Loop group ($LG=C^{\infty }(S^1,G)$)

The loop group of a Lie group is the group of smooth maps from the circle to that group, with pointwise multiplication and the smooth-function topology. Rotation of the source circle acts by group automorphisms. For $G=U(1)$, every loop has the form $z^w{e}^{ih(z)}$, where $w\in \mathbb{Z}$ is its winding number and $h$ is a smooth real periodic function determined up to addition of an integer multiple of $2\pi$.

Such a continuous projective unitary representation admits strongly continuous rotations whose self-adjoint generator is bounded below and whose adjoint action rotates the loops. In the basic fermionic construction, integer one-particle modes are filled below zero. The rotation energy counts nonnegative particle energies and positive hole energies. It has finite-dimensional integer eigenspaces and implements circle rotations. Tensor products give higher nonnegative integer levels.

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.