A Conventions
375
where N L and N R are the numbers of left- and right-chirality supersymmetries. The
last form is used when it is not important to know the chirality of the supercharges.
The variation of a field φ(x) is defined by
δφ(x) = φ
(x) − φ(x).
(A.21)
Given an internal symmetry with parameters α a , the Noether current in Lorentzian
signature is given by
J
μ
a = λ
∂L
∂(∂ μ φ)
δφ
δα a ,
∇ μ J
μ
a = 0,
(A.22)
where L is the Lagrangian, which does not include the factor
√ g for curved spaces,
and λ is some normalization. 5 The conserved charges Q a associated to the currents
J
μ
a for a fixed spatial slice t = cst are
Q a =
1
λ
d
D−1 x
√
h J
0
a ,
(A.23)
where is a spatial slice and h is the induced metric. One sets λ = 2π in two
dimensions, otherwise λ = 1. The variation of a field under a transformation
generated by Q is
δ α a φ(x) = iα
a
[Q a , φ(x)].
(A.24)
In Euclidean signature, the current and variation are
J
μ
a = iλ
∂L
∂(∂ μ φ)
δφ
δα a ,
(A.25a)
δ α a φ(x) = −α
a
[Q a , φ(x)].
(A.25b)
Note that the charge is still given by (A.23). The factor of i in (A.25a) can be
understood as follows. 6 First, the time component J 0
a of the current transforms
like time such that J 0
a → iJ 0
a , which implies that the charge also gets a factor i,
Q a → iQ a . This explains the minus sign in (A.25b). Then, one needs to make
this consistent with the formula (B.9) for the charge associated to a general surface.
Given a spacelike n μ , the integration measure includes the time which transforms
with a factor of i: one can interpret it as coming from the spatial components of
the current, J i
a → iJ i
a , while working with a Euclidean region. Another way to
5 Including the
√
g would give the current density
√ gJ
μ
a . The simple derivative of the latter
vanishes ∂ μ (
√ gJ
μ
a ) = 0 in view of the identity (B.4).
6 We stress that these formulas and arguments do not apply to the energy–momentum tensor.
375
where N L and N R are the numbers of left- and right-chirality supersymmetries. The
last form is used when it is not important to know the chirality of the supercharges.
The variation of a field φ(x) is defined by
δφ(x) = φ
(x) − φ(x).
(A.21)
Given an internal symmetry with parameters α a , the Noether current in Lorentzian
signature is given by
J
μ
a = λ
∂L
∂(∂ μ φ)
δφ
δα a ,
∇ μ J
μ
a = 0,
(A.22)
where L is the Lagrangian, which does not include the factor
√ g for curved spaces,
and λ is some normalization. 5 The conserved charges Q a associated to the currents
J
μ
a for a fixed spatial slice t = cst are
Q a =
1
λ
d
D−1 x
√
h J
0
a ,
(A.23)
where is a spatial slice and h is the induced metric. One sets λ = 2π in two
dimensions, otherwise λ = 1. The variation of a field under a transformation
generated by Q is
δ α a φ(x) = iα
a
[Q a , φ(x)].
(A.24)
In Euclidean signature, the current and variation are
J
μ
a = iλ
∂L
∂(∂ μ φ)
δφ
δα a ,
(A.25a)
δ α a φ(x) = −α
a
[Q a , φ(x)].
(A.25b)
Note that the charge is still given by (A.23). The factor of i in (A.25a) can be
understood as follows. 6 First, the time component J 0
a of the current transforms
like time such that J 0
a → iJ 0
a , which implies that the charge also gets a factor i,
Q a → iQ a . This explains the minus sign in (A.25b). Then, one needs to make
this consistent with the formula (B.9) for the charge associated to a general surface.
Given a spacelike n μ , the integration measure includes the time which transforms
with a factor of i: one can interpret it as coming from the spatial components of
the current, J i
a → iJ i
a , while working with a Euclidean region. Another way to
5 Including the
√
g would give the current density
√ gJ
μ
a . The simple derivative of the latter
vanishes ∂ μ (
√ gJ
μ
a ) = 0 in view of the identity (B.4).
6 We stress that these formulas and arguments do not apply to the energy–momentum tensor.