×

\((3+1)\)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces. (English) Zbl 1380.83091

Summary: We apply the recently suggested strategy to lift state spaces and operators for \((2+1)\)-dimensional topological quantum field theories to state spaces and operators for a \((3+1)\)-dimensional TQFT with defects. We start from the \((2+1)\)-dimensional TuraevViro theory and obtain a state space, consistent with the state space expected from the Crane-Yetter model with line defects. This work has important applications for quantum gravity as well as the theory of topological phases in \((3+1)\) dimensions. It provides a self-dual quantum geometry realization based on a vacuum state peaked on a homogeneously curved geometry. The state spaces and operators we construct here provide also an improved version of the Walker-Wang model, and simplify its analysis considerably. We in particular show that the fusion bases of the \((2+1)\)-dimensional theory lead to a rich set of bases for the \((3+1)\)-dimensional theory. This includes a quantum deformed spin network basis, which in a loop quantum gravity context diagonalizes spatial geometry operators. We also obtain a dual curvature basis, that diagonalizes the Walker-Wang Hamiltonian. Furthermore, the construction presented here can be generalized to provide state spaces for the recently introduced dichromatic four-dimensional manifold invariants.

MSC:

83C45 Quantization of the gravitational field
53Z05 Applications of differential geometry to physics
81T45 Topological field theories in quantum mechanics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] V.G. Turaev and O.Y. Viro, State sum invariants of 3 manifolds and quantum 6j symbols, Topology31 (1992) 865 [INSPIRE]. · Zbl 0779.57009 · doi:10.1016/0040-9383(92)90015-A
[2] J.W. Barrett and B.W. Westbury, Invariants of piecewise linear three manifolds, Trans. Am. Math. Soc.348 (1996) 3997 [hep-th/9311155] [INSPIRE]. · Zbl 0865.57013 · doi:10.1090/S0002-9947-96-01660-1
[3] M. Bärenz and J. Barrett, Dichromatic state sum models for four-manifolds from pivotal functors, arXiv:1601.03580 [INSPIRE]. · Zbl 1414.57017
[4] R. König, G. Kuperberg and B.W. Reichardt, Quantum computation with Turaev-Viro codes, Ann. Phys.325 (2010) 2707 [arXiv:1002.2816]. · Zbl 1206.81033 · doi:10.1016/j.aop.2010.08.001
[5] Y. Hu, N. Geer and Y.-S. Wu, Full dyon excitation spectrum in generalized Levin-Wen models, arXiv:1502.03433 [INSPIRE].
[6] B. Dittrich and M. Geiller, Quantum gravity kinematics from extended TQFTs, New J. Phys.19 (2017) 013003 [arXiv:1604.05195] [INSPIRE]. · Zbl 1512.81079 · doi:10.1088/1367-2630/aa54e2
[7] C. Delcamp, B. Dittrich and A. Riello, On entanglement entropy in non-Abelian lattice gauge theory and 3D quantum gravity, JHEP11 (2016) 102 [arXiv:1609.04806] [INSPIRE]. · Zbl 1390.83061 · doi:10.1007/JHEP11(2016)102
[8] C. Delcamp, B. Dittrich and A. Riello, Fusion basis for lattice gauge theory and loop quantum gravity, JHEP02 (2017) 061 [arXiv:1607.08881] [INSPIRE]. · Zbl 1377.83027 · doi:10.1007/JHEP02(2017)061
[9] B. Dittrich and S. Steinhaus, Time evolution as refining, coarse graining and entangling, New J. Phys.16 (2014) 123041 [arXiv:1311.7565] [INSPIRE]. · Zbl 1451.81068 · doi:10.1088/1367-2630/16/12/123041
[10] A. Ashtekar and C.J. Isham, Representations of the holonomy algebras of gravity and nonAbelian gauge theories, Class. Quant. Grav.9 (1992) 1433 [hep-th/9202053] [INSPIRE]. · Zbl 0773.53033 · doi:10.1088/0264-9381/9/6/004
