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 2cocycle representing such an element is a bilinear function of two elements of , which we will denote and . We have to remember that the 2cocycle takes values in which is identified with the space of Taylor series at the unit of the group manifold. Let denote the group element.
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 1form on the worldsheet. Substitution of into intertwines the with the de Rham differential . (The verification of this statement uses the fact that satisfies the zerocurvature equations (40).)

