8.2 BRST in the CFT Formalism
183
As usual, Q B ∼ Q B,L when considering only the holomorphic sectors such that we
generally omit the index.
8.2.1 OPE
The OPE of the BRST current with T is
T (z)j B (w) ∼
c m
2
− 4 − 6κ
c(w)
(z − w) 4 + (3 − 2κ)
∂c(w)
(z − w) 3
+
j B (w)
(z − w) 2 +
∂j B (w)
z − w
.
(8.6)
Hence, the BRST current is a primary operator only if
c m = 26,
κ =
3
2
.
(8.7)
The BRST current must be primary; otherwise, the BRST symmetry is anomalous,
which means that the theory is not consistent. This provides another derivation of
the critical dimension. In this case, the OPE becomes
T (z)j B (w) ∼
j B (w)
(z − w) 2 +
∂j B (w)
z − w
.
(8.8)
Remark 8.1 (Critical Dimension in 2d Gravity) The value c m = 26 (critical dimension) was obtained in Sect. 2.3 by requiring that the Liouville field decouples from
the path integral. In 2d gravity, where this condition is not necessary (nor even
desirable), the Liouville field is effectively part of the matter, such that c L +c m = 26.
One can also study the BRST cohomology in this case.
The OPE of j B (z) with the ghosts is
j B (z)b(w) ∼
2κ
(z − w) 3 +
j (w)
(z − w) 2 +
T (w)
z − w
,
(8.9a)
j B (z)c(w) ∼
: c(w)∂c(w) :
z − w
.
(8.9b)
Similarly, the OPE with any matter weight h primary field φ is
j B (z)φ(w) ∼ h
c(w)φ(w)
(z − w) 2 +
: h ∂c(w)φ(w) + c(w)∂φ(w) :
z − w
,
(8.9c)
using that c(w) 2 = 0 to cancel one term.