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Loop quantum gravity boundary dynamics and \(\mathrm{SL}(2,\mathbb{C})\) gauge theory. (English) Zbl 1481.83045

Summary: In the context of the quest for a holographic formulation of quantum gravity, we investigate the basic boundary theory structure for loop quantum gravity. In 3 + 1 space-time dimensions, the boundary theory lives on the 2 + 1-dimensional time-like boundary and is supposed to describe the time evolution of gravitational boundary modes – ‘edge modes’ – living on the two-dimensional boundary of space, i.e. the space-time corner. We do not analyse the boundary structures of general relativity and their quantization, and focus instead on investigating which boundary theories can be supported by the standard loop quantum gravity formalism and spin network states. Focussing on ‘electric’ excitations – quanta of area – living on the corner, we formulate their dynamics in terms of classical spinor variables and we show that the coupling constants of a polynomial Hamiltonian can be understood as the components of a background metric on the 2 + 1-dimensional time-like boundary. This leads to a conjecture of a deeper correspondence between boundary Hamiltonian and boundary metric states. We further show that one can reformulate the quanta of area data in terms of a \(\mathrm{SL}(2,\mathbb{C})\) connection, transporting the spinors on the boundary surface and whose SU(2) component would define ‘magnetic’ excitations (tangential Ashtekar-Barbero connection), thereby opening the door to writing the loop quantum gravity boundary dynamics as a 2 + 1-dimensional \(\mathrm{SL}(2,\mathbb{C})\) gauge theory.

MSC:

83C45 Quantization of the gravitational field
58J32 Boundary value problems on manifolds
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
70S15 Yang-Mills and other gauge theories in mechanics of particles and systems
58A15 Exterior differential systems (Cartan theory)
81T35 Correspondence, duality, holography (AdS/CFT, gauge/gravity, etc.)
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[1] Freidel, L.; Yokokura, Y., Non-equilibrium thermodynamics of gravitational screens, Class. Quantum Grav., 32 (2015) · Zbl 1329.83044 · doi:10.1088/0264-9381/32/21/215002
[2] Freidel, L.; Perez, A., Quantum gravity at the corner (2015)
[3] Donnelly, W.; Freidel, L., Local subsystems in gauge theory and gravity, J. High Energy Phys. (2016) · Zbl 1390.83016 · doi:10.1007/jhep09(2016)102
[4] Freidel, L.; Perez, A.; Pranzetti, D., Loop gravity string, Phys. Rev. D, 95 (2017) · doi:10.1103/physrevd.95.106002
[5] De Paoli, E.; Speziale, S., A gauge-invariant symplectic potential for tetrad general relativity, J. High Energy Phys. (2018) · Zbl 1395.83046 · doi:10.1007/JHEP07(2018)040
[6] Freidel, L.; Livine, E.; Pranzetti, D., Gravitational edge modes: from Kac-Moody charges to Poincaré networks, Class. Quantum Grav., 36 (2019) · Zbl 1478.83089 · doi:10.1088/1361-6382/ab40fe
[7] Freidel, L.; Livine, E. R.; Pranzetti, D., Kinematical gravitational charge algebra, Phys. Rev. D, 101 (2020) · doi:10.1103/physrevd.101.024012
[8] Harlow, D.; Wu, J-Q, Covariant phase space with boundaries, J. High Energy Phys. (2020) · Zbl 1461.83007 · doi:10.1007/jhep10(2020)146
[9] Takayanagi, T.; Tamaoka, K., Gravity edges modes and Hayward term, J. High Energy Phys. (2020) · Zbl 1435.83186 · doi:10.1007/jhep02(2020)167
[10] Freidel, L.; Geiller, M.; Pranzetti, D., Edge modes of gravity. Part I. Corner potentials and charges, J. High Energy Phys. (2020) · Zbl 1460.83030 · doi:10.1007/JHEP11(2020)026
[11] Freidel, L.; Geiller, M.; Pranzetti, D., Edge modes of gravity. Part II. Corner metric and Lorentz charges, J. High Energy Phys. (2020) · Zbl 1456.83024 · doi:10.1007/JHEP11(2020)027
