We study the quantization of the corner symmetry algebra of 3D gravity, that is, the algebra of observables associated with 1D spatial boundaries. In the continuum field theory, at the classical level, this symmetry algebra is given by the central extension of the Poincaré loop algebra. At the quantum level, we construct a discrete current algebra based on a quantum symmetry group given by the Drinfeld double DSU(2). Those discrete currents depend on an integer N, a discreteness parameter, understood as the number of quanta of geometry on the 1D boundary: low N is the deep quantum regime, while large N should lead back to a continuum picture. We show that this algebra satisfies two fundamental properties. First, it is compatible with the quantum space-time picture given by the Ponzano–Regge state-sum model, which provides discrete path integral amplitudes for 3D quantum gravity. The integer N then counts the flux lines attached to the boundary. Second, we analyze the refinement, coarse-graining, and fusion processes as N changes, and we show that the N limit is a classical limit where we recover the Poincaré current algebra. Identifying such a discrete current algebra on quantum boundaries is an important step toward understanding how conformal field theories arise on spatial boundaries in quantized space-times such as in loop quantum gravity.

1.
B.
Czech
,
L.
Lamprou
,
S.
McCandlish
,
B.
Mosk
, and
J.
Sully
, “
A stereoscopic look into the bulk
,”
J. High Energy Phys.
2016
(
7
),
129
; arXiv:1604.03110.
2.
W.
Donnelly
, “
Quantum gravity tomography
,” arXiv:1806.05643.
3.
W.
Donnelly
and
L.
Freidel
, “
Local subsystems in gauge theory and gravity
,”
J. High Energy Phys.
2016
(
9
),
102
; arXiv:1601.04744.
4.
G.
Amelino-Camelia
, “
Doubly special relativity
,”
Nature
418
,
34
35
(
2002
); arXiv:gr-qc/0207049.
5.
J.
Kowalski-Glikman
and
S.
Nowak
, “
Doubly special relativity theories as different bases of κ-Poincaré algebra
,”
Phys. Lett. B
539
,
126
132
(
2002
); arXiv:hep-th/0203040.
6.
G.
Amelino-Camelia
,
L.
Freidel
,
J.
Kowalski-Glikman
, and
L.
Smolin
, “
The principle of relative locality
,”
Phys. Rev. D
84
,
084010
(
2011
); arXiv:1101.0931.
7.
A.
Ghosh
and
A.
Perez
, “
Black hole entropy and isolated horizons thermodynamics
,”
Phys. Rev. Lett.
107
,
241301
(
2011
); arXiv:1107.1320;
[PubMed]
8.
V.
Bonzom
,
R.
Gurau
,
J. P.
Ryan
, and
A.
Tanasa
, “
The double scaling limit of random tensor models
,”
J. High Energy Phys.
2014
(
9
),
51
; arXiv:1404.7517.
9.
R.
Gurau
, “
The complete 1/N expansion of a SYK-like tensor model
,”
Nucl. Phys. B
916
,
386
401
(
2017
); arXiv:1611.04032.
10.
L.
Freidel
and
A.
Perez
, “
Quantum gravity at the corner
,” arXiv:1507.02573.
11.
L.
Freidel
,
A.
Perez
, and
D.
Pranzetti
, “
Loop gravity string
,”
Phys. Rev. D
95
(
10
),
106002
(
2017
); arXiv:1611.03668.
12.
L.
Freidel
,
E. R.
Livine
, and
D.
Pranzetti
, “
Gravitational edge modes: From Kac–Moody charges to Poincaré networks
,”
Classical Quantum Gravity
36
(
19
),
195014
(
2019
); arXiv:1906.07876.
13.
L.
Freidel
,
E. R.
Livine
, and
D.
Pranzetti
, “
Kinematical gravitational charge algebra
,”
Phys. Rev. D
101
(
2
),
024012
(
2020
); arXiv:1910.05642.
14.
L.
Freidel
,
M.
Geiller
, and
D.
Pranzetti
, “
Edge modes of gravity–I: Corner potentials and charges
,” arXiv:2006.12527.
15.
L.
Freidel
,
M.
Geiller
, and
D.
Pranzetti
, “
Edge modes of gravity–II: Corner metric and Lorentz charges
,” arXiv:2007.03563.
16.
L.
Freidel
,
M.
Geiller
, and
D.
Pranzetti
, “
Edge modes of gravity–III: Corner simplicity constraints
,” arXiv:2007.12635.
