String Theory Demystified

CHAPTER 5 Conformal Field Theory Part I
We’ve shown that Eq. (5.14) is an equivalent way to write the Polyakov action.
But it’s much simpler, and it’s much simpler to derive the equations of motion using
this form. We can do this by varying the action [Eq. (5.14)] with respect to the
coordinate X µ. This is done by letting X → X + δ X . Then
µ µ µ
1 2 µ
1 2 µ
S→d z∂X ∂ X + X = d z∂X ∂ X +∂
2 ′ ∫ ( µ δ µ )
πα
2πα
′ ∫ ( µ δ Xµ
)
1 2 µ 1 2 µ
= dz∂X∂ Xµ + dz∂X( ∂ δXµ ) = S+
δ
2πα
′ ∫ 2πα
′ ∫
S
We can obtain the equations of classical motion by requiring that δS = 0. Integrating
by parts and discarding the boundary term:
1 2 µ
δS
= d z X δ Xµ
πα ′ ∫ ∂ ( ∂ )
2
1 2 µ
=− dz∂∂X δ Xµ
2πα
′ ∫ ( )
We have used the fact that partial derivatives commute. This term must vanish
for the action to be invariant. Therefore it must be the case that
97
µ
∂∂ X (, z z)
= 0 (5.15)
µ µ
We’ve written X = X (, z z)to
emphasize that in general the coordinates can be
a function of z and z.
However, as you might guess from your studies of complex
variables there is a special case of interest, that of analytic or holomorphic functions.
A function f (, z z)is
holomorphic if
That is, f = f( z)
only. On the other hand, if
∂
∂ =
f
0 (5.16)
z
∂
∂ =
f
0 (5.17)
z
and f = f( z),
then we say that f is antiholomorphic. In string theory, if ∂( ∂ X ) =
µ
0
then ∂X µ is a holomorphic function which is called left moving. In the other case,
where ∂( ∂ X ) =
µ
0 , the function ∂X µ is antiholomorphic and is called right moving.