60
2 Worldsheet Path Integral: Vacuum Amplitudes
symmetry), which is simpler. A last possibility is to introduce a Lagrange multiplier.
These aspects are related to the question of introducing a ghost for the Weyl
symmetry, which is described in Sect. 2.4.2.
The equations of motion are
(P 1 c) ab = ∇ a c b +∇ b c a −g ab ∇ c c
c
= 0,
(P
†
1 b) a = −2∇
b b ab = 0.
(2.146)
Hence, the classical solutions of b and c are respectively mapped to the zero-modes
of the operators P
†
1 and P 1 , and they are thus associated to the CKV and Teichmüller
parameters.
The energy–momentum tensor is
T
gh
ab = −b ac ∇ b c
c
− b bc ∇ a c
c
+ c
c
∇ c b ab + g ab b cd ∇
c c
d .
(2.147)
Its trace vanishes off-shell (i.e. without using the b and c equations of motion)
g
ab T
gh
ab = 0,
(2.148)
which shows that the action (2.145) is invariant under Weyl transformations
S gh [e
2ω g, b, c] = S gh [g, b, c].
(2.149)
The action (2.145) also has a U(1) global symmetry. The associated conserved
charge is called the ghost number and counts the number of c ghosts minus the
number of b ghosts, i.e.
N gh (b) = −1,
N gh (c) = 1.
(2.150a)
The matter fields are inert under this symmetry:
N gh (() = 0.
(2.150b)
In terms of actions, the path integral (2.136) can be rewritten as
Z g =
M g
d
M g t
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
d ˆ
g d
ˆ
g b d
ˆ
g c e
−S m [ ˆ
g,,]−S gh [ ˆ
g,b,c] .
(2.151)
One can use (2.136) or (2.151) indifferently: the first is more appropriate when
using spectral analysis to compute the determinant explicitly, while the second is
more natural in the context of CFTs.
2 Worldsheet Path Integral: Vacuum Amplitudes
symmetry), which is simpler. A last possibility is to introduce a Lagrange multiplier.
These aspects are related to the question of introducing a ghost for the Weyl
symmetry, which is described in Sect. 2.4.2.
The equations of motion are
(P 1 c) ab = ∇ a c b +∇ b c a −g ab ∇ c c
c
= 0,
(P
†
1 b) a = −2∇
b b ab = 0.
(2.146)
Hence, the classical solutions of b and c are respectively mapped to the zero-modes
of the operators P
†
1 and P 1 , and they are thus associated to the CKV and Teichmüller
parameters.
The energy–momentum tensor is
T
gh
ab = −b ac ∇ b c
c
− b bc ∇ a c
c
+ c
c
∇ c b ab + g ab b cd ∇
c c
d .
(2.147)
Its trace vanishes off-shell (i.e. without using the b and c equations of motion)
g
ab T
gh
ab = 0,
(2.148)
which shows that the action (2.145) is invariant under Weyl transformations
S gh [e
2ω g, b, c] = S gh [g, b, c].
(2.149)
The action (2.145) also has a U(1) global symmetry. The associated conserved
charge is called the ghost number and counts the number of c ghosts minus the
number of b ghosts, i.e.
N gh (b) = −1,
N gh (c) = 1.
(2.150a)
The matter fields are inert under this symmetry:
N gh (() = 0.
(2.150b)
In terms of actions, the path integral (2.136) can be rewritten as
Z g =
M g
d
M g t
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
d ˆ
g d
ˆ
g b d
ˆ
g c e
−S m [ ˆ
g,,]−S gh [ ˆ
g,b,c] .
(2.151)
One can use (2.136) or (2.151) indifferently: the first is more appropriate when
using spectral analysis to compute the determinant explicitly, while the second is
more natural in the context of CFTs.