17.
G.
Ponzano
and
T.
Regge
, “
Semiclassical limit of Racah coefficients
,” in
Spectroscopic and Group Theoretical Methods in Physics
, edited by
E. F.
Bloch
(
North-Holland Publishing Co.
,
Amsterdam
,
1968
), pp.
1
58
.
18.
L.
Freidel
and
D.
Louapre
, “
Ponzano–Regge model revisited: I. Gauge fixing, observables and interacting spinning particles
,”
Classical Quantum Gravity
21
,
5685
5726
(
2004
); arXiv:hep-th/0401076.
19.
L.
Freidel
and
D.
Louapre
, “
Ponzano-Regge model revisited II: Equivalence with Chern-Simons
,” arXiv:gr-qc/0410141.
20.
L.
Freidel
and
E. R.
Livine
, “
Ponzano–Regge model revisited: III. Feynman diagrams and effective field theory
,”
Classical Quantum Gravity
23
,
2021
2062
(
2006
); arXiv:hep-th/0502106.
21.
J. W.
Barrett
and
I.
Naish-Guzman
, “
The Ponzano–Regge model
,”
Classical Quantum Gravity
26
,
155014
(
2009
); arXiv:0803.3319.
22.
C.
Goeller
, “
Quasi-local 3D quantum gravity: Exact amplitude and holography
,” Ph.D. thesis,
Ecole Normale Superieure, Perimeter Institute for Theoretical Physics
,
Lyon
,
2019
; arXiv:2005.09985.
23.
V. G.
Turaev
and
O. Y.
Viro
, “
State sum invariants of 3-manifolds and quantum 6j-symbols
,”
Topology
31
(
4
),
865
902
(
1992
).
24.
N.
Reshetikhin
and
V. G.
Turaev
, “
Invariants of three manifolds via link polynomials and quantum groups
,”
Invent. Math.
103
,
547
597
(
1991
).
25.
H.
Ooguri
, “
Partition functions and topology changing amplitudes in the three-dimensional lattice gravity of Ponzano and Regge
,”
Nucl. Phys. B
382
,
276
304
(
1992
); arXiv:hep-th/9112072.
26.
E.
Witten
, “
2 + 1 dimensional gravity as an exactly soluble system
,”
Nucl. Phys. B
311
,
46
(
1988
).
27.
C.
Meusburger
and
K.
Noui
, “
The Hilbert space of 3d gravity: Quantum group symmetries and observables
,”
Adv. Theor. Math. Phys.
14
(
6
),
1651
1715
(
2010
); arXiv:0809.2875.
28.
M.
Dupuis
,
L.
Freidel
, and
F.
Girelli
, “
Discretization of 3d gravity in different polarizations
,”
Phys. Rev. D
96
(
8
),
086017
(
2017
); arXiv:1701.02439.
29.
M.
Dupuis
,
E. R.
Livine
, and
Q.
Pan
, “
q-deformed 3D loop gravity on the torus
,”
Classical Quantum Gravity
37
(
2
),
025017
(
2020
); arXiv:1907.11074.
30.
C.
Rovelli
, “
The basis of the Ponzano-Regge-Turaev-Viro-Ooguri quantum gravity model in the loop representation basis
,”
Phys. Rev. D
48
,
2702
2707
(
1993
); arXiv:hep-th/9304164.
31.
V.
Bonzom
and
L.
Freidel
, “
The Hamiltonian constraint in 3d Riemannian loop quantum gravity
,”
Classical Quantum Gravity
28
,
195006
(
2011
); arXiv:1101.3524.
32.
V.
Bonzom
and
E. R.
Livine
, “
A new Hamiltonian for the topological BF phase with spinor networks
,”
J. Math. Phys.
53
,
072201
(
2012
); arXiv:1110.3272.
33.
V.
Bonzom
,
M.
Dupuis
, and
F.
Girelli
, “
Towards the Turaev-Viro amplitudes from a Hamiltonian constraint
,”
Phys. Rev. D
90
(
10
),
104038
(
2014
); arXiv:1403.7121.
34.
M.
Geiller
, “
Edge modes and corner ambiguities in 3d Chern–Simons theory and gravity
,”
Nucl. Phys. B
924
,
312
365
(
2017
); arXiv:1703.04748.
35.
D.
Grumiller
,
W.
Merbis
, and
M.
Riegler
, “
Most general flat space boundary conditions in three-dimensional Einstein gravity
,”
Classical Quantum Gravity
34
(
18
),
184001
(
2017
); arXiv:1704.07419.
36.
M.
Banados
,
T.
Brotz
, and
M. E.
Ortiz
, “
Boundary dynamics and the statistical mechanics of the 2 + 1-dimensional black hole
,”
Nucl. Phys. B
545
,
340
370
(
1999
); arXiv:hep-th/9802076.
37.
M.
Bañados
and
M. E.
Ortiz
, “
The central charge in three-dimensional anti-de Sitter space
,”
Classical Quantum Gravity
16
,
1733
1736
(
1999
); arXiv:hep-th/9806089.
