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  1. A very short note and a question.

    Given a space XX with an action of a group GG on it, an U(1)U(1)-cocycle on the quotient groupoid X//GX// G, i.e., a functor X//GBU(1)X//G \to \mathbf{B} U(1), can be explicitly described as a function λ:G×XU(1)\lambda: G\times X \to U(1) such that

    λ(hg,x)=λ(h,gx)λ(g,x) \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 Σ\Sigma and take as XX the space of Riemannian metrics on Σ\Sigma, and as GG the additive group of smooth real-valued fucnctions on Σ\Sigma, acting on metrics by conformal rescaling: (f,g ij)e fg ij(f,g_{i j})\mapsto e^f g_{i j}. Then the Liouville cocycle is the function

    λ:C (Σ,)×Met(Σ)U(1) \lambda: C^\infty(\Sigma,\mathbb{R})\times Met(\Sigma) \to U(1)

    defined by

    λ(f,g)=exp(i2 Σ(df* gdf+2fR gdμ g)), \lambda(f,g)=exp(\frac{i}{2} \int_\Sigma(d f\wedge *_g d f +2 f R_g d \mu_g)),

    where * g*_g is the Hodge star operator defined by the Riemannian metric gg, R gR_g is the scalar curvarure and dμ gd \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.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 27th 2010

    I don’t know the answer. But thanks for the source code for Liouville cocycle! :-)

  2. added a few lins to Liouville cocycle, and created a stub for projective representation.