14
1 Introduction
Finally, the external states must be specified: this amounts to prescribe boundary
conditions for the path integral or to insert the corresponding wave functions. Under
the conformal mapping which brings the external legs to punctures located at z i , the
states are mapped to local operators V i (k i , z i ) inserted at the points z i . The latter
are built from the CFT fields and are called vertex operators: they are characterized
by a momentum k μ which comes from the Fourier transformation of the X μ fields
representing the non-compact dimensions. These operators are inserted inside the
path integral with integrals over the positions z i in order to describe all possible
conformal mappings.
Ultimately, the amplitude (amputated Green function) is computed as
A n (k 1 , . . . , k n ) =
g≥0
g
n−2+2g
s
A g,n ,
(1.23)
where
A g,n =
n
i=1
d
2 z i
dg ab d e
−S cft [g ab ,,]
n
i=1
V i (k i , z i )
(1.24)
is the g-loop n-point amplitude (for simplicity we omit the dependence on the states
beyond the momentum). denotes collectively the CFT fields and g ab is the metric
on the surface.
The integration over the metrics and over the puncture locations contain a huge
redundancy due to the invariance under reparametrizations, which means that one
integrates over many equivalent surfaces. To avoid this, Faddeev–Popov ghosts must
be introduced and the integral is restricted to only finitely many (real) parameters t λ .
They form the moduli space M g,n of the Riemann surfaces g,n whose dimension
is
dim R M g,n = 6g − 6 + 2n.
(1.25)
The computation of the amplitude A g,n can be summarized as
A g,n =
M g,n
6g−6+2n
λ=1
dt λ F (t).
(1.26)
The function F (t) is a correlation function in the worldsheet CFT defined on the
Riemann surface g,n
F (t) =
n
i=1
V i × ghosts × super-ghosts
g,n
.
(1.27)
Note that the (super)ghost part is independent of the choice of the matter CFT.
1 Introduction
Finally, the external states must be specified: this amounts to prescribe boundary
conditions for the path integral or to insert the corresponding wave functions. Under
the conformal mapping which brings the external legs to punctures located at z i , the
states are mapped to local operators V i (k i , z i ) inserted at the points z i . The latter
are built from the CFT fields and are called vertex operators: they are characterized
by a momentum k μ which comes from the Fourier transformation of the X μ fields
representing the non-compact dimensions. These operators are inserted inside the
path integral with integrals over the positions z i in order to describe all possible
conformal mappings.
Ultimately, the amplitude (amputated Green function) is computed as
A n (k 1 , . . . , k n ) =
g≥0
g
n−2+2g
s
A g,n ,
(1.23)
where
A g,n =
n
i=1
d
2 z i
dg ab d e
−S cft [g ab ,,]
n
i=1
V i (k i , z i )
(1.24)
is the g-loop n-point amplitude (for simplicity we omit the dependence on the states
beyond the momentum). denotes collectively the CFT fields and g ab is the metric
on the surface.
The integration over the metrics and over the puncture locations contain a huge
redundancy due to the invariance under reparametrizations, which means that one
integrates over many equivalent surfaces. To avoid this, Faddeev–Popov ghosts must
be introduced and the integral is restricted to only finitely many (real) parameters t λ .
They form the moduli space M g,n of the Riemann surfaces g,n whose dimension
is
dim R M g,n = 6g − 6 + 2n.
(1.25)
The computation of the amplitude A g,n can be summarized as
A g,n =
M g,n
6g−6+2n
λ=1
dt λ F (t).
(1.26)
The function F (t) is a correlation function in the worldsheet CFT defined on the
Riemann surface g,n
F (t) =
n
i=1
V i × ghosts × super-ghosts
g,n
.
(1.27)
Note that the (super)ghost part is independent of the choice of the matter CFT.