38.
B.
Oblak
, “
BMS particles in three dimensions
,” Ph.D. thesis,
University of Brussels
,
2016
; arXiv:1610.08526.
39.
K.
Krasnov
, “
Pure connection action principle for general relativity
,”
Phys. Rev. Lett.
106
,
251103
(
2011
); arXiv:1103.4498.
40.
L.
Freidel
and
S.
Speziale
, “
On the relations between gravity and BF theories
,”
SIGMA
8
(
032
),
15
(
2012
); arXiv:1201.4247.
41.
C.
Delcamp
,
L.
Freidel
, and
F.
Girelli
, “
Dual loop quantizations of 3d gravity
,” arXiv:1803.03246.
42.
C.
Rovelli
and
L.
Smolin
, “
Spin networks and quantum gravity
,”
Phys. Rev. D
52
,
5743
5759
(
1995
); arXiv:gr-qc/9505006.
43.
E. R.
Livine
, “
The spinfoam framework for quantum gravity
,” Habilitation Thesis (
ENS Lyon
,
2010
, Vol. 10.
44.
A.
Perez
, “
The spin foam approach to quantum gravity
,”
Living Rev. Relativ.
16
(
3
),
3
(
2013
); arXiv:1205.2019.
45.
N.
Bodendorfer
, “
An elementary introduction to loop quantum gravity
,” arXiv:1607.05129.
46.
A.
Blommaert
,
T. G.
Mertens
, and
H.
Verschelde
, “
The Schwarzian theory—A Wilson line perspective
,”
J. High Energy Phys.
2018
(
12
),
22
(
2018
); arXiv:1806.07765.
47.
T.
Thiemann
, “
Modern canonical quantum general relativity
,” arXiv:gr-qc/0110034.
48.
K.
Noui
and
A.
Perez
, “
Three-dimensional loop quantum gravity: Coupling to point particles
,”
Classical Quantum Gravity
22
,
4489
4514
(
2005
); arXiv:gr-qc/0402111.
49.
K.
Noui
and
A.
Perez
, “
Three-dimensional loop quantum gravity: Physical scalar product and spin foam models
,”
Classical Quantum Gravity
22
,
1739
1762
(
2005
); arXiv:gr-qc/0402110.
50.
R.
Dowdall
,
H.
Gomes
, and
F.
Hellmann
, “
Asymptotic analysis of the Ponzano–Regge model for handlebodies
,”
J. Phys. A: Math. Theor.
43
,
115203
(
2010
); arXiv:0909.2027.
51.
R. J.
Dowdall
and
W. J.
Fairbairn
, “
Observables in 3d spinfoam quantum gravity with fermions
,”
Gen. Relativ. Gravitation
43
,
1263
1307
(
2011
); arXiv:1003.1847.
52.
J. W.
Barrett
and
F.
Hellmann
, “
Holonomy observables in Ponzano–Regge-type state sum models
,”
Classical Quantum Gravity
29
,
045006
(
2012
); arXiv:1106.6016.
53.
B.
Dittrich
,
C.
Goeller
,
E. R.
Livine
, and
A.
Riello
, “
Quasi-local holographic dualities in non-perturbative 3D quantum gravity
,”
Classical Quantum Gravity
35
(
13
),
13LT01
(
2018
); arXiv:1803.02759.
54.
B.
Dittrich
,
C.
Goeller
,
E. R.
Livine
, and
A.
Riello
, “
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
); arXiv:1710.04202.
55.
B.
Dittrich
,
C.
Goeller
,
E. R.
Livine
, and
A.
Riello
, “
Quasi-local holographic dualities in non-perturbative 3d quantum gravity II—From coherent quantum boundaries to BMS3 characters
,”
Nucl. Phys. B
938
,
878
934
(
2019
); arXiv:1710.04237.
56.
C.
Goeller
,
E. R.
Livine
, and
A.
Riello
, “
Non-perturbative 3D quantum gravity: Quantum boundary states and exact partition function
,”
Gen. Relativ. Gravitation
52
(
3
),
24
(
2020
); arXiv:1912.01968.
57.
A.
Ashtekar
,
V.
Husain
,
C.
Rovelli
,
J.
Samuel
, and
L.
Smolin
, “
2 + 1-quantum gravity as a toy model for the 3 + 1 theory
,”
Classical Quantum Gravity
6
,
L185
(
1989
).
58.
E.
Batista
and
S.
Majid
, “
Noncommutative geometry of angular momentum space U(su(2))
,”
J. Math. Phys.
44
,
107
137
(
2003
); arXiv:hep-th/0205128.
59.
I.
Gutierrez-Sagredo
,
A.
Ballesteros
, and
F. J.
Herranz
, “
Drinfel’d double structures for Poincaré and Euclidean groups
,”
J. Phys. Conf. Ser.
1194
(
1
),
012041
(
2019
); arXiv:1812.02075.
60.
E.
Joung
,
J.
Mourad
, and
K.
Noui
, “
Three dimensional quantum geometry and deformed Poincare symmetry
,”
J. Math. Phys.
50
,
052503
