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Liouville cocycle

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Contents

Idea

The Liouville cocycle is a degree 1 cocycle in the groupoid cohomology of the action groupoid of the action of conformal rescalings on the space of Riemannian metrics on a surface.

Definition

Recall that given a space XX with an action of a group GG on it, a U(1)U(1)-cocycle on the action 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).

Now 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 real-valued smooth functions 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} \,.

The Liouville cocycle with central charge cc\in \mathbb{R} 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(ic2 Σ(df* gdf+4fR gdμ g)), \lambda(f,g)=exp(\frac{i c}{2} \int_\Sigma(d f\wedge *_g d f +4 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.

In conformal field theory

In conformal field theory, the Liouville cocycle appears when one moves from genuine representations of 2-dimensional conformal cobordisms to projective representations. The obstruction for such a projective representation to be a genuine representation is precisely given by the central charge cc; when c0c\neq 0, one says that the conformal field theory has a conformal anomaly.

Revised on May 28, 2010 15:48:29 by Urs Schreiber (131.211.36.96)