String Theory Demystified

106 String Theory Demystifi ed
Worldsheet of
closed string
τ 1
z=e t+is
The energy-momentum tensor of the worldsheet is a conserved quantity, meaning
that
µ
∂ T = 0 (5.31)
µν
The energy-momentum tensor is also traceless. This means that in the coordinates
(, )
z z , the components T zz = 0. The conservation condition [Eq. (5.31)] implies that
∂ zTzz =∂ zTzz = 0 (5.32)
That is, the energy-momentum tensor is composed of a holomorphic and
antiholomorphic functions given by Tzz and Tzz,
respectively. A holomorphic
function has a Laurent series expansion, which we write as
∞
Lm
Tzz ()= z
m+
z
∑
m=−∞
2 (5.33)
We have written this expression in a way anticipating that the Laurent coeffi cients
are the Viarasoro generators. The antiholomorphic component also has a Laurent
expansion:
∞
Lm
Tzz ( z)=
m+
z
R
z plane
Figure 5.1. The worldsheet of a closed string is mapped to the z plane. A slice
through the cylinder, at a constant time, is mapped to a circle of a
fi xed radius in the z plane. The radius of the circle in the z plane
corresponds to (Euclidean) time on the worldsheet.
∑
m=−∞
2 (5.34)