In page 47 of the book "Loop groups", by Pressley and Segal, we have the following proposition:
In the construction, they define a map $C:LX \longrightarrow \mathbb{T}$, from the space of loops in $X$ to the group $\mathbb{T}$, to be given by
$$l \longmapsto exp(\mathsf{i} \int\limits_{\delta} \omega)$$
where $\delta$ is a piece of surface in $X$ bounded by the loop $l$. They then define $\tilde{\Gamma}$ to be the group of all triples $(\gamma,p,u)$, where $\gamma$ is in $\Gamma$, $p$ is a path from a base-point $x_0$ in $X$ to $\gamma \cdot x_0$ and $u$ is an element of $\mathbb{T}$. Two elements $(\gamma,p,u)$ and $(\gamma^\prime,p^\prime,u^\prime)$ are equivalent if:
$\hspace{5mm}$ $\bullet$ $\gamma = \gamma^\prime$, and
$\hspace{5mm}$ $\bullet$ $u^\prime = C(p^\prime * p^{-1}) u.$
The group multiplication is also defined as:
$$(\gamma,p,u) \cdot (\gamma^\prime,p^\prime,u^\prime) = (\gamma \gamma^\prime,p * \gamma p^\prime,u u^\prime).$$
My understanding from above construction is that the element $u$ that appeared in $(\gamma,p,u)$ is an arbitrary element in $\mathbb{T}$. Is it right? In that case, how can we define the $2$-cocycle $c:\Gamma \times \Gamma \longrightarrow \mathbb{T}$ associated to $\tilde{\Gamma}$? After defining a set-theoretic section $s:\Gamma \longrightarrow \tilde{\Gamma}$, the map $c$ is defined as
$$c(\gamma,\gamma^\prime) = \frac{s(\gamma)s(\gamma^\prime)}{s(\gamma \gamma^\prime)}.$$
So, the only components in triples $(\gamma,p,u)$ and $(\gamma^\prime,p^\prime,u^\prime)$ that can be used to obtain an element in $\mathbb{T}$, are $u$ and $u^\prime$.