[11] A. Ashtekar and J. Lewandowski, Representation theory of analytic holonomy C∗algebras, in Knots and quantum gravity, J. Baez ed., Oxford University Press, Oxford U.K. (1994), gr-qc/9311010 [INSPIRE]. · Zbl 0827.46055
[12] A. Ashtekar and J. Lewandowski, Projective techniques and functional integration for gauge theories, J. Math. Phys.36 (1995) 2170 [gr-qc/9411046] [INSPIRE]. · Zbl 0844.58009
[13] A. Ashtekar and J. Lewandowski, Differential geometry on the space of connections via graphs and projective limits, J. Geom. Phys.17 (1995) 191 [hep-th/9412073] [INSPIRE]. · Zbl 0851.53014 · doi:10.1016/0393-0440(95)00028-G
[14] G.T. Horowitz, Exactly soluble diffeomorphism invariant theories, Commun. Math. Phys.125 (1989) 417 [INSPIRE]. · Zbl 0684.53075 · doi:10.1007/BF01218410
[15] B. Dittrich and M. Geiller, A new vacuum for loop quantum gravity, Class. Quant. Grav.32 (2015) 112001 [arXiv:1401.6441] [INSPIRE]. · Zbl 1320.83030 · doi:10.1088/0264-9381/32/11/112001
[16] B. Dittrich and M. Geiller, Flux formulation of loop quantum gravity: classical framework, Class. Quant. Grav.32 (2015) 135016 [arXiv:1412.3752] [INSPIRE]. · Zbl 1327.83111 · doi:10.1088/0264-9381/32/13/135016
[17] B. Bahr, B. Dittrich and M. Geiller, A new realization of quantum geometry, arXiv:1506.08571 [INSPIRE]. · Zbl 1320.83030
[18] S. Major and L. Smolin, Quantum deformation of quantum gravity, Nucl. Phys.B 473 (1996) 267 [gr-qc/9512020] [INSPIRE]. · Zbl 0925.83019
[19] B. Bahr and B. Dittrich, Improved and Perfect Actions in Discrete Gravity, Phys. Rev.D 80 (2009) 124030 [arXiv:0907.4323] [INSPIRE].
[20] B. Bahr and B. Dittrich, Regge calculus from a new angle, New J. Phys.12 (2010) 033010 [arXiv:0907.4325] [INSPIRE]. · Zbl 1360.83016 · doi:10.1088/1367-2630/12/3/033010
[21] M. Dupuis and F. Girelli, Observables in loop quantum gravity with a cosmological constant, Phys. Rev.D 90 (2014) 104037 [arXiv:1311.6841] [INSPIRE].
[22] C. Rovelli and F. Vidotto, Compact phase space, cosmological constant and discrete time, Phys. Rev.D 91 (2015) 084037 [arXiv:1502.00278] [INSPIRE].
[23] C. Perini, C. Rovelli and S. Speziale, Self-energy and vertex radiative corrections in LQG, Phys. Lett.B 682 (2009) 78 [arXiv:0810.1714] [INSPIRE]. · doi:10.1016/j.physletb.2009.10.076
[24] A. Riello, Self-energy of the Lorentzian Engle-Pereira-Rovelli-Livine and Freidel-Krasnov model of quantum gravity, Phys. Rev.D 88 (2013) 024011 [arXiv:1302.1781] [INSPIRE].
[25] V. Bonzom and B. Dittrich, Bubble divergences and gauge symmetries in spin foams, Phys. Rev.D 88 (2013) 124021 [arXiv:1304.6632] [INSPIRE].
[26] L.-Q. Chen, Bulk amplitude and degree of divergence in 4d spin foams, Phys. Rev.D 94 (2016) 104025 [arXiv:1602.01825] [INSPIRE].
[27] B. Dittrich, M. Martín-Benito and E. Schnetter, Coarse graining of spin net models: dynamics of intertwiners, New J. Phys.15 (2013) 103004 [arXiv:1306.2987] [INSPIRE].
[28] B. Dittrich and W. Kaminski, Topological lattice field theories from intertwiner dynamics, arXiv:1311.1798 [INSPIRE]. · Zbl 0948.83024
[29] B. Dittrich, M. Martin-Benito and S. Steinhaus, Quantum group spin nets: refinement limit and relation to spin foams, Phys. Rev.D 90 (2014) 024058 [arXiv:1312.0905] [INSPIRE].
[30] B. Dittrich, S. Mizera and S. Steinhaus, Decorated tensor network renormalization for lattice gauge theories and spin foam models, New J. Phys.18 (2016) 053009 [arXiv:1409.2407] [INSPIRE]. · doi:10.1088/1367-2630/18/5/053009
[31] S. Steinhaus, Coupled intertwiner dynamics: A toy model for coupling matter to spin foam models, Phys. Rev.D 92 (2015) 064007 [arXiv:1506.04749] [INSPIRE].
