String Theory and M-Theory

68 Conformal field theory and string interactions
which are known as the descendant states, gives a representation of the
(holomorphic) Virasoro algebra known as a Verma module.
Highest-weight states appeared in Chapter 2, where we learned that the
physical open-string states of the bosonic string theory satisfy
and
(L0 − 1)|φ〉 = 0 (3.48)
Ln|φ〉 = 0 with n > 0. (3.49)
Therefore, physical open-string states of the bosonic string theory correspond
to highest-weight states with h = 1. This construction has a straightforward
generalization to primary fields Φ(z, ¯z) of dimension (h, ˜ h). In this
case one has
and
(L0 − h)|Φ〉 = ( L0 − ˜ h)|Φ〉 = 0 (3.50)
Ln|Φ〉 = Ln|Φ〉 = 0 with n > 0. (3.51)
Therefore, physical closed-string states of the bosonic string theory correspond
to highest-weight states with h = ˜ h = 1.
Kac–Moody algebras
Particularly interesting examples of conformal fields are the two-dimensional
currents J A α (z, ¯z), A = 1, 2, . . . , dim G, associated with a compact Lie group
symmetry G in a conformal field theory. Current conservation implies that
there is a holomorphic component J A (z) and an antiholomorphic component
J A (¯z), just as was shown for T earlier. Let us consider the holomorphic
current J A (z) only. The zero modes J A 0 are the generators of the Lie algebra
of G with
[J A 0 , J B 0 ] = if AB CJ C 0 . (3.52)
The algebra of the currents J A (z) is an infinite-dimensional extension of this,
known as an affine Lie algebra or a Kac–Moody algebra G. These currents
have conformal dimension h = 1, and therefore the mode expansion is
J A (z) =
∞
n=−∞
The Kac–Moody algebra is given by the OPE
J A n
A = 1, 2, . . . , dim G . (3.53)
zn+1 J A (z)J B (w) ∼ kδAB
2(z − w) 2 + if AB CJ C (w)
+ . . . (3.54)
z − w