10.2 Open String Field Expansion, Parity and Ghost Number
221
A quantum string field generally contains components n of all worldsheet
ghost numbers n:
=
n∈Z
n ,
N gh (( n ) = n.
(10.50)
The projections on the positive and negative (cylinder) ghost numbers are denoted
by ± :
= + + − ,
, + =
n>1
n ,
, − =
n≤1
n .
(10.51)
The shift in the indices is explained by the relation (B.56) between the cylinder and
plane ghost numbers.
For a field n of fixed ghost number, coefficients of the expansion vanish
whenever the ghost number of the basis state does not match the one of the field:
∀n r = n : ψ r = 0.
(10.52)
Another possibility to define the field n is to insert a delta function:
| n = δ(N gh − n) | =
r
δ(n r − n) ψ r |φ r .
(10.53)
According to (10.45), a string field can also be separated in terms of the ghost
zero-modes:
| = | ↓ + | ↑ = | ↓ + c 0 |
↓ ,
(10.54a)
| ↑ = c 0 |
↓ ,
|
↓ = b 0 | ↑ ,
(10.54b)
where the components satisfy the constraints
b 0 | ↓ = 0,
c 0 | ↑ = 0,
b 0 |
↓ = 0.
(10.55)
The fields | ↓ and | ↑ (or |
↓ ) are called the down and top components, and
they can be expanded as
| ↓ =
r
ψ ↓,r |φ ↓,r ,
| ↑ =
r
ψ ↑,r |φ ↑,r .
(10.56)