Integrated Vertex in AdS
Finally, we are now ready to discuss our construction of the integrated vertex.
1 Construction of the integrated vertex
Suppose that we are given an unintegrated vertex, and we want to construct the corresponding integrated vertex.
Unintegrated vertices correspond to the elements of 
.
The 2-cocycle
representing such an element is a bilinear function of two elements of 
,
which we will denote 
and 
. We have to remember that the 2-cocycle takes
values in 
which is identified with the space of Taylor series at the
unit of the group manifold. Let 
denote the group element.
Therefore, the element
is a function
of two elements of
, which we will call
and
, taking values in the functions of
:
Remember that our generalized Lax operators (
38), (
39) have the form:
where
are operators of the conformal dimension 1 taking values in
.
Theorem 8: The integrated vertex operator is obtained from (42)
by replacing and with :
2 Proof of the descent procedure
We have to verify the descent procedure (5), (4).
The proof is a variation of the famous differential geometry formula 
.
Consider 
. This is a -valued differential
1-form on the worldsheet. Substitution of 
into 
intertwines the 
with the de Rham
differential 
. (The verification of this statement uses the fact that satisfies
the zero-curvature equations (40).)
It is useful to introduce the free Grassmann parameters
and
. Let us formally introduce the operator
as follows:
This operator simply replaces
with
—
cp. Eq. (
27).
Eq. (
41) implies that the BRST differential acts as follows:
But
, therefore:
This was the first step (
5) of the descent procedure.
To make the second step (
4),
we have to remember that
is a
relative cocycle, which means that it always vanishes whenever
one of the ghosts falls into
. This implies:
and further:
Notice that the unintegrated vertex
is obtained from the cocycle
by replacing the ghost
fields with
,
as it should be.
