32
3 String Theory
To understand the geometric origin of the ghost number 2 condition, we return
to the inner product (3.7). A more careful inspection of (3.7) reveals that the
correlation function . . . is zero unless we saturate it with three c-ghost and three
˜
c-ghost insertions, in other words, the correlator . . . has ghost number −6. These
ghost zero modes correspond to the vector fields that generate the action of the
automorphism group SL(2, C) of S 2 . Since they leave the reference metric dzd ¯
z
invariant, they play no role in the gauge fixing procedure. This is why they show
up as zero modes of the ghost action. On the other hand, the punctures of the two
ψ-insertions are not invariant. One way to take care of this is to attach each insertion
of ψ to a c ¯
c-pair. This is equivalent to restricting gauge fixing to diffeomorphisms
that preserve the insertion point. Thus, an equivalence class ψ of physical states has
a representative of the form
ψ(0) = ψ(φ(0), ∂ z φ(0), . . .)c(0) ¯
c(0) .
This takes care of four ghost zero modes in (3.7). Nonphysical states, on the other
hand, do not correspond to conformal primaries, as we have already mentioned,
therefore they depend not just on the position of the operator, but also on the
holomorphic reparametrization of the disk. Consequently, such states may involve
arbitrary polynomials of derivatives of the ghost fields.
The two remaining zero modes correspond to rotation and scaling of the unit
disc, respectively. We have encountered the latter already for the point particle and
it corresponds to time translation generated by Q, whereas the former is simply
redundant due to the constraint L
−
0 = 0. Furthermore, using b
†
0 = b 0 , one easily
checks that the inner product 1 , ψ 2 is degenerate on states containing b
−
0 . To
remedy this using [b
−
0 , c
−
0 ] = 1, we redefine the inner product as
Ω(ψ 1 , ψ 2 ) := =ψ 1 , c
−
0 ψ 2 .
Due to the c
−
0 -insertion, Ω is graded anti-symmetric,
Ω(A, B) = (−1)
(|A|+1)(|B|+1) Ω(B, A) .
Furthermore, Ω(QA, B) = (−1) |A| Ω(A, QB). Now we have all necessary tools to
write down the quadratic action for the closed string field. It is simply
S[ψ] =
1
2
Ω(ψ, Qψ) .
(3.9)
Just like the point particle, the string has a propagator, that is a homotopy inverse
for Q, which we denote by Q −1 . In order to define it, repeating the steps in
Remark 2.6, we fix the gauge, which amounts to fixing a representative for every
element of the cohomology H . More precisely, the gauge fixing determines a map
i : H → V ,
3 String Theory
To understand the geometric origin of the ghost number 2 condition, we return
to the inner product (3.7). A more careful inspection of (3.7) reveals that the
correlation function . . . is zero unless we saturate it with three c-ghost and three
˜
c-ghost insertions, in other words, the correlator . . . has ghost number −6. These
ghost zero modes correspond to the vector fields that generate the action of the
automorphism group SL(2, C) of S 2 . Since they leave the reference metric dzd ¯
z
invariant, they play no role in the gauge fixing procedure. This is why they show
up as zero modes of the ghost action. On the other hand, the punctures of the two
ψ-insertions are not invariant. One way to take care of this is to attach each insertion
of ψ to a c ¯
c-pair. This is equivalent to restricting gauge fixing to diffeomorphisms
that preserve the insertion point. Thus, an equivalence class ψ of physical states has
a representative of the form
ψ(0) = ψ(φ(0), ∂ z φ(0), . . .)c(0) ¯
c(0) .
This takes care of four ghost zero modes in (3.7). Nonphysical states, on the other
hand, do not correspond to conformal primaries, as we have already mentioned,
therefore they depend not just on the position of the operator, but also on the
holomorphic reparametrization of the disk. Consequently, such states may involve
arbitrary polynomials of derivatives of the ghost fields.
The two remaining zero modes correspond to rotation and scaling of the unit
disc, respectively. We have encountered the latter already for the point particle and
it corresponds to time translation generated by Q, whereas the former is simply
redundant due to the constraint L
−
0 = 0. Furthermore, using b
†
0 = b 0 , one easily
checks that the inner product 1 , ψ 2 is degenerate on states containing b
−
0 . To
remedy this using [b
−
0 , c
−
0 ] = 1, we redefine the inner product as
Ω(ψ 1 , ψ 2 ) := =ψ 1 , c
−
0 ψ 2 .
Due to the c
−
0 -insertion, Ω is graded anti-symmetric,
Ω(A, B) = (−1)
(|A|+1)(|B|+1) Ω(B, A) .
Furthermore, Ω(QA, B) = (−1) |A| Ω(A, QB). Now we have all necessary tools to
write down the quadratic action for the closed string field. It is simply
S[ψ] =
1
2
Ω(ψ, Qψ) .
(3.9)
Just like the point particle, the string has a propagator, that is a homotopy inverse
for Q, which we denote by Q −1 . In order to define it, repeating the steps in
Remark 2.6, we fix the gauge, which amounts to fixing a representative for every
element of the cohomology H . More precisely, the gauge fixing determines a map
i : H → V ,