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.

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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
i{D̂n,D̂m}=(nm)D̂n+m+c12n(n21)δn+m,0.
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
R12=k(δkI)(1k)(1I),R13=k(δkI)(1I)(1k),R23=k(1I)(δkI)(1k).
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
ĜijĜpq=ĜijĜpqĜij1.
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
αa(θj+1)+αa(θj)=2αa(θj)+O1Nandαa(θj+1)αa(θj)=2πNθαa|θj+O1N2
with the fact that ĝ01 and ĝa1N to show that
i4ĝja(αj+1aαja)O1N2N0,P[α.+1+α.,φj]2P[α,φ]k=1N(αj+1aαja)φjaĝj0=2πNk=1Nθαa|θjφa(θj)φdα.
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.

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