242
11 Introduction to Off-Shell String Theory
Since there is no constraint on the states, the ghost number of φ r is arbitrary and
denoted as
n r := N gh (φ r ) ∈ Z
(11.29)
(the ghost number is restricted for states in the cohomology of Q B ). The Grassmann
parity of a state φ r is denoted as |φ r |. When there are no fermions in the matter sector
(usually the case for the bosonic string), only ghosts are odd. Then, the Grassmann
parity of a state is odd (resp. even) if its ghost number is odd (resp. even):
|φ r | := N gh (φ r ) mod 2 =
0 N gh (φ r ) even,
1 N gh (φ r ) odd.
(11.30)
The dual basis {|φ c
r } is defined from the BPZ inner product:
φ
c
r |φ s = δ rs .
(11.31)
Denoting the ghost numbers of the dual states by
n
c
r := N gh (φ
c
r ),
(11.32)
the product is non-vanishing if
n
c
r + n r = 6,
(11.33)
due to the ghost number anomaly on the sphere. This condition cannot be satisfied
if the dual state φ c
r is simply taken to be the BPZ conjugate φ t
r since the BPZ
conjugation does not change the ghost number. This implies that
φ r |φ s = 0
(11.34)
from the ghost anomaly for every state, except for the closed string states with
N gh = 3 (in fact, the inner product of these states is also zero as can be seen after
investigation). One can show that
φ r |φ
c
s = (−1)
|φ r | δ rs .
(11.35)
Hence, the resolution of the identity can be written in the two equivalent ways
1 =
r
|φ r φ
c
r | =
r
(−1)
|φ r |
|φ
c
r φ r | .
(11.36)