(
2009
); arXiv:0806.4121.
61.
L.
Freidel
and
S.
Majid
, “
Noncommutative harmonic analysis, sampling theory and the Duflo map in 2 + 1 quantum gravity
,”
Classical Quantum Gravity
25
,
045006
(
2008
); arXiv:hep-th/0601004.
62.
A.
Ashtekar
and
J. D.
Romano
, “
Chern-Simons and palatini actions and (2 + 1) gravity
,”
Phys. Lett. B
229
,
56
60
(
1989
).
63.
R.
Kashaev
, “
Heisenberg double and pentagon relation
,” arXiv:q-alg/9503005 (
1995
).
64.
N.
Aghaei
and
M.
Pawelkiewicz
, “
Heisenberg double and Drinfeld double of the quantum superplane
,” e-Print: arXiv:1909.04565 [math.QA].
65.
A.
Maitland
, “
A first taste of quantum gravity effects: Deforming phase spaces with the Heisenberg double
,” Master Thesis (
University of Waterloo, ON, Canada
,
2014
), available at https://uwspace.uwaterloo.ca/handle/10012/8866.
66.
D.
Kazhdan
and
G.
Lusztig
, “
Tensor structures arising from affine Lie algebras
,”
J. Am. Math. Soc.
6
(
4
),
905
947
(
1993
).
67.
L.
Faddeev
and
A. Y.
Volkov
, “
Abelian current algebra and the Virasoro algebra on the lattice
,”
Phys. Lett. B
315
,
311
318
(
1993
); arXiv:hep-th/9307048.
68.
L.
Faddeev
and
A. Y.
Volkov
, “
Shift operator for nonAbelian lattice current algebra
,”
Publ. Res. Inst. Math. Sci.
40
,
1113
1125
(
2004
); arXiv:hep-th/9606088.
69.
L. D.
Faddeev
and
A. Y.
Volkov
, “
Algebraic quantization of integrable models in discrete space-time
,” arXiv:hep-th/9710039.
70.
L. D.
Faddeev
,
R. M.
Kashaev
, and
A. Y.
Volkov
, “
Strongly coupled quantum discrete Liouville theory. I: Algebraic approach and duality
,”
Commun. Math. Phys.
219
,
199
219
(
2001
); arXiv:hep-th/0006156.
71.
Y.
Zou
,
A.
Milsted
, and
G.
Vidal
, “
Conformal fields and operator product expansion in critical quantum spin chains
,”
Phys. Rev. Lett.
124
(
4
),
040604
(
2020
); arXiv:1901.06439.
72.
L.
Grans-Samuelsson
,
J. L.
Jacobsen
, and
H.
Saleur
, “
The action of the Virasoro algebra in quantum spin chains. Part I. The non-rational case
,”
J. High Energy Phys.
2021
(
2
),
130
; arXiv:2010.12819.
73.
L.
Grans-Samuelsson
,
L.
Liu
,
Y.
He
,
J. L.
Jacobsen
, and
H.
Saleur
, “
The action of the Virasoro algebra in the two-dimensional Potts and loop models at generic Q
,”
J. High Energy Phys.
2020
(
10
),
109
; arXiv:2007.11539.
74.
A.
Milsted
and
G.
Vidal
, “
Tensor networks as conformal transformations
,” arXiv:1805.12524.
75.
F.
Pastawski
,
B.
Yoshida
,
D.
Harlow
, and
J.
Preskill
, “
Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence
,”
J. High Energy Phys.
2015
(
6
),
149
; arXiv:1503.06237.
76.
L.
Freidel
and
K.
Krasnov
, “
2D conformal field theories and holography
,”
J. Math. Phys.
45
,
2378
2404
(
2004
); arXiv:hep-th/0205091.
77.
K.
Krasnov
, “
Holography and Riemann surfaces
,”
Adv. Theor. Math. Phys.
4
,
929
979
(
2000
); arXiv:hep-th/0005106.
78.
K.
Krasnov
, “
Twistors, CFT and holography
,” arXiv:hep-th/0311162.
79.
K.
Krasnov
, “
Holography for the Lorentz group Racah coefficients
,”
Classical Quantum Gravity
22
,
1933
1944
(
2005
); arXiv:gr-qc/0412104.
80.
L.
Freidel
,
K.
Krasnov
, and
E. R.
Livine
, “
Holomorphic factorization for a quantum tetrahedron
,”
Commun. Math. Phys.
297
,
45
93
(
2010
); arXiv:0905.3627.
81.
B.
Dittrich
and
J.
Hnybida
, “
Ising model from intertwiners
,” arXiv:1312.5646.
82.
V.
Bonzom
,
F.
Costantino
, and
E. R.
Livine
, “
Duality between spin networks and the 2D Ising model
,”
Commun. Math. Phys.
344
(
2
),
531
579
(
2016
); arXiv:1504.02822.
83.
E. A.
Mazenc
,
V.
Shyam
, and
R. M.
Soni
, “
A TT̄ deformation for curved spacetimes from 3d gravity
,” arXiv:1912.09179.
84.

