String Theory and M-Theory

98 Conformal field theory and string interactions
this takes the form
I =
+1/2
−1/2
∞
dx √
1−x2 +1/2
dy
=
y2 −1/2
dx
√ 1 − x 2
= π
3 ,
where we have set τ = x + iy. ✷
3.6 The linear-dilaton vacuum and noncritical strings
An interesting example of a nontrivial background that preserves conformal
symmetry is one in which the dilaton field depends linearly on the spatial
coordinates. Letting y denote the direction along which it varies and x µ the
other D − 1 space-time coordinates, the linear dilaton background is
Φ(X µ , Y ) = kY (z, ¯z), (3.132)
where k is a constant. After fixing the conformal gauge, the dilaton term
no longer contributes to the world-sheet action, which remains independent
of k, but it does contribute to the energy–momentum tensor.
The energy–momentum tensor for the linear-dilaton background is derived
by varying the action with respect to the world-sheet metric before fixing
the conformal gauge. The result is
T (z) = −2(∂X µ ∂Xµ + ∂Y ∂Y ) + k∂ 2 Y. (3.133)
This expression gives a T T OPE that still has the correct structure to define
a CFT. One peculiarity is that the OPE of T with Y has an extra term
(proportional to k), which implies that ∂Y does not satisfy the definition of
a conformal field.
Calling D the total space-time dimension (including Y ), the central charge
determined by the T T OPE turns out to be
c = ˜c = D + 3k 2 . (3.134)
Thus, the required value c = 26 can be achieved for D < 26 by choosing
26 − D
k = . (3.135)
3
Of course, there is Lorentz invariance in only D − 1 dimensions, since the
Y direction is special. Theories with k = 0 are called noncritical string
theories.
The extra term in T contributes to L0, and hence to the equation of motion
for the free tachyon field t(x µ , y). For simplicity, let us consider solutions