[12] Freidel, L.; Geiller, M.; Pranzetti, D., Edge modes of gravity. Part III. Corner simplicity constraints (2020)
[13] Bahr, B.; Dittrich, B.; Geiller, M., A new realization of quantum geometry (2015)
[14] Charles, C.; Livine, E. R., The Fock space of Loopy spin networks for quantum gravity, Gen. Relativ. Gravit., 48, 113 (2016) · Zbl 1381.83002 · doi:10.1007/s10714-016-2107-5
[15] Delcamp, C.; Dittrich, B., Towards a dual spin network basis for (3 + 1)d lattice gauge theories and topological phases, J. High Energy Phys. (2018) · Zbl 1402.81204 · doi:10.1007/JHEP10(2018)023
[16] Freidel, L.; Livine, E. R., Bubble networks: framed discrete geometry for quantum gravity, Gen. Relativ. Gravit., 51, 9 (2019) · Zbl 1409.83063 · doi:10.1007/s10714-018-2493-y
[17] Livine, E. R., Area propagator and boosted spin networks in loop quantum gravity, Class. Quantum Grav., 36 (2019) · Zbl 1478.83094 · doi:10.1088/1361-6382/ab32d4
[18] Freidel, L.; Speziale, S., Twisted geometries: a geometric parametrisation of SU(2) phase space, Phys. Rev. D, 82 (2010) · doi:10.1103/physrevd.82.084040
[19] Freidel, L.; Speziale, S., From twistors to twisted geometries, Phys. Rev. D, 82 (2010) · doi:10.1103/physrevd.82.084041
[20] Freidel, L.; Krasnov, K.; Livine, E. R., Holomorphic factorization for a quantum tetrahedron, Commun. Math. Phys., 297, 45-93 (2010) · Zbl 1211.22013 · doi:10.1007/s00220-010-1036-5
[21] Borja, E. F.; Freidel, L.; Garay, I.; Livine, E. R., U(N) tools for loop quantum gravity: the return of the spinor, Class. Quantum Grav., 28 (2011) · Zbl 1211.83013 · doi:10.1088/0264-9381/28/5/055005
[22] Livine, E.; Tambornino, J., Spinor representation for loop quantum gravity, J. Math. Phys., 53 (2012) · Zbl 1273.83073 · doi:10.1063/1.3675465
[23] Speziale, S.; Wieland, W. M., The twistorial structure of loop-gravity transition amplitudes, Phys. Rev. D, 86 (2012) · doi:10.1103/physrevd.86.124023
[24] Livine, E. R.; Tambornino, J., Holonomy operator and quantization ambiguities on spinor space, Phys. Rev. D, 87 (2013) · doi:10.1103/physrevd.87.104014
[25] Alesci, E.; Lewandowski, J.; Mäkinen, I., Coherent 3j-symbol representation for the loop quantum gravity intertwiner space, Phys. Rev. D, 94 (2016) · doi:10.1103/physrevd.94.084028
[26] Calcinari, A.; Freidel, L.; Livine, E.; Speziale, S., Twisted geometries coherent states for loop quantum gravity (2020)
[27] Wieland, W., Quasi-local gravitational angular momentum and centre of mass from generalised Witten equations, Gen. Relativ. Gravit., 49, 38 (2017) · Zbl 1380.83172 · doi:10.1007/s10714-017-2200-4
[28] Wieland, W., New boundary variables for classical and quantum gravity on a null surface, Class. Quantum Grav., 34 (2017) · Zbl 1380.83103 · doi:10.1088/1361-6382/aa8d06
[29] Feller, A.; Livine, E. R., Quantum surface and intertwiner dynamics in loop quantum gravity, Phys. Rev. D, 95 (2017) · doi:10.1103/physrevd.95.124038
[30] Haggard, H. M.; Han, M.; Kaminski, W.; Riello, A., SL(2, C) Chern-Simons theory, flat connections, and four-dimensional quantum geometry (2015)
[31] Han, M.; Huang, Z., SU(2) flat connection on a Riemann surface and 3D twisted geometry with a cosmological constant, Phys. Rev. D, 95 (2017) · doi:10.1103/physrevd.95.044018
[32] Charles, C.; Livine, E. R., The closure constraint for the hyperbolic tetrahedron as a Bianchi identity, Gen. Relativ. Gravit., 49, 92 (2017) · Zbl 1381.83003 · doi:10.1007/s10714-017-2255-2
[33] Delcamp, C.; Dittrich, B.; Riello, A., Fusion basis for lattice gauge theory and loop quantum gravity, J. High Energy Phys. (2017) · Zbl 1377.83027 · doi:10.1007/JHEP02(2017)061
[34] Ashtekar, A.; Baez, J.; Corichi, A.; Krasnov, K., Quantum geometry and black hole entropy, Phys. Rev. Lett., 80, 904-907 (1998) · Zbl 0949.83024 · doi:10.1103/physrevlett.80.904
