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 .
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
, taking values in the functions of
Remember that our generalized Lax operators (38
) have the form:
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
. Let us formally introduce the operator
This operator simply replaces
cp. Eq. (27
) implies that the BRST differential acts as follows:
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:
Notice that the unintegrated vertex
is obtained from the cocycle
by replacing the ghost
, as it should be