[32] B. Dittrich, E. Schnetter, C.J. Seth and S. Steinhaus, Coarse graining flow of spin foam intertwiners, Phys. Rev.D 94 (2016) 124050 [arXiv:1609.02429] [INSPIRE].
[33] C. Delcamp and B. Dittrich, Towards a phase diagram for spin foams, arXiv:1612.04506 [INSPIRE]. · Zbl 1380.83090
[34] C. Delcamp and B. Dittrich, From 3D TQFTs to 4D models with defects, arXiv:1606.02384 [INSPIRE]. · Zbl 1368.81140
[35] F. Markopoulou and L. Smolin, Quantum geometry with intrinsic local causality, Phys. Rev.D 58 (1998) 084032 [gr-qc/9712067] [INSPIRE].
[36] N. Reshetikhin and V.G. Turaev, Invariants of three manifolds via link polynomials and quantum groups, Inv. Math.103 (1991) 547. · Zbl 0725.57007 · doi:10.1007/BF01239527
[37] L. Crane and D. Yetter, A categorical construction of 4 − D topological quantum field theories, in the proceedings of the Quantum topology, October 30-November 1, Dayton, U.S.A. (1992), hep-th/9301062 [INSPIRE]. · Zbl 0841.57030
[38] L. Crane, L.H. Kauffman and D.N. Yetter, State sum invariants of four manifolds. 1., hep-th/9409167 [INSPIRE]. · Zbl 0883.57021
[39] J.W. Barrett, J.M. Garcia-Islas and J.F. Martins, Observables in the Turaev-Viro and Crane-Yetter models, J. Math. Phys.48 (2007) 093508 [math/0411281] [INSPIRE]. · Zbl 1152.81328
[40] H.M. Haggard, M. Han, W. Kaminski and A. Riello, SL(2, ℂ) Chern-Simons theory, a non-planar graph operator and 4D quantum gravity with a cosmological constant: semiclassical geometry, Nucl. Phys.B 900 (2015) 1 [arXiv:1412.7546] [INSPIRE]. · Zbl 1331.83025 · doi:10.1016/j.nuclphysb.2015.08.023
[41] H.M. Haggard, M. Han and A. Riello, Encoding Curved Tetrahedra in Face Holonomies: Phase Space of Shapes from Group-Valued Moment Maps, Annales Henri Poincaré17 (2016) 2001 [arXiv:1506.03053] [INSPIRE]. · Zbl 1345.83015 · doi:10.1007/s00023-015-0455-4
[42] H.M. Haggard, M. Han, W. Kaminski and A. Riello, Four-dimensional quantum gravity with a cosmological constant from three-dimensional holomorphic blocks, Phys. Lett.B 752 (2016) 258 [arXiv:1509.00458] [INSPIRE]. · Zbl 1360.83052 · doi:10.1016/j.physletb.2015.11.058
[43] H.M. Haggard, M. Han, W. Kaminski and A. Riello, SL(2, ℂ) Chern-Simons theory, flat connections and four-dimensional quantum geometry, arXiv:1512.07690 [INSPIRE]. · Zbl 1331.83025
[44] M. Han and Z. Huang, SU(2) flat connection on a Riemann surface and 3D twisted geometry with a cosmological constant, Phys. Rev.D 95 (2017) 044018 [arXiv:1610.01246] [INSPIRE].
[45] K. Walker and Z. Wang, (3 + 1)-TQFTs and Topological Insulators, arXiv:1104.2632 [INSPIRE].
[46] M.A. Levin and X.-G. Wen, String net condensation: a physical mechanism for topological phases, Phys. Rev.B 71 (2005) 045110 [cond-mat/0404617] [INSPIRE].
