8.3 BRST Cohomology: Two Flat Directions
193
can be rewritten as
m
2
,L = −2 0 p
+
L p
−
L ,
L
0 = N
+
+ N
−
+ N
b
+ N
c .
(8.61)
The expression for the sum of the Virasoro operators (7.65) easily follows
from (8.57):
L
0
m + L
1
m = 0
n
: α
+
n α
−
m−n : = 0
n =0,m
: α
+
n α
−
m−n : + 0
α
−
0 α
+
m + α
+
m α
−
m
.
(8.62)
Computation: Equation (8.56)
For the modes α ±
m , we have
α
+
m , α
±
n
=
1
2
α
0
m +
i
√ 0
α
1
m
,
α
0
n ±
i
√
0
α
1
n
=
1
2
α
0
m , α
0
n
∓
1
0
α
1
m , α
1
n
=
0
2
m δ m+n,0 (1 ∓ 1),
where we used (7.72). The other commutators follow similarly from (7.73), for
example,
x
−
L , p
±
L
=
1
2
x
0
L −
i
√ 0
x
1
L
,
p
0
L ±
i
√
0
p
1
L
=
1
2
x
0
L , p
0
L
± 0
x
1
L , p
1
L
=
0
2
(1 ± 1).
Computation: Equation (8.57)
For the modes α ±
m , we have
n
α
+
n α
−
m−n =
1
2
n
α
0
n +
i
√
0
α
1
n
α
0
m−n −
i
√
0
α
1
m−n
=
1
2
n
α
0
n α
0
m−n + 0 α
1
n α
1
m−n +
i
√ 0
α
0
m−n α
1
n − α
0
n α
1
m−n
.
The last two terms in parenthesis cancel as can be seen by shifting the sum
n → m − n in one of the terms. Note that, for m = 2n, there is no cross-term
only after summing over n.
The relations for the zero-modes follow simply by observing that expressions in both coordinates can be rewritten in terms of the 2-dimensional
(spacetime) flat metric.