54
2 Worldsheet Path Integral: Vacuum Amplitudes
where S L is the Liouville action
S L [ ˆ
g, φ] :=
1
4π
d
2 σ
ˆ
g
ˆ
g
ab ∂ a φ∂ b φ + ˆ
Rφ
,
(2.119)
where ˆ
R is the Ricci scalar of the metric ˆ
g ab . These relations require to introduce
counter-terms, discussed further in Sect. 2.3.4. The coefficients c m and c gh are the
central charges respectively of the matter and ghost systems, with
c gh = −26.
(2.120)
This value will be derived in Sect. 7.2.
The inner products between φ i and μ j , and between the ψ i , and the CKV volume
are independent of φ [26, sec. 14.2.2, 11, p. 931]
det(φ i , μ j ) e 2φ ˆ
g = det( ˆ
φ i , ˆ
μ j ) ˆ
g ,
det(ψ i , ψ j ) e 2φ ˆ
g = det(ψ i , ψ j ) ˆ
g ,
ckv [e
2φ
ˆ
g] = ckv [ ˆ
g].
(2.121)
Remark 2.13 (Weyl and Gravitational Anomalies) The Weyl anomaly translates
into a non-zero trace of the quantum energy–momentum tensor
g
μν T μν =
c
12
R,
(2.122)
where c is the central charge of the theory. The Weyl anomaly can be traded
for a gravitational anomaly, which means that diffeomorphisms are broken at the
quantum level [20].
Inserting (2.118) in (2.117) yields
Z g =
M g
d
M g t
Diff [ ˆ
g]
gauge [ ˆ
g]
det(φ i , ˆ
μ j ) ˆ
g
det(φ i , φ j ) ˆ
g
ckv [ ˆ
g] −1
det(ψ i , ψ j ) ˆ
g
FP [ ˆ
g] Z m [ ˆ
g]
×
d g φ e
−
c L
6 S L [ ˆ
g,φ] ,
(2.123)
with the Liouville central charge
c L := 26 − c m .
(2.124)
The critical “dimension” is defined to be the value of the matter central charge c m
such that the Liouville central charge cancels
c L = 0 ⇒ c m = 26.
(2.125)