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.







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).)






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