[47] T. Lan and X.-G. Wen, Topological quasiparticles and the holographic bulk-edge relation in (2 + 1)-dimensional string-net models, Phys. Rev.B 90 (2014) 115119 [arXiv:1311.1784] [INSPIRE]. · doi:10.1103/PhysRevB.90.115119
[48] J.C. Baez, Four-Dimensional BF theory with cosmological term as a topological quantum field theory, Lett. Math. Phys.38 (1996) 129 [q-alg/9507006] [INSPIRE]. · Zbl 0858.57022
[49] C.W. von Keyserlingk, F.J. Burnell and S.H. Simon, Three-dimensional topological lattice models with surface anyons, Phys. Rev.B 87 (2013) 045107 [arXiv:1208.5128] [INSPIRE]. · doi:10.1103/PhysRevB.87.045107
[50] C. Delcamp and B. Dittrich, One fusion basis to rule them all, to appear. · Zbl 1368.81140
[51] C. Rovelli and L. Smolin, Discreteness of area and volume in quantum gravity, Nucl. Phys.B 442 (1995) 593 [Erratum ibid.B 456 (1995) 753] [gr-qc/9411005] [INSPIRE]. · Zbl 0925.83013
[52] C. Rovelli and L. Smolin, Spin networks and quantum gravity, Phys. Rev.D 52 (1995) 5743 [gr-qc/9505006] [INSPIRE]. · Zbl 0925.83013
[53] J.W. Barrett, Geometrical measurements in three-dimensional quantum gravity, Int. J. Mod. Phys.A 18S2 (2003) 97 [gr-qc/0203018] [INSPIRE]. · Zbl 1080.83510
[54] L. Freidel and D. Louapre, Ponzano-Regge model revisited II: equivalence with Chern-Simons, gr-qc/0410141 [INSPIRE]. · Zbl 1060.83013
[55] L. Freidel and E.R. Livine, Ponzano-Regge model revisited III: Feynman diagrams and effective field theory, Class. Quant. Grav.23 (2006) 2021 [hep-th/0502106] [INSPIRE]. · Zbl 1091.83503 · doi:10.1088/0264-9381/23/6/012
[56] L. Freidel, K. Noui and P. Roche, 6j symbols duality relations, J. Math. Phys.48 (2007) 113512 [hep-th/0604181] [INSPIRE]. · Zbl 1153.81359 · doi:10.1063/1.2803507
[57] B. Dittrich, From the discrete to the continuous: towards a cylindrically consistent dynamics, New J. Phys.14 (2012) 123004 [arXiv:1205.6127] [INSPIRE]. · Zbl 1448.83037 · doi:10.1088/1367-2630/14/12/123004
[58] B. Dittrich, The continuum limit of loop quantum gravity — A framework for solving the theory, in 100 Years of General Relativity, A. Ashtekar and J. Pullin eds., World Scientific, Singapore (2014), arXiv:1409.1450 [INSPIRE].
[59] A.N. Kirillov and N.Y. Reshetikhin, Representations of the algebra U q(SU(2)), q-orthogonal polynomials and invariants of links, in Infinite dimensional Lie algebras and groups, V.G. Kac ed., World Scientific, Singapore (1989). · Zbl 0742.17018
[60] L.H. Kauffman and S. Lins, Temperley-Lieb recoupling theory and invariants of 3-manifolds, Princeton University Press, Princeton U.K. (1994). · Zbl 0821.57003
[61] J.S. Carter, D.E. Flath and M. Saito, The classical and quantum 6j-symbols, Princeton University Press, Princeton U.S.A. (1995). · Zbl 0851.17001
[62] A. Kirillov, Jr, String-net model of Turaev-Viro invariants, arXiv:1106.6033 [INSPIRE].
[63] P.A. Horvathy, Bogomolny-type equations in curved space, [INSPIRE]. · Zbl 0651.53053
[64] G. Alagic, S. P. Jordan, R. Koenig and B.W. Reichardt, Approximating Turaev-Viro 3-manifold invariants is universal for quantum computation, Phys. Rev.A 82 (2010) 040302. · doi:10.1103/PhysRevA.82.040302
[65] A. Yu. Kitaev, Fault tolerant quantum computation by anyons, Annals Phys.303 (2003) 2 [quant-ph/9707021] [INSPIRE]. · Zbl 1012.81006
[66] J. Johnson, Notes on Heegaard Splittings, http://users.math.yale.edu/ jj327/notes.pdf. · Zbl 1310.83014
[67] R.E. Gompf and A. Stipsicz, 4-manifolds and Kirby Calculus, Graduate studies in mathematics, American Mathematical Society, U.S.A. (1999). · Zbl 0933.57020
[68] A. Kirillov, Jr. and B. Balsam, Turaev-Viro invariants as an extended TQFT, arXiv:1004.1533 [INSPIRE].
