B Summary of Important Formulas
389
The OPEs between the ghosts and different currents are
c(z)b(w) ∼
1
z − w
,
b(z)c(w) ∼
1
z − w
, b(z)b(w) ∼ 0, c(z)c(w) ∼ 0,
(B.54a)
T (z)b(w) ∼
2b(w)
(z − w) 2 +
∂b(w)
z − w
, T (z)c(w) ∼
−c(w)
(z − w) 2 +
∂c(w)
z − w
.
(B.54b)
j (z)b(w) ∼ −
b(w)
z − w
, j (z)c(w) ∼
c(w)
z − w
. j(z)O(w) ∼ N gh (O)
O(w)
z − w
,
(B.54c)
j (z)j (w) ∼
1
(z − w) 2 .
(B.54d)
T (z)j (w) ∼
−3
(z − w) 3 +
j (w)
(z − w) 2 +
∂j (w)
z − w
,
(B.54e)
any operator O(z) is defined by
The OPE (B.54e) implies that the ghost number is not conserved on a curved
space:
N
c
− N
b
= 3 − 3g
(B.55)
and leads to a shift between the ghost numbers on the plane and the cylinder:
N gh,L = N
cyl
gh,L +
3
2
.
(B.56)
The SL(2, C) vacuum |0 is defined by
∀n > −2 : b n |0 = 0,
∀n > 1 : c n |0 = 0.
(B.57)
The mode c 1 does not annihilate the vacuum, and the two degenerate energy vacua
are
| ↓↓ := c 1 |0 ,
| ↑↑ := c 0 c 1 |0 .
(B.58)
The zero-point energy of these states is
L 0 | ↓↓ = a gh | ↓↓ ,
L 0 | ↑↑ = a gh | ↑↑ ,
a gh = −1.
(B.59)