String Theory and M-Theory

3.1 Conformal field theory 67
Such fields play an important role in the next chapter. A free fermi field has
T (z) = − 1
: ψ(z)∂ψ(z) : (3.41)
2
which leads to c = 1/2.
The fact that a pair of fermi fields gives c = 1 is significant. When a free
scalar field takes values on a circle of suitable radius, there is an equivalent
theory in which the scalar field is replaced by a pair of fermi fields. The
replacement of a boson by a pair of fermions is called fermionization, and
its (more common) inverse is called bosonization. It is not our purpose to
explore this in detail here, just to point out that the central charges match
up. In fact, in the simplest case the formulas take the form 5
ψ± = : exp(±iφ) : . (3.42)
Here φ is a boson normalized in the usual way, so that the normal-ordered
operator has dimension 1/2. Clearly, for this expression to be single-valued,
φ should have period 2π.
Given a holomorphic primary field Φ(z) of dimension h, one can associate
a state |Φ〉 that satisfies
L0|Φ〉 = h|Φ〉 and Ln|Φ〉 = 0, n > 0. (3.43)
Such a state is called a highest-weight state. This state–operator correspondence
is another very useful concept in conformal field theory. The relevant
definition is
|Φ〉 = lim
z→0 Φ(z)|0〉, (3.44)
where |0〉 denotes the conformal vacuum. Recall that z = 0 corresponds to
the infinite past in Euclidean time. Writing a mode expansion
the way this works is that
Φ(z) =
∞
n=−∞
Φn
, (3.45)
zn+h Φn|0〉 = 0 for n > −h and Φ−h|0〉 = |Φ〉. (3.46)
A highest-weight state |Φ〉, taken together with the infinite collection of
states of the form
L−n1L−n2 . . . L−nk |Φ〉, (3.47)
5 Strictly speaking, the right-hand side of this equation should contain another factor called a
cocycle. However, this can often be ignored.