[35] Ashtekar, A.; Baez, J. C.; Krasnov, K., Quantum geometry of isolated horizons and black hole entropy, Adv. Theor. Math. Phys., 4, 1-94 (2000) · Zbl 0981.83028 · doi:10.4310/atmp.2000.v4.n1.a1
[36] Domagala, M.; Lewandowski, J., Black-hole entropy from quantum geometry, Class. Quantum Grav., 21, 5233-5243 (2004) · Zbl 1062.83053 · doi:10.1088/0264-9381/21/22/014
[37] Agulló, I.; Borja, E. F.; Díaz-Polo, J., Computing black hole entropy in loop quantum gravity from a conformal field theory perspective, J. Cosmol. Astropart. Phys. (2009) · doi:10.1088/1475-7516/2009/07/016
[38] Asin, O.; Ben Achour, J.; Geiller, M.; Noui, K.; Perez, A., Black holes as gases of punctures with a chemical potential: Bose-Einstein condensation and logarithmic corrections to the entropy, Phys. Rev. D, 91 (2015) · doi:10.1103/physrevd.91.084005
[39] Freidel, L.; Livine, E. R., The fine structure of SU(2) intertwiners from U(N) representations, J. Math. Phys., 51 (2010) · Zbl 1312.83017 · doi:10.1063/1.3473786
[40] Freidel, L.; Livine, E. R., U(N) coherent states for loop quantum gravity, J. Math. Phys., 52 (2011) · Zbl 1317.83035 · doi:10.1063/1.3587121
[41] Livine, E. R., Deformations of polyhedra and polygons by the unitary group, J. Math. Phys., 54 (2013) · Zbl 1290.83023 · doi:10.1063/1.4840635
[42] Livine, E. R.; Speziale, S., A new spinfoam vertex for quantum gravity, Phys. Rev. D, 76 (2007) · doi:10.1103/physrevd.76.084028
[43] Livine, E. R.; Speziale, S., Consistently solving the simplicity constraints for spinfoam quantum gravity, Europhys. Lett., 81 (2008) · doi:10.1209/0295-5075/81/50004
[44] Freidel, L.; Krasnov, K., A new spin foam model for 4D gravity, Class. Quantum Grav., 25 (2008) · Zbl 1144.83007 · doi:10.1088/0264-9381/25/12/125018
[45] Dupuis, M.; Livine, E. R., Holomorphic simplicity constraints for 4D spinfoam models, Class. Quantum Grav., 28 (2011) · Zbl 1230.83076 · doi:10.1088/0264-9381/28/21/215022
[46] Dupuis, M.; Freidel, L.; Livine, E. R.; Speziale, S., Holomorphic Lorentzian simplicity constraints, J. Math. Phys., 53 (2012) · Zbl 1274.83050 · doi:10.1063/1.3692327
[47] Freidel, L.; Hnybida, J., On the exact evaluation of spin networks, J. Math. Phys., 54 (2013) · Zbl 1288.83020 · doi:10.1063/1.4830008
[48] Banburski, A.; Chen, L-Q; Freidel, L.; Hnybida, J., Pachner moves in a 4D Riemannian holomorphic spin foam model, Phys. Rev. D, 92 (2015) · doi:10.1103/physrevd.92.124014
[49] Oriti, D., The universe as a quantum gravity condensate, C. R. Phys., 18, 235-245 (2017) · doi:10.1016/j.crhy.2017.02.003
[50] Oriti, D.; Pranzetti, D.; Sindoni, L., Black holes as quantum gravity condensates, Phys. Rev. D, 97 (2018) · doi:10.1103/physrevd.97.066017
[51] Carrozza, S.; Gielen, S.; Oriti, D., Editorial for the special issue ‘progress in group field theory and related quantum gravity formalisms’, Universe, 6, 19 (2020) · doi:10.3390/universe6010019
[52] Livine, E. R., Deformation operators of spin networks and coarse-graining, Class. Quantum Grav., 31 (2014) · Zbl 1291.83111 · doi:10.1088/0264-9381/31/7/075004
[53] Girelli, F.; Livine, E. R., Reconstructing quantum geometry from quantum information: spin networks as harmonic oscillators, Class. Quantum Grav., 22, 3295-3313 (2005) · Zbl 1075.83019 · doi:10.1088/0264-9381/22/16/011
[54] Girelli, F.; Sellaroli, G., SO*(2N) coherent states for loop quantum gravity, J. Math. Phys., 58 (2017) · Zbl 1370.83028 · doi:10.1063/1.4993223
[55] ’t Hooft, G., The Evolution of gravitating point particles in (2 + 1)-dimensions, Class. Quantum Grav., 10, 1023-1038 (1993) · doi:10.1088/0264-9381/10/5/019
[56] ’t Hooft, G., Canonical quantization of gravitating point particles in (2 + 1)-dimensions, Class. Quantum Grav., 10, 1653-1664 (1993) · Zbl 0787.53078 · doi:10.1088/0264-9381/10/8/022
