String Theory Demystified

64 String Theory Demystifi ed
µ ν
Moving to the last line, we dropped the term ∂αb ∂βb
. This is because we are
assuming that b µ is a small displacement, and so we neglect terms in second order.
You will recognize that the fi rst term in the last line is just the original lagrangian
(density). So we separate the result as
L
P
ν
( )
T αβ µ ν µ ν µ
→− −hh ∂αX ∂ β X +∂αX ∂ βb
+∂αb ∂βX
2
ν
η ( ββ X )
T αβ µ ν T αβ µ ν µ
=− −hh ∂αX ∂βX µν − −hh ∂αX ∂ βb
+∂αb ∂
2 2
= L + δ L
P p
The second term δ LPwill be associated with the conserved current. To get it, we want
to peel off terms involving b µ . In order to do this, we will need to get the same
indices α, β , µ ,and ν on both terms. This is easy because we can exploit the symmetry
of the metric. Take a look at the fi rst term. We are going to manipulate it to get the
form we want in three steps. First, recalling that repeated indices are dummy
indices that we can call what we want, we swap the labels µ ↔ ν.
Then we exploit
the symmetry of the metric to write it the way it originally was, and then we
lower an index:
T
αβ µ ν T
αβ ν µ
− −hh ∂ X ∂ b η = − −hh ∂ X ∂ b η (relabel dummy indices µ ↔ ν)
α β µν
α β νµ
2 2
( ) ( )
So now
T
2
αβ ν µ
=− −hh ∂ X ∂ b
α ββ
T
αβ
= − −hh ∂ X ∂ b
α µ β
2
( ) η (symmetry of the metric η = η )
µν µν νµ
µ ( )
η
µν
(lower an index)
T αβ
µ T αβ µ ν
δLP =− −hh ( ∂α Xµ ∂βb)
− −hh ∂αb∂βXη 2 2
( )
Now we work on the second term, in two steps. First we lower an index, and
then we swap the labels used for the dummy indices α ↔ β and again exploit the
µν
η
µν