arXiv:hep-th/9304011 v1 Apr 5 1993

The stress-energy tensor following from (3.2), T µν
Feigin–Fuks form
T z¯z = 0
T zz = − 1 2 (∂φ)2 + 1 2 Q∂2 φ
= −2π δS
δg µν , takes the familiar
(3.7)
T¯z¯z = − 1 2 (∂φ)2 + 1 2 Q∂2 φ ,
where we have used the equations of motion in the first line.
Since φ is a component of a metric, its transformation law under conformal transformations
z → w = f(z),
φ → φ + 1 γ log ∣ ∣∣ dw
dz
∣ 2 , (3.8)
is more complicated than that of an ordinary scalar field. In particular, the U(1) current
∂ z φ measuring the Liouville momentum transforms as
and the stress tensor T zz transforms as
∂ z φ → dw
dz ∂ wφ + d 1
∣ ∣∣
dz γ log dw
∣ , (3.9)
dz
T zz →
( dw
) 2Tww
+ 1 S[w; z] . (3.10)
dz γ2 The object S[w; z] is called the “Schwartzian derivative” and has many equivalent definitions.
8 Combining (3.7) and (3.9), we see that
S[w; z] = − 1 (
∂ z log∣ dw
) 2 ∣
d2
∣ ∣∣ + 2 dz dz log dw
2 ∣
dz
= T zz
(φ = log∣ dw
)
∣
(3.11)
dz
= w′′′
w ′ − 3 ( w
′′ ) 2
.
2 w ′
The unusual transformation laws (3.9) and (3.10) have counterparts in the quantum theory,
where they result in shifted formulae for conformal charges and weights in Liouville theory.
8 It may be considered, for example, as the integrated version of the conformal anomaly, or it
may be defined in terms of “projective connections.” For a discussion of the latter concept see
[40].
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