172
7 CFT Systems
The vacua are annihilated by different combinations of the zero-modes:
b 0 | ↓↓↓ = ¯
b 0 | ↓↓↓ = 0,
c 0 | ↑↓↓ = ¯
b 0 | ↑↓↓ = 0,
b 0 | ↓↑↑ = ¯
c 0 | ↓↑↑ = 0,
c 0 | ↑↑↑ = ¯
c 0 | ↑↑↑ = 0.
(7.155b)
In these manipulations, one has to be careful to correctly anti-commute the modes
with the ones hidden in the definitions of the vacua.
There is a second basis which is more natural when using the zero-modes c
±
0 and
b
±
0 (7.132):
| ↓↓↓ , |++ , |−− , | ↑↑↑
,
(7.156)
where the two vacua |±± are combinations of the | ↓↑↑ and | ↑↓↓ vacua:
|±± = | ↑↓↓ ± | ↓↑↑ .
(7.157)
The different vacua are naturally related by acting with c
±
0 and b
±
0 which act as
raising and lowering operators:
c
±
0 | ↓↓↓ =
1
2
|±± ,
c
∓
0 |±± = ± | ↑↑↑ ,
b
±
0 |±± = ±2 | ↓↓↓ ,
b
∓
0 | ↑↑↑ = ± |±± .
(7.158a)
From the previous relations, it follows that the different vacua are annihilated by the
zero-modes as follow:
b
+
0 | ↓↓↓ = b
−
0 | ↓↓↓ = 0,
c
−
0 |−− = b
+
0 |−− = 0,
c
+
0 |++ = b
−
0 |++ = 0
c
+
0 | ↑↑↑ = c
−
0 | ↑↑↑ = 0.
(7.158b)
This also means that we have
c
−
0 c
+
0 | ↓↓↓ =
1
2
| ↑↑↑ ,
b
+
0 b
−
0 | ↑↑↑ = 2 | ↓↓↓ .
(7.158c)
Computation: Equation (7.158)
2 c
+
0 |±± = (c 0 + ¯
c 0 ) | ↑↓↓ ± (c 0 + ¯
c 0 ) | ↓↑↑ = ¯
c 0 | ↑↓↓ ± c 0 | ↓↑↑ = (−1 ± 1) | ↑↑↑
b
+
0 |±± = (b 0 + ¯
b 0 ) | ↑↓↓ ± (b 0 + ¯
b 0 ) | ↓↑↑ = b 0 | ↑↓↓ ± ¯
b 0 | ↓↑↑ = (1 ± 1) | ↓↓↓
2 c
±
0 | ↓↓↓ = (c 0 ± ¯
c 0 ) | ↓↓↓ = c 0 | ↓↓↓ ± ¯
c 0 | ↓↓↓ = | ↑↓↓ ± | ↓↑↑ = |±±
b
±
0 | ↑↑↑ = (b 0 ± ¯
b 0 ) | ↑↑↑ = b 0 | ↑↑↑ ± ¯
b 0 | ↑↑↑ = | ↓↑↑ ∓ | ↑↓↓ = ∓ |∓∓