[57] Loll, R., Quantum gravity from causal dynamical triangulations: a review, Class. Quantum Grav., 37 (2020) · Zbl 1478.83095 · doi:10.1088/1361-6382/ab57c7
[58] Livine, E. R., The spinfoam framework for quantum gravity, Habilitation Thesis (2010)
[59] Perez, A., The spin-foam approach to quantum gravity, Living Rev. Relativ., 16, 3 (2013) · Zbl 1320.83008 · doi:10.12942/lrr-2013-3
[60] Dittrich, B.; Goeller, C.; Livine, E. R.; Riello, A., Quasi-local holographic dualities in non-perturbative 3D quantum gravity I—convergence of multiple approaches and examples of Ponzano-Regge statistical duals, Nucl. Phys. B, 938, 807-877 (2019) · Zbl 1409.83059 · doi:10.1016/j.nuclphysb.2018.06.007
[61] Dittrich, B.; Goeller, C.; Livine, E. R.; Riello, A., Quasi-local holographic dualities in non-perturbative 3D quantum gravity, Class. Quantum Grav., 35 (2018) · Zbl 1409.83130 · doi:10.1088/1361-6382/aac606
[62] Goeller, C.; Livine, E. R.; Riello, A., Non-perturbative 3D quantum gravity: quantum boundary states and exact partition function, Gen. Relativ. Gravit., 52, 24 (2020) · Zbl 1442.83012 · doi:10.1007/s10714-020-02673-3
[63] Buffenoir, E.; Noui, K.; Roche, P., Hamiltonian quantization of Chern-Simons theory with SL(2, C) group, Class. Quantum Grav., 19, 4953 (2002) · Zbl 1021.83016 · doi:10.1088/0264-9381/19/19/313
[64] Dupuis, M.; Girelli, F., Quantum hyperbolic geometry in loop quantum gravity with cosmological constant, Phys. Rev. D, 87 (2013) · doi:10.1103/physrevd.87.121502
[65] Bonzom, V.; Dupuis, M.; Girelli, F.; Livine, E. R., Deformed phase space for 3D loop gravity and hyperbolic discrete geometries (2014)
[66] Bonzom, V.; Dupuis, M.; Girelli, F., Towards the Turaev-Viro amplitudes from a Hamiltonian constraint, Phys. Rev. D, 90 (2014) · doi:10.1103/physrevd.90.104038
[67] Dupuis, M.; Girelli, F.; Livine, E. R., Deformed spinor networks for loop gravity: towards hyperbolic twisted geometries, Gen. Relativ. Gravit., 46, 1802 (2014) · Zbl 1308.83062 · doi:10.1007/s10714-014-1802-3
[68] Dupuis, M.; Livine, E. R.; Pan, Q., q-deformed 3D loop gravity on the torus, Class. Quantum Grav., 37 (2020) · Zbl 1478.83084 · doi:10.1088/1361-6382/ab5d4f
[69] Markopoulou, F.; Smolin, L., Disordered locality in loop quantum gravity states, Class. Quantum Grav., 24, 3813-3823 (2007) · Zbl 1129.83321 · doi:10.1088/0264-9381/24/15/003
[70] Bianchi, E.; Dona, P.; Speziale, S., Polyhedra in loop quantum gravity, Phys. Rev. D, 83 (2011) · doi:10.1103/physrevd.83.044035
[71] Livine, E. R.; Speziale, S.; Tambornino, J., Twistor networks and covariant twisted geometries, Phys. Rev. D, 85 (2012) · doi:10.1103/physrevd.85.064002
[72] Amelino-Camelia, G.; Freidel, L.; Kowalski-Glikman, J.; Smolin, L., The principle of relative locality, Phys. Rev. D, 84 (2011) · Zbl 1263.83064 · doi:10.1103/physrevd.84.084010
[73] Amelino-Camelia, G.; Freidel, L.; Kowalski-Glikman, J.; Smolin, L., Relative locality: a deepening of the relativity principle, Gen. Relativ. Gravit., 43, 2547-2553 (2011) · Zbl 1228.83040 · doi:10.1007/s10714-011-1212-8
[74] Freidel, L.; Rempel, T., Scalar field theory in curved momentum space (2013)
[75] Isidro, J. M.; Labastida, J. M F.; Ramallo, A. V., Coset constructions in Chern-Simons gauge theory, Phys. Lett. B, 282, 63-72 (1992) · doi:10.1016/0370-2693(92)90480-r
[76] Frittelli, S.; Lehner, L.; Rovelli, C., The complete spectrum of the area from recoupling theory in loop quantum gravity, Class. Quantum Grav., 13, 2921-2931 (1996) · Zbl 0861.47050 · doi:10.1088/0264-9381/13/11/008
[77] Rovelli, C.; Upadhya, P., Loop quantum gravity and quanta of space: a primer (1998)
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