String Theory and M-Theory

86 Conformal field theory and string interactions
identity
α µ
1
−m =
π
suggests that we simply replace
α µ
−m →
z −m ∂X µ dz (3.100)
2i
(m − 1)! ∂m X µ , m > 0. (3.101)
This is not an identity, of course. The right-hand side contains α µ
−m plus an
infinite series of z-dependent terms with positive and negative powers. So,
according to this proposal, a general closed-string vertex operator is given
by an expression of the form
Vφ(z, ¯z) = :
∂ mi
µi X (z) ¯∂ nj νj ik·X(z,¯z)
X (¯z)e :, (3.102)
or a superposition of such terms, where
i
k 2
8
j
= 1 − mi = 1 −
nj. (3.103)
i
It is not at all obvious that this ensures that Vφ has conformal dimension
(1, 1). In fact, this is only the case if the original Fock-space state satisfies
the Virasoro constraints.
Vertex operators can also be introduced in the formalism with Faddeev–
Popov ghosts. In this case the physical state condition is QB|φ〉 = QB|φ〉 =
0. Physical states are BRST closed, but not exact. The corresponding
statement for vertex operators is that if φ is BRST closed, then [QB, Vφ] =
[ QB, Vφ] = 0. Similarly, if φ is BRST exact, then Vφ can be written as the
anticommutator of QB or QB with some operator.
The operator correspondences for the ghosts are
and
b−m →
c−m →
j
1
(m − 2)! ∂m−1 b, m ≥ 2 (3.104)
1
(m + 1)! ∂m+1 c, m ≥ −1. (3.105)
These rules reflect the fact that b is dimension 2 and c is dimension −1. In
particular, the unit operator is associated with a state that is annihilated
by bm with m ≥ −1 and by cm with m ≥ 2. Such a state is uniquely
(up to normalization) given by b−1| ↓〉, which has ghost number −3/2. Let
us illustrate the implications of this by considering the tachyon. Since one