String Theory and M-Theory

74 Conformal field theory and string interactions
=
dw
2πi
c
12 (m3 − m)w m+n−1 + 2(m + 1)w n+m+1 T (w) + w m+n+2 ∂T (w)
Performing the integral over w on a path encircling the origin, yields the
Virasoro algebra
[Lm, Ln] = (m − n)Lm+n + c
12 (m3 − m)δm+n,0.
EXERCISE 3.3
Verify that the expressions (3.38) and (3.39) for the transformation of the
energy–momentum tensor under conformal transformations are consistent
with Eq. (3.37) for an infinitesimal transformation w(z) = z + ε(z).
SOLUTION
Under the infinitesimal transformation f(z) = z+ε(z), Eqs (3.38) and (3.39)
reduce to T (z) → T (z) + δεT (z) with
δεT (z) = −2∂ε(z)T (z) − ε(z)∂T (z) − c
12 ∂3 ε(z).
On the other hand, using Eq. (3.30), the change of T (w) under an infinitesimal
conformal transformation is given by
dz
dz
δεT (w) = ε(z)[T (z), T (w)] = ε(z)T (z)T (w),
2πi 2πi
where the integration contour C is the one displayed in Fig. 3.2. Using
Eq. (3.37), this becomes
dz
2πi ε(z)
c/2 2T (w) ∂T (w)
+ +
(z − w) 4 (z − w) 2 z − w
C
= 2∂ε(w)T (w) + ε(w)∂T (w) + c
12 ∂3 ε(w).
But δεT (w) = −δεT (z), since z ∼ w − ε(w). This shows that both methods
yield the same result for ∂εT (z) to first order in ε. ✷
EXERCISE 3.4
Show that Eqs (3.38) and (3.39) satisfy the group property by considering
two successive conformal transformations.
C
.
✷