[69] B. Balsam, Turaev-Viro invariants as an extended TQFT II, arXiv:1010.1222. · Zbl 1360.83016
[70] B. Balsam, Turaev-Viro invariants as an extended TQFT III, arXiv:1012.0560. · Zbl 0934.83020
[71] V. Turaev and A. Virelizier, On two approaches to 3-dimensional TQFTs, arXiv:1006.3501 [INSPIRE]. · Zbl 1254.57012
[72] C. Rovelli, Quantum gravity, Cambridge University Press, Cambridge U.K. (2004). · Zbl 1091.83001 · doi:10.1017/CBO9780511755804
[73] T. Thiemann, Introduction to modern canonical quantum general relativity, Cambridge University Press, Cambridge U.K. (2007). · Zbl 1129.83004
[74] A. Perez, The spin foam approach to quantum gravity, Living Rev. Rel.16 (2013) 3 [arXiv:1205.2019] [INSPIRE]. · Zbl 1320.83008 · doi:10.12942/lrr-2013-3
[75] F.J. Burnell, X. Chen, L. Fidkowski and A. Vishwanath, Exactly soluble model of a three-dimensional symmetry-protected topological phase of bosons with surface topological order, Phys. Rev.B 90 (2014) 245122 [arXiv:1302.7072] [INSPIRE]. · doi:10.1103/PhysRevB.90.245122
[76] S.L. Kokkendorff, Polar duality and the generalized law of sines, J. Geom.86 (2007) 140. · Zbl 1115.51010 · doi:10.1007/s00022-006-1858-7
[77] D.J. Williamson and Z. Wang, Hamiltonian realizations of (3 + 1)-TQFTs, arXiv:1606.07144 [INSPIRE].
[78] L. Smolin, Linking topological quantum field theory and nonperturbative quantum gravity, J. Math. Phys.36 (1995) 6417 [gr-qc/9505028] [INSPIRE]. · Zbl 0856.58055
[79] J.C. Baez and A. Perez, Quantization of strings and branes coupled to BF theory, Adv. Theor. Math. Phys.11 (2007) 451 [gr-qc/0605087] [INSPIRE]. · Zbl 1147.83015
[80] E.R. Livine, Deformation operators of spin networks and coarse-graining, Class. Quant. Grav.31 (2014) 075004 [arXiv:1310.3362] [INSPIRE]. · Zbl 1291.83111 · doi:10.1088/0264-9381/31/7/075004
[81] L. Freidel and K. Krasnov, Spin foam models and the classical action principle, Adv. Theor. Math. Phys.2 (1999) 1183 [hep-th/9807092] [INSPIRE]. · Zbl 0948.83024 · doi:10.4310/ATMP.1998.v2.n6.a1
[82] L. Freidel and K. Krasnov, Discrete space-time volume for three-dimensional BF theory and quantum gravity, Class. Quant. Grav.16 (1999) 351 [hep-th/9804185] [INSPIRE]. · Zbl 0934.83020 · doi:10.1088/0264-9381/16/2/003
[83] E.R. Livine, 3d quantum gravity: coarse-graining and q-deformation, Annales Henri Poincaré18 (2017) 1465 [arXiv:1610.02716] [INSPIRE]. · Zbl 1366.83023 · doi:10.1007/s00023-016-0535-0
[84] A. Ashtekar and J. Lewandowski, Quantum theory of geometry. 2. Volume operators, Adv. Theor. Math. Phys.1 (1998) 388 [gr-qc/9711031] [INSPIRE]. · Zbl 0866.58077
[85] M. Dupuis, F. Girelli and E.R. Livine, Deformed spinor networks for loop gravity: towards hyperbolic twisted geometries, Gen. Rel. Grav.46 (2014) 1802 [arXiv:1403.7482] [INSPIRE]. · Zbl 1308.83062 · doi:10.1007/s10714-014-1802-3
[86] C. Charles and E.R. Livine, Closure constraints for hyperbolic tetrahedra, Class. Quant. Grav.32 (2015) 135003 [arXiv:1501.00855] [INSPIRE]. · Zbl 1323.83011 · doi:10.1088/0264-9381/32/13/135003
[87] C. Charles and E.R. Livine, The closure constraint for the hyperbolic tetrahedron as a Bianchi identity, arXiv:1607.08359 [INSPIRE]. · Zbl 1381.83003
[88] A. Ashtekar, New variables for classical and quantum gravity, Phys. Rev. Lett.57 (1986) 2244 [INSPIRE]. · doi:10.1103/PhysRevLett.57.2244
[89] A. Ashtekar, New Hamiltonian formulation of general relativity, Phys. Rev.D 36 (1987) 1587 [INSPIRE].
