String Theory Demystified

CHAPTER 3 Symmetries and Worldsheet Currents 65
symmetry of the metric, but this time it’s the worldsheet metric we are talking
about:
T
δLP =−
2
αβ
µ T
−hh ( ∂α Xµ ∂βb)
−
2
αβ µ ν
−hh ∂αb∂βXη T
=−
2
αβ
µ T
−hh ( ∂α Xµ ∂βb)
−
2
αβ µ
−hh ( ∂αb∂β Xµ
)
T
= −
2
αβ
µ T
−hh ( ∂α Xµ ∂βb)
−
2
βα µ
−hh ( ∂βb∂α Xµ
)
T
= −
2
αβ
µ T
−hh ( ∂α Xµ ∂βb)
−
2
αβ µ
−hh ( ∂βb∂α Xµ
)
T
= −
2
ααβ
µ T
−hh
( ∂α Xµ ∂βb
) −
2
αβ
µ
−hh ( ∂α Xµ ∂βb)
=−T αβ
µ
−hh ( ∂αXµ
)∂ βb
( )
µ
Now notice we have a term multiplied by ∂βb , which is the small parameter we used
to vary X µ . The rest of this expression is the conserved current we’re looking for:
β αβ
P =−T −hh ( ∂ X ) (3.20)
µ
If we use reparameterization and Weyl invariance to take hαβ → ηαβ
then we have
β
P =−T∂X µ
α µ
α µ
τ
σ
⇒ P =−T∂ X P =−T∂ X
µ
τ µ µ
The conservation equation for the current is
σ µ
α
∂ P = 0 (3.21)
α µ
Dropping the string tension T and ignoring the minus sign we see that the
conservation equation for the current becomes the equation of motion for the string
worldsheet:
τ σ
∂ P +∂ P = 0 (3.22)
τ µ
σ µ
µν