1 Construction of the integrated vertex 2 Proof of the descent procedure

Finally, we are now ready to discuss our construction of the integrated vertex.

#### 1Construction 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 :
 (42)
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 :

#### 2Proof 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:
 where
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:
 where
Notice that the unintegrated vertex is obtained from the cocycle by replacing the ghost fields with , as it should be.