13.2 Properties of Forms
277
The Beltrami form for this vector is
B(V ) = i
C i
dw i w i b(w i ) − i
C i
d ¯
w i ¯
w i ¯
b( ¯
w i ),
(13.36)
where D i is kept to the left.
In the p-form (13.11), the ith operator V i is inserted in D i and encircled by C i .
Because there is no other operator inside D i , the contribution of this disk to the form
is
B(V )V i (0) = i
C i
dw i w i b(w i )V i (0) − i
C i
d ¯
w i ¯
w i ¯
b( ¯
w i )V i (0).
(13.37)
The state–operator correspondence allows to rewrite this result as
i(b 0 − ¯
b 0 ) |V i ,
(13.38)
since the contour integral picks the zero-modes of b and of ¯
b. Requiring that the
form vanishes implies the ghost counter-part of the level-matching condition
b
−
0 |V i = 0.
(13.39)
Hence, consistency of off-shell amplitudes implies that
V i ∈ H
− ,
(13.40)
where H − is defined in (11.38).
Reparametrization of w i A reparametrization of the local coordinate w i keeping
the origin of D i fixed reads
w i −→ f (w i ),
f (0) = 0.
(13.41)
The function can be expanded in series
f (w i ) =
m≥0
p m w
m+1
i
.
(13.42)
Because the transformation is holomorphic, it can be extended on C i . Each
parameter p m provides a coordinate of P g,n and whose deformation corresponds
to a vector field
v m = w
m+1
i
,
¯
v m = 0.
(13.43)