= Loop group two-cocycle for U(1)
{title2=$c((w,h),(v,k))=e^{ivh_0-i\omega(h,k)/2}$}
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 $\omega(h,k)=(2\pi)^{-1}\int_0^{2\pi}h'k$, choose implementers $U(w,h)=S^w e^{iA(h)}$, where $S$ increases charge by one, $SQS^*=Q-I$, 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>.
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