[90] J.F. Barbero G., Real Ashtekar variables for Lorentzian signature space times, Phys. Rev.D 51 (1995) 5507 [gr-qc/9410014] [INSPIRE].
[91] G. Immirzi, Real and complex connections for canonical gravity, Class. Quant. Grav.14 (1997) L177 [gr-qc/9612030] [INSPIRE]. · Zbl 0887.53059
[92] B. Dittrich and J.P. Ryan, Phase space descriptions for simplicial 4d geometries, Class. Quant. Grav.28 (2011) 065006 [arXiv:0807.2806] [INSPIRE]. · Zbl 1214.83010 · doi:10.1088/0264-9381/28/6/065006
[93] B. Dittrich and J.P. Ryan, Simplicity in simplicial phase space, Phys. Rev.D 82 (2010) 064026 [arXiv:1006.4295] [INSPIRE].
[94] B. Dittrich and J.P. Ryan, On the role of the Barbero-Immirzi parameter in discrete quantum gravity, Class. Quant. Grav.30 (2013) 095015 [arXiv:1209.4892] [INSPIRE]. · Zbl 1269.83030 · doi:10.1088/0264-9381/30/9/095015
[95] C. Charles and E.R. Livine, Ashtekar-Barbero holonomy on the hyperboloid: Immirzi parameter as a cutoff for quantum gravity, Phys. Rev.D 92 (2015) 124031 [arXiv:1507.00851] [INSPIRE].
[96] A. Ashtekar, A.P. Balachandran and S. Jo, The CP problem in quantum gravity, Int. J. Mod. Phys.A 4 (1989) 1493 [INSPIRE]. · doi:10.1142/S0217751X89000649
[97] D.J. Rezende and A. Perez, The Θ parameter in loop quantum gravity: Effects on quantum geometry and black hole entropy, Phys. Rev.D 78 (2008) 084025 [arXiv:0711.3107] [INSPIRE].
[98] N. Bodendorfer, Some notes on the Kodama state, maximal symmetry and the isolated horizon boundary condition, Phys. Rev.D 93 (2016) 124042 [arXiv:1602.05499] [INSPIRE].
[99] B. Dittrich and P.A. Hohn, Canonical simplicial gravity, Class. Quant. Grav.29 (2012) 115009 [arXiv:1108.1974] [INSPIRE]. · Zbl 1246.83006 · doi:10.1088/0264-9381/29/11/115009
[100] B. Dittrich, Diffeomorphism symmetry in quantum gravity models, Adv. Sci. Lett.2 (2008) 151 [arXiv:0810.3594] [INSPIRE]. · doi:10.1166/asl.2009.1022
[101] B. Bahr and B. Dittrich, (Broken) gauge symmetries and constraints in Regge calculus, Class. Quant. Grav.26 (2009) 225011 [arXiv:0905.1670] [INSPIRE]. · Zbl 1181.83062
[102] B. Dittrich and S. Steinhaus, Path integral measure and triangulation independence in discrete gravity, Phys. Rev.D 85 (2012) 044032 [arXiv:1110.6866] [INSPIRE].
[103] B. Dittrich, W. Kaminski and S. Steinhaus, Discretization independence implies non-locality in 4D discrete quantum gravity, Class. Quant. Grav.31 (2014) 245009 [arXiv:1404.5288] [INSPIRE]. · Zbl 1310.83014 · doi:10.1088/0264-9381/31/24/245009
[104] B. Bahr, B. Dittrich and S. Steinhaus, Perfect discretization of reparametrization invariant path integrals, Phys. Rev.D 83 (2011) 105026 [arXiv:1101.4775] [INSPIRE].
[105] B. Dittrich, How to construct diffeomorphism symmetry on the lattice, PoS (QGQGS2011) 012 [arXiv:1201.3840] [INSPIRE].
[106] B. Bahr, On background-independent renormalization of spin foam models, Class. Quant. Grav.34 (2017) 075001 [arXiv:1407.7746] [INSPIRE]. · Zbl 1368.83031 · doi:10.1088/1361-6382/aa5e13
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.