More precisely, it ensures that the leakage of symplectic flux can be reabsorbed into a boundary redefinition of the symplectic form ΩΣΩΣ=ΩΣ+SωR.

85.

In order to clearly visualize the deformation of the boundary vector fields, one can introduce an arbitrary parameter γ and define the vector fields on Σ as ξXX∂θ + γXr. This parameter does not affect the action of the diffeomorphisms and, in the end, simply rescales the central charges λ and μ.

86.
If one introduces a shift in the vacuum energy and defines D̂n=Dnc24δn,0, one recovers the usual expression
87.

We could imagine a fine-tuned scenario where the radial fields Ar and er still rotate while keeping their scalar products constant, but this does not seem at a first glance to affect the charge conservation in an interesting way.

88.

As a Lie algebra, this is simply the Euclidean algebra iso(3), where X̂ plays the role of angular momenta, while Ĝa=λPa plays the role of momenta. The parameter λ with the physical dimension of a length plays the role of the Planck length in 3D gravity.58,60

89.

This completion contains only finite sums of the exponential function. It is possible to include a wider class of functions for this completion; see Ref. 61. For the purpose of this paper, we only need to consider the group algebra.

90.
With the standard tensorial notations, the three R-matrices acting on the triple tensorial product are explicitly
91.

The comma between the indices is introduced in order to easily distinguish the index. It will appear in the notation when needed.

92.
The action of Ĝij on another elements Ĝpq is the action by conjugation
93.

From a broader point of view, this is reminiscent of a celebrated and fundamental result of mathematical physics anticipated by Moore and Seiberg and due to Kazhdan and Lusztig,66 who proved the equivalence as the braided tensor category between the category of positive energy representations of the loop algebra L(g) at level and the category of representations of the quantum group Uq(g) for q=eiπκ with κ=+ȟQ0, where ȟ is the dual coxeter number. We leave for future investigation the precise dictionary and correspondence between the discrete and continuum objects and algebras, which is clearly the missing part of the present study. Our goal would then be to establish in a mathematically rigorous way the equivalent of the Kazhdan–Lusztig result at the discrete level.

94.

The notation is such that P[α.+1+α.,φ]=j=1N[αj+1+αj,φj]cĝjc.

95.
We simply use the fact that
with the fact that ĝ01 and ĝa1N to show that
96.

With the usual convention where = 1.

97.

The floor function ⌊·⌋ is defined by 2p+12=p and 2p2=p for an integer p.

98.

All operators trivially commute with Ĵn0.

You do not currently have access to this content.