A. Ghosh, D. Pranzetti / Nuclear Physics B 889 (2014) 1–24 5
Fig. 1. Hole H around the source p. D is a disk around H where the connection is flat. Its boundary ∂D consists of two
components, an inner boundary ∂H and an outer boundary B.
where the connection A = A
i
μ
dx
μ
τ
i
has curvature F(A) =dA + A ∧ A and the generators {τ
i
}
form a basis of the su(2) Lie algebra with normalisation tr (τ
i
τ
j
) =−(1/2)δ
ij
and [τ
i
, τ
j
] =
ij k
τ
k
. The phase space of the theory (1) is described by the equal time Poisson brackets
A
i
a
(x), A
j
b
(y)
=δ
ij
ab
4π
k
δ
2
(x −y), a,b = 1, 2; x
0
=y
0
(2)
with the convention
012
≡
12
=−
21
= 1. Interaction with a static non-abelian charged source
at a point p in D with the CS field is encoded in the interaction action [31,39]
S
int
=λ
j
c
tr
τ
3
Λ
−1
dΛ +Λ
−1
AΛ
, (3)
where c is a Wilson-line (a timelike path), λ
j
is a coupling constant in which the label j refers to
the spin-j unitary irreducible representation of SU(2) and Λ is a group element in the fundamen-
tal representation. Let us expand the Lie algebra element Λ
−1
τ
i
Λ = L
ik
τ
k
where the matrix L
is such that L
−1
= L
T
and det L = 1. In the Hamiltonian framework, variation of the two terms
(1), (3) leads to the constraint
F
i
12
A(x)
=−
2π
k
S
i
δ
2
(x −p), (4)
where S
i
=
1
2
λ
j
L
i3
is the conjugate variable to Λ (the phase space of the source is T
∗
(SU(2)))
in the sense that it satisfies the Poisson brackets
S
i
,Λ
=τ
i
Λ. (5)
In presence of a point source the connection A(x) is not well defined as x approaches p and,
therefore, a regularisation is needed [40]. This can be done by piercing a hole H containing
p and eventually shrinking the hole to p (see Fig. 1). Such regularisation is the analog of the
framing technique needed in the co
variant context to solve the ambiguities appearing in the
computation of the expectation values of the Wilson lines [31]; it represents a standard procedure
to deal with the conical singularity produced by a particle in 3-dim gravity [33]. Also in the
context of isolated horizons, such regularisation is not a n
ovel feature of our approach. In fact,
already in [3], as an intermediary step to the construction of the horizon Hilbert space for a set
of punctures on a 2-sphere, an analog set of paths was introduced. More precisely, in order to
construct the quantum phase space consisting of generalised connections that are flat e
verywhere
on the 2-sphere except at the punctures, it was necessary to introduce (at the classical level)
some extra structure associated to certain equivalence class of paths. It was proven that such
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