)
[32] Mack, G., All unitary ray representations of the conformal group SU(2,2) with positive energy, Commun. Math. Phys., 55, 1, (1977) · Zbl 0352.22012
[33] Minwalla, S., Restrictions imposed by superconformal invariance on quantum field theories, Adv. Theor. Math. Phys., 2, 781, (1998) · Zbl 1041.81534
[34] Capper, DM; Duff, MJ, Trace anomalies in dimensional regularization, Nuovo Cim., A 23, 173, (1974)
[35] S. Deser, M.J. Duff and C.J. Isham, Nonlocal Conformal Anomalies, Nucl. Phys.B 111 (1976) 45 [INSPIRE]. · Zbl 0967.81529
[36] Deser, S.; Schwimmer, A., Geometric classification of conformal anomalies in arbitrary dimensions, Phys. Lett., B 309, 279, (1993)
[37] Wess, J.; Zumino, B., Consequences of anomalous Ward identities, Phys. Lett., B 37, 95, (1971)
[38] Bardeen, WA; Zumino, B., Consistent and covariant anomalies in gauge and gravitational theories, Nucl. Phys., B 244, 421, (1984)
[39] Cappelli, A.; Coste, A., On the stress tensor of conformal field theories in higher dimensions, Nucl. Phys., B 314, 707, (1989)
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