String Theory and M-Theory

3.1 Conformal field theory 61
rotations are required to satisfy ωµν = −ωνµ. Altogether there are
D + 1
1
D(D − 1) + 1 + D = (D + 2)(D + 1) (3.8)
2 2
linearly independent infinitesimal conformal transformations, so this is the
number of generators of the conformal group.
The number of conformal-group generators in D dimensions is the same
as for the group of rotations in D + 2 dimensions. In fact, by commuting
the infinitesimal conformal transformations one can derive the Lie algebra,
and it turns out to be a noncompact form of SO(D + 2). In the case of
Lorentzian signature, the Lie algebra is SO(D, 2), while if the manifold is
Euclidean it is SO(D + 1, 1).
When D > 2 the algebras discussed above generate the entire conformal
group, except that an inversion is not infinitesimally generated. Because of
the inversion element, the groups have two disconnected components. When
D = 2, the SO(2, 2) or SO(3, 1) algebra is a subalgebra of a much larger
algebra.
The conformal group in two dimensions
As has already been remarked, conformal transformations in two dimensions
consist of analytic coordinate transformations
z → f(z) and ¯z → ¯ f(¯z). (3.9)
These are angle-preserving transformations wherever f and its inverse function
are holomorphic, that is, f is biholomorphic.
To exhibit the generators, consider infinitesimal conformal transformations
of the form
z → z ′ = z − εnz n+1
and ¯z → ¯z ′ = ¯z − ¯εn¯z n+1 , n ∈
The corresponding infinitesimal generators are 2
. (3.10)
ℓn = −z n+1 ∂ and ¯ ℓn = −¯z n+1 ¯ ∂, (3.11)
where ∂ = ∂/∂z and ¯ ∂ = ∂/∂¯z. These generators satisfy the classical
Virasoro algebras
[ℓm, ℓn] = (m − n)ℓm+n and ¯ℓm, ¯
ℓn = (m − n) ℓm+n, ¯ (3.12)
while ℓm, ¯
ℓn = 0. In the quantum case the Virasoro algebra can acquire
2 For n < −1 these are defined on the punctured plane, which has the origin removed. Similarly,
for n > 1, the point at infinity is removed. Note that ℓ−1, ℓ0 and ℓ1 are special in that they
are defined globally on the Riemann sphere.