Gravity and Strings

30 Noether’s theorems
First, observe that, in general, in the local case, the transformations contain derivatives of
the local parameters. We eliminate these derivatives of the transformation parameters by
integration by parts in Eq. (2.16). We obtain an identity of the form4
d d
x ∂µj µ
N2 (σ ) + σ I
δS
DI , (2.23)
δϕ
where DI are operators containing derivatives acting on the equations of motion. This identity
is true for arbitrary parameters. We can choose parameters such that j µ
N2 (σ ) vanishes on
the boundary. Then, we obtain the off-shell identities that do not involve the transformation
parameters
δS
DI = 0, (2.24)
δϕ
that relate the equations of motion, so not all of them are independent. These identities are
called gauge or Bianchi identities. Since they are identically true for arbitrary values of the
parameters, we obtain the off-shell “conservation law” 5
∂µj µ
N2 (σ ) = 0. (2.25)
Since this is an identity that holds independently of the equations of motion, it follows
that the current density j µ
N2 (σ ) can always be written as the divergence of a two-index
antisymmetric tensor, usually called the superpotential, that is,
j µ
N2
νµ
(σ ) = ∂νj (σ ), jνµ (σ ) =−jµν(σ
). (2.26)
N2
This identity for the vector densities is written in terms of the vectors j µ
N2 = √ |g| j µ
N2 :
j µ
N2 (σ ) =∇ν j νµ
N2 (σ ),
νµ
µν
jN2 (σ ) =−j N2 (σ ). (2.27)
Observe that the difference between j µ
N1 and jµ
N2 is always a term proportional to the
equations of motion, i.e. it vanishes on-shell. Thus, these two currents are identical on-
shell. In general we are free to add any term that vanishes on-shell to the current j µ
N1 since
it is conserved only on-shell. We have just seen that there is a specific on-shell vanishing
term that relates j µ µ µ
N1 to j N2 . j N2 cannot be modified in this way because its defining property
is that it is conserved off-shell. However, we could add to both currents terms of the form
∂ν νµ , where µν = [µν] , which would change the superpotential. If νµ is of the form
∂ρU ρνµ , with U ρνµ = U [ρν]µ , then ∂ν νµ = 0 and the change in the superpotential will not
change the Noether current.
It is easy to see that Noether currents are sensitive to the addition of total derivatives
to the Lagrangian even if these do not modify the equations of motion: on adding to the
action (2.1)
S = d
d x ∂µL µ , (2.28)
4 This expression is just symbolic. We need a more explicit form of the infinitesimal transformations in order
to obtain more explicit expressions. We will find several examples in the following chapters.
5 Strong conservation law in the language of [110].
N2
N2