 Loop group two-cocycle for U(1)

Write a loop as $z^w{e}^{ih}$ with $h$ smooth real periodic and $h_0$ its mean. For the level-one representation with modes filled below zero and $\omega (h,k)=(2\pi {)}^{-1}{\int }_0^{2\pi }{h}^{\prime }k$, choose implementers $U(w,h)=S^w{e}^{iA(h)}$, where $S$ increases charge by one, $SQ{S}^{\ast }=Q-I$, and $[A(h),A(k)]=i\omega (h,k)I$. Their product differs from $U(w+v,h+k)$ by the displayed two-cocycle. Its phase is bilinear, which verifies the cocycle identity; integer $v$ makes it independent of the $2\pi$ ambiguity in $h_0$. Rephasing implementers changes it by a group coboundary.