7.1 Free Scalar
145
The solution is
G
(r) = −
2 ln r
(7.13)
and the form (7.9) follows by writing
ln r =
1
2
ln r
2
=
1
2
ln |σ − σ
|
2 .
(7.14)
The action (7.1) is obviously invariant under constant translations of X:
X −→ X + a,
a ∈ R.
(7.15)
The associated U(1) current 2 is conserved and reads
J
μ
:= 2π i
∂L
∂(∂ μ X)
=
i
2 g
μν ∂ ν X,
∇ μ J
μ
= 0.
(7.16)
On flat space, the charge follows from (A.23):
p =
1
2π
dσ J
0
=
i
2ππ 2
dσ ∂
0 X.
(7.17)
This charge is called momentum because it corresponds to the spacetime momentum
in string theory.
Moreover, there is a another topological current
J
μ
:= −i
μν J ν =
1
2
μν ∂ ν X,
(7.18)
which is identically conserved:
∇ μ
J
μ
∝
μν
[∇ μ , ∇ ν ]X = 0
(7.19)
since [∇ μ , ∇ ν ] = 0 when acting on a scalar field. Note that ˜
J μ is the Hodge dual of
J μ . The conserved charge is called the winding number and reads on flat space:
w =
1
2π
dσ
J
0
=
1
2ππ 2
2π
0
dσ ∂ 1 X =
1
2ππ 2
X(τ, 2π) − X(τ, 0)
.
(7.20)
Remark 7.1 (Normalization of the Current) The definition of the current (7.16) may
look confusing. The factor of i is due to the Euclidean signature, see (A.25a), and the
2 The group is R but the algebra is u(1) (since locally there is no difference between the real line
and the circle).