Weyl quadratic gravity as a gauge theory and non-metricity vs torsion duality (2312.13384v4)
Abstract: We review (non-supersymmetric) gauge theories of four-dimensional space-time symmetries and their quadratic action. The only true gauge theory of such a symmetry (with a physical gauge boson) that has an exact geometric interpretation, generates Einstein gravity in its spontaneously broken phase and is anomaly-free, is that of Weyl gauge symmetry (of dilatations). Gauging the full conformal group does not generate a true gauge theory of physical (dynamical) associated gauge bosons. Regarding the Weyl gauge symmetry, it is naturally realised in Weyl conformal geometry, where it admits two different but equivalent geometric formulations, of same quadratic action: one non-metric but torsion-free, the other Weyl gauge-covariant and metric (with respect to a new differential operator). To clarify the origin of this intriguing result, a third equivalent formulation of this gauge symmetry is constructed using the standard, modern approach on the tangent space (uplifted to space-time by the vielbein), which is metric but has vectorial torsion. This shows an interesting duality vectorial non-metricity vs vectorial torsion of the corresponding formulations, related by a projective transformation. We comment on the physical meaning of these results.
- D. Z. Freedman and A. Van Proeyen, “Supergravity,” Cambridge Univ. Press, 2012, ISBN 978-1-139-36806-3, 978-0-521-19401-3 doi:10.1017/CBO9781139026833;
- Hermann Weyl “Eine neue Erweiterung der Relativitätstheorie” (“A new extension of the theory of relativity”), Ann. Phys. (Leipzig) (4) 59 (1919), 101-133;
- E. Scholz, “The unexpected resurgence of Weyl geometry in late 20-th century physics,” Einstein Stud. 14 (2018), 261-360 doi:10.1007/978-1-4939-7708-6_11;
- D. M. Ghilencea, “Spontaneous breaking of Weyl quadratic gravity to Einstein action and Higgs potential,” JHEP 1903 (2019) 049 doi:10.1007/JHEP03(2019)049 [arXiv:1812.08613 [hep-th]]; D. M. Ghilencea, “Stueckelberg breaking of Weyl conformal geometry and applications to gravity,” Phys. Rev. D 101 (2020) no.4, 045010 doi:10.1103/PhysRevD.101.045010 [arXiv:1904.06596]; See also Section 2.1 in [6];
- D. M. Ghilencea, “Standard Model in Weyl conformal geometry,” Eur. Phys. J. C 82 (2022) no.1, 23 doi:10.1140/epjc/s10052-021-09887-y [arXiv:2104.15118 [hep-ph]];
- D. M. Ghilencea, “Weyl conformal geometry vs Weyl anomaly,” JHEP 10 (2023), 113 doi:10.1007/JHEP10(2023)113 [arXiv:2309.11372 [hep-th]];
- P. G. Ferreira, C. T. Hill, J. Noller and G. G. Ross, “Scale-independent R2superscript𝑅2R^{2}italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT inflation,” Phys. Rev. D 100 (2019) no.12, 123516 doi:10.1103/PhysRevD.100.123516 [arXiv:1906.03415 [gr-qc]];
- D. M. Ghilencea, “Weyl R22{}^{2}start_FLOATSUPERSCRIPT 2 end_FLOATSUPERSCRIPT inflation with an emergent Planck scale,” JHEP 10 (2019), 209 doi:10.1007/JHEP10(2019)209 [arXiv:1906.11572 [gr-qc]];
- A. A. Starobinsky, “A New Type of Isotropic Cosmological Models Without Singularity,” Phys. Lett. B 91 (1980), 99-102 doi:10.1016/0370-2693(80)90670-X;
- M. Crăciun and T. Harko, “Testing Weyl geometric gravity with the SPARC galactic rotation curves database,” [arXiv:2311.16893 [gr-qc]]; P. Burikham, T. Harko, K. Pimsamarn and S. Shahidi, “Dark matter as a Weyl geometric effect,” Phys. Rev. D 107 (2023) no.6, 064008 doi:10.1103/PhysRevD.107.064008 [arXiv:2302.08289 [gr-qc]];
- M. Kaku, P. K. Townsend and P. van Nieuwenhuizen, “Gauge Theory of the Conformal and Superconformal Group,” Phys. Lett. B 69 (1977), 304-308; J. T. Wheeler, “The Auxiliary field in conformal gauge theory,” Phys. Rev. D 44 (1991), 1769-1773 doi:10.1103/PhysRevD.44.1769;
- I. L. Buchbinder, I. L. Shapiro, “Introduction to QFT with applications to Quantum Gravity”, Oxford University Press, Oxford 2021;
- W. Drechsler and H. Tann, “Broken Weyl invariance and the origin of mass,” Found. Phys. 29 (1999), 1023-1064 doi:10.1023/A:1012851715278 [arXiv:gr-qc/9802044];
- W. Jia and M. Karydas, “Obstruction tensors in Weyl geometry and holographic Weyl anomaly,” Phys. Rev. D 104 (2021) no.12, 126031 doi:10.1103/PhysRevD.104.126031 [arXiv:2109.14014 [hep-th]];
- D. M. Ghilencea and C. T. Hill, “Standard Model in conformal geometry: Local vs gauged scale invariance,” Annals Phys. 460 (2024), 169562 doi:10.1016/j.aop.2023.169562 [arXiv:2303.02515 [hep-th]];
- D. M. Ghilencea, “Non-metric geometry as the origin of mass in gauge theories of scale invariance,” Eur. Phys. J. C 83, no.2, 176 (2023) doi:10.1140/epjc/s10052-023-11237-z [arXiv:2203.05381 [hep-th]];
- D. S. Klemm and L. Ravera, “Einstein manifolds with torsion and nonmetricity,” Phys. Rev. D 101 (2020) no.4, 044011 doi:10.1103/PhysRevD.101.044011 [arXiv:1811.11458];
- D. Iosifidis, A. C. Petkou and C. G. Tsagas, “Torsion/non-metricity duality in f(R) gravity,” Gen. Rel. Grav. 51 (2019) no.5, 66 doi:10.1007/s10714-019-2539-9 [arXiv:1810.06602 [gr-qc]];
- S. R. Coleman and J. Mandula, “All Possible Symmetries of the S Matrix,” Phys. Rev. 159 (1967), 1251-1256 doi:10.1103/PhysRev.159.1251;
- K. Hayashi and T. Kugo, “Everything about the Weyl’s gauge field,” Prog. Theor. Phys. 61 (1979), 334 doi:10.1143/PTP.61.334; K. Hayashi, M. Kasuya and T. Shirafuji, “Elementary Particles and Weyl’s Gauge Field,” Prog. Theor. Phys. 57 (1977), 431 [erratum: Prog. Theor. Phys. 59 (1978), 681] doi:10.1143/PTP.57.431;