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A very short note and a question.
Given a space $X$ with an action of a group $G$ on it, an $U(1)$-cocycle on the quotient groupoid $X// G$, i.e., a functor $X//G \to \mathbf{B} U(1)$, can be explicitly described as a function $\lambda: G\times X \to U(1)$ such that
$\lambda(h g,x)=\lambda(h, g x)\lambda(g,x)$An extremely nice and nontrivial example of this is the Liouville cocycle. Fix a Riemann surface $\Sigma$ and take as $X$ the space of Riemannian metrics on $\Sigma$, and as $G$ the additive group of smooth real-valued fucnctions on $\Sigma$, acting on metrics by conformal rescaling: $(f,g_{i j})\mapsto e^f g_{i j}$. Then the Liouville cocycle is the function
$\lambda: C^\infty(\Sigma,\mathbb{R})\times Met(\Sigma) \to U(1)$defined by
$\lambda(f,g)=exp(\frac{i}{2} \int_\Sigma(d f\wedge *_g d f +2 f R_g d \mu_g)),$where $*_g$ is the Hodge star operator defined by the Riemannian metric $g$, $R_g$ is the scalar curvarure and $d \mu_g$ is the volume form. It would be interesting if under suitable hypothesis (e.g., compatibility with glueing of Riemann surfaces in view of CFT), the Lioville cocycle would be the only possibility (up to a scalar factor: central charge). Maybe Zoran knows the answer to this.
I donâ€™t know the answer. But thanks for the source code for Liouville cocycle! :-)
added a few lins to Liouville cocycle, and created a stub for projective representation.
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