Abstract
We study the categorical generalizations of a BF theory to 2BF and 3BF theories, corresponding to 2-groups and 3-groups, in the framework of higher gauge theory. In particular, we construct the constrained 3BF actions describing the correct dynamics of Yang-Mills, Klein-Gordon, Dirac, Weyl, and Majorana fields coupled to Einstein-Cartan gravity. The action is naturally split into a topological sector and a sector with simplicity constraints, adapted to the spinfoam quantization programme. In addition, the structure of the 3-group gives rise to a novel gauge group which specifies the spectrum of matter fields present in the theory, just like the ordinary gauge group specifies the spectrum of gauge bosons in the Yang-Mills theory. This allows us to rewrite the whole Standard Model coupled to gravity as a constrained 3BF action, facilitating the nonperturbative quantization of both gravity and matter fields. Moreover, the presence and the properties of this new gauge group open up a possibility of a nontrivial unification of all fields and a possible explanation of fermion families and all other structure in the matter spectrum of the theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Rovelli, Quantum gravity, Cambridge University Press, Cambridge, U.K. (2004).
C. Rovelli and F. Vidotto, Covariant loop quantum gravity, Cambridge University Press, Cambridge, U.K. (2014).
T. Thiemann, Modern canonical quantum general relativity, Cambridge University Press, Cambridge, U.K. (2007).
G. Ponzano and T. Regge, Semiclassical limit of Racah coefficients, in Spectroscopic and group theoretical methods in physics, F. Block ed., North Holland, Amsterdam, The Netherlands (1968).
J.W. Barrett and L. Crane, Relativistic spin networks and quantum gravity, J. Math. Phys.39 (1998) 3296 [gr-qc/9709028] [INSPIRE].
J.W. Barrett and L. Crane, A Lorentzian signature model for quantum general relativity, Class. Quant. Grav.17 (2000) 3101 [gr-qc/9904025] [INSPIRE].
H. Ooguri, Topological lattice models in four-dimensions, Mod. Phys. Lett.A 7 (1992) 2799 [hep-th/9205090] [INSPIRE].
J. Engle, E. Livine, R. Pereira and C. Rovelli, LQG vertex with finite Immirzi parameter, Nucl. Phys.B 799 (2008) 136 [arXiv:0711.0146] [INSPIRE].
L. Freidel and K. Krasnov, A new spin foam model for 4d gravity, Class. Quant. Grav.25 (2008) 125018 [arXiv:0708.1595] [INSPIRE].
E. Bianchi, M. Han, C. Rovelli, W. Wieland, E. Magliaro and C. Perini, Spinfoam fermions, Class. Quant. Grav.30 (2013) 235023 [arXiv:1012.4719] [INSPIRE].
J.C. Baez and J. Huerta, An invitation to higher gauge theory, Gen. Rel. Grav.43 (2011) 2335 [arXiv:1003.4485] [INSPIRE].
A. Mikovíc and M. Vojinović, Poincaré 2-group and quantum gravity, Class. Quant. Grav.29 (2012) 165003 [arXiv:1110.4694] [INSPIRE].
M. Celada, D. González and M. Montesinos, BF gravity, Class. Quant. Grav.33 (2016) 213001 [arXiv:1610.02020] [INSPIRE].
C. Rovelli, Zakopane lectures on loop gravity, PoS(QGQGS2011)003 (2011) [arXiv:1102.3660] [INSPIRE].
J.F. Plebanski, On the separation of Einsteinian substructures, J. Math. Phys.18 (1977) 2511 [INSPIRE].
F. Girelli, H. Pfeiffer and E.M. Popescu, Topological higher gauge theory — from BF to BFCG theory, J. Math. Phys.49 (2008) 032503 [arXiv:0708.3051] [INSPIRE].
J.F. Martins and A. Miković, Lie crossed modules and gauge-invariant actions for 2-BF theories, Adv. Theor. Math. Phys.15 (2011) 1059 [arXiv:1006.0903] [INSPIRE].
L. Crane and M.D. Sheppeard, 2-categorical Poincaré representations and state sum applications, math.QA/0306440 [INSPIRE].
M. Vojinovíc, Causal dynamical triangulations in the spincube model of quantum gravity, Phys. Rev.D 94 (2016) 024058 [arXiv:1506.06839] [INSPIRE].
A. Mikovíc, Spin-cube models of quantum gravity, Rev. Math. Phys.25 (2013) 1343008 [arXiv:1302.5564] [INSPIRE].
A. Miković and M.A. Oliveira, Canonical formulation of Poincaré BFCG theory and its quantization, Gen. Rel. Grav.47 (2015) 58 [arXiv:1409.3751] [INSPIRE].
A. Miković, M.A. Oliveira and M. Vojinovic, Hamiltonian analysis of the BFCG theory for a generic Lie 2-group, arXiv:1610.09621 [INSPIRE].
A. Mikovíc, M.A. Oliveira and M. Vojinovic, Hamiltonian analysis of the BFCG formulation of general relativity, Class. Quant. Grav.36 (2019) 015005 [arXiv:1807.06354] [INSPIRE].
J.F. Martins and R. Picken, The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module, Differ. Geom. Appl. J.29 (2011) 179 [arXiv:0907.2566] [INSPIRE].
W. Wang, On 3-gauge transformations, 3-curvatures and Gray-categories, J. Math. Phys.55 (2014) 043506 [arXiv:1311.3796] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1904.07566
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Radenković, T., Vojinović, M. Higher gauge theories based on 3-groups. J. High Energ. Phys. 2019, 222 (2019). https://doi.org/10.1007/JHEP10(2019)222
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2019)222