JHEP12(2018)093
vortex endpoints of any (3 + 1) dimensional cosmic string, in order for the string to obey
fractional statistics.
2 Fractional statistics of Abelian vortices in (2 + 1) dimensions
Equipping vortices with an electric charge generally leads to a contribution to the effective
action S of the Hopf form [21]
S ⊇ S
Hopf
= β
Z
j
(vortex)µ
µνλ
∂
ν
∂
2
J
(el)λ
, (2.1)
obtained after integrating out any mediating gauge fields after appropriate gauge fix-
ing. Here, β is a theory dependent coefficient. The Hopf term describes the interac-
tion of a (topological) vortex current j
(vortex)µ
whose charge corresponds to the winding
number, with an electric current J
(el)λ
, also localized on a vortex. In its presence, in-
terchanging two vortices with electric charges q
(i)
e
and winding numbers n
(i)
implies an
Aharonov-Bohm phase
∆S =
β
2
n
(1)
q
(2)
e
+ n
(2)
q
(1)
e
. (2.2)
For suitably chosen β, this equation entails the existence of anyonic vortices.
In the following, we discuss two explicit realizations of this setup, which are in part
well known results (see e.g. [9]). In the first scenario, vortices are equipped with electric
charge by means of a Chern-Simons term in their action. In that case, the electric current
above is identical to the vortex current. In another setup, the vortex is explicitly charged
under an additional gauge field.
2.1 Electrically charged Chern-Simons vortices
Electrically charged static Abelian vortices in (2+1) dimensional Minkowski spacetime can
be obtained numerically as static lowest energy topologically non-trivial configurations in
the theory with a Chern-Simons term added to the Nielsen-Olesen Lagrangian [7]:
L = −
1
4
F
µν
F
µν
+
1
2
(D
µ
φ)
†
(D
µ
φ) −
λ
4
φ
†
φ − v
2
2
+ µ
µνα
A
µ
∂
ν
A
α
. (2.3)
Here F
µν
≡ ∂
[µ
A
ν]
, D
µ
φ ≡ ∂
µ
φ − ieA
µ
φ and φ is a complex scalar field which can be
parametrized as
φ(x) = ρ(x)e
iθ
(v)
(x)
, (2.4)
with the two real valued functions ρ(x) and θ
(v)
(x). This Lagrangian is invariant under the
U(1) transformation φ → e
iw(x)
φ, A
µ
→ A
µ
+
1
e
∂
µ
w(x) with w(x) the gauge parameter.
Electrically charged vortices with the electric current J
µ
given by
J
µ
=
ie
2
φ (D
µ
φ)
†
− φ
†
D
µ
φ
, (2.5)
have been found in [7] numerically by minimizing the energy functional which corresponds
to this Lagrangian after using appropriate ansatz-functions and boundary conditions for
– 4 –
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