String Theory and M-Theory

72 Conformal field theory and string interactions
which is the desired result. The first nontrivial case is m = 1, which has
c = 1/2. It has been proved that these are all of the unitary representations
of the Virasoro algebra with c < 1.
To understand the structure of these unitary minimal models, one should
also determine all of their highest-weight states. Equivalently, one can identify
the primary fields that give rise to the highest-weight states by acting
on the conformal vacuum |0〉. Since |0〉, itself, is a highest-weight state, the
identity operator I is a primary field (with h = 0). Using the known SU(2)k
representations, one can work out all of the primary fields of these minimal
models. The result is a collection of conformal fields φpq with conformal
dimensions hpq given by
hpq = [(m + 3)p − (m + 2)q]2 − 1
, 1 ≤ p ≤ m + 1 and 1 ≤ q ≤ p.
4(m + 2)(m + 3)
(3.75)
Because of the symmetry (p, q) → (m + 2 − p, m + 3 − q), an equivalent
labeling is to allow 1 ≤ p ≤ m + 1, 1 ≤ q ≤ m + 2 and to restrict p − q to
even values. For example, the m = 1 theory, with c = 1/2, describes the
two-dimensional Ising model at the critical point. It has primary fields with
dimensions h11 = 0 (the identity operator), h21 = 1/2 (a free fermion), and
h22 = 1/16 (a spin field).
Note that the minimal models have c < 1 and accumulate at c = 1.
This limiting value c = 1 can be realized by a free boson X. There are
actually a continuously infinite number of possibilities for c = 1 unitary
representations, since the coordinate X can describe a circle of any radius. 7
EXERCISES
EXERCISE 3.1
Use the oscillator expansion in Eq. (3.21) to derive the OPE:
SOLUTION
∂X µ (z)∂X ν (w) = − 1 η
4
µν
+ . . .
(z − w) 2
Since the singular part of the OPE of the two fields ∂X µ (z) and ∂X ν (w)
7 Chapter 6 shows that radius R and radius α ′ /R are equivalent.