178
7 CFT Systems
Hence, b n is anti-Hermitian if = −1. The BPZ conjugates of the modes are
b
t
n = (−1)
λ b −n ,
c
t
n = (−1)
1−λ c −n ,
(7.179)
using I + (z) with (6.111).
In the rest of this section, we consider only the case = 1 and λ ∈ N. The
adjoints of the vacuum read
| ↓↓
‡
= =0|c 1−λ · · · c −1 ,
| ↑↑
‡
= =0|c 1−λ · · · c −1 c 0 .
(7.180)
The BPZ conjugates of the vacua are
| := | ↓↓
t
= (−1)
(1−λ) 2 0|c −1 · · · c 1−λ ,
↑ | := | ↑↑
t
= (−1)
λ(1−λ)
0|c 0 c −1 · · · c 1−λ .
(7.181)
The signs are inconvenient but will disappear when considering both the left and
right vacua together as in (7.154). We have the following relations:
| = (−1)
a λ +(1−λ)(2−λ)
| ↓↓
‡ ,
| = (−1)
a λ | ↑↑
‡ ,
(7.182)
where a λ is the zero-point energy (7.152).
Computation: Equation (7.182)
To prove the relation, we can start from the BPZ conjugate ↓ | and reorder the
modes to bring them in the same order as the adjoint:
| = (−1)
(1−λ) 2 +
1
2 (2−λ)(1−λ)
| ↓↓
‡
= (−1)
−a λ +(1−λ)(2−λ)
| ↓↓
‡ .
The reordering gives a factor (−1) to the power:
λ−2
i=1
i =
1
2
(2 − λ)(1 − λ) = −a λ + 1 − λ.
Similarly, for the second vacuum:
| = (−1)
λ(1−λ)−
1
2 λ(1−λ)
| ↑↑
‡
= (−1)
1
2 λ(1−λ)
| ↑↑
‡ .
We can identify the power with (7.152).
Then, we have the following relations:
↑ |b 0 = =↓ |,
↓ |c 0 = =↑ |,
↓ |b 0 = 0,
↑ |c 0 = 0.
(7.183)