62
2 Worldsheet Path Integral: Vacuum Amplitudes
and thus c w is nothing else than the divergence of the c a field: the Weyl ghost is a
composite field (this makes connection with Remark 2.17)—see also (2.65c). The
energy–momentum tensor of the ghosts with action (2.153) is
T
gh
ab = −
b ac ∇ b c
c
+ b bc ∇ a c
c
+ 2b ab c w
− ∇ c (b ab c
c )
+
1
2
g ab g
cd
b ce ∇ d c
e
+ b de ∇ c c
e
+ 2b cd c w
.
(2.156)
The trace of this tensor
g
ab T
gh
ab = −g
ab
∇ c (b ab c
c )
(2.157)
does not vanish off-shell, but it does on-shell since g ab b ab = 0. This implies that
the theory is Weyl invariant even if the action is not. It is interesting to contrast this
with the trace (2.148) when the Weyl ghost has been integrated out.
The equations of motion (2.146) and energy–momentum tensor (2.147) for the
action (2.145) can be easily derived by replacing c w by its solution in the previous
formulas.
Computation: Equation (2.156)
The first parenthesis comes from varying g ab , the second from the covariant
derivatives and the last from the
√
g. The second term comes from
g
ab
b ac δ∇ b c
c
+ b bc δ∇ a c
c
= 2g
ab b ac δ∇ b c
c
= 2g
ab b ac δδ
c
bd c
d
= g
ab b ac g
ce
∇ b δg de + ∇ d δg be − ∇ e δg bd
c
d
= b
ab
∇ a δg bc + ∇ c δg ab − ∇ b δg ac
c
c
= b
ab
∇ c δg ab c
c ,
where two terms have cancelled due to the symmetry of b ab . Integrating by
part gives the term in the previous equation.
Note that the integration on the Weyl ghost yields a delta function
d g c w e
−(c w ,g ab b ab ) g = δ
g
ab b ab
.
(2.158)
2.4.3 Zero-Modes
The path integral (2.151) excludes the zero-modes of the ghosts. One can expect
them to be related to the determinants of elements of ker P 1 and ker P
†
1 with
Grassmann coefficients. They can be included after few simple manipulations (see
also Appendix C.1.3).
2 Worldsheet Path Integral: Vacuum Amplitudes
and thus c w is nothing else than the divergence of the c a field: the Weyl ghost is a
composite field (this makes connection with Remark 2.17)—see also (2.65c). The
energy–momentum tensor of the ghosts with action (2.153) is
T
gh
ab = −
b ac ∇ b c
c
+ b bc ∇ a c
c
+ 2b ab c w
− ∇ c (b ab c
c )
+
1
2
g ab g
cd
b ce ∇ d c
e
+ b de ∇ c c
e
+ 2b cd c w
.
(2.156)
The trace of this tensor
g
ab T
gh
ab = −g
ab
∇ c (b ab c
c )
(2.157)
does not vanish off-shell, but it does on-shell since g ab b ab = 0. This implies that
the theory is Weyl invariant even if the action is not. It is interesting to contrast this
with the trace (2.148) when the Weyl ghost has been integrated out.
The equations of motion (2.146) and energy–momentum tensor (2.147) for the
action (2.145) can be easily derived by replacing c w by its solution in the previous
formulas.
Computation: Equation (2.156)
The first parenthesis comes from varying g ab , the second from the covariant
derivatives and the last from the
√
g. The second term comes from
g
ab
b ac δ∇ b c
c
+ b bc δ∇ a c
c
= 2g
ab b ac δ∇ b c
c
= 2g
ab b ac δδ
c
bd c
d
= g
ab b ac g
ce
∇ b δg de + ∇ d δg be − ∇ e δg bd
c
d
= b
ab
∇ a δg bc + ∇ c δg ab − ∇ b δg ac
c
c
= b
ab
∇ c δg ab c
c ,
where two terms have cancelled due to the symmetry of b ab . Integrating by
part gives the term in the previous equation.
Note that the integration on the Weyl ghost yields a delta function
d g c w e
−(c w ,g ab b ab ) g = δ
g
ab b ab
.
(2.158)
2.4.3 Zero-Modes
The path integral (2.151) excludes the zero-modes of the ghosts. One can expect
them to be related to the determinants of elements of ker P 1 and ker P
†
1 with
Grassmann coefficients. They can be included after few simple manipulations (see
also Appendix C.1.3).