cohomology

# 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 $X$ with an action of a group $G$ on it, a $U\left(1\right)$-cocycle on the action groupoid $X//G$, i.e. a functor $X//G\to BU\left(1\right)$, can be explicitly described as a function $\lambda :G×X\to U\left(1\right)$ such that

$\lambda \left(hg,x\right)=\lambda \left(h,gx\right)\lambda \left(g,x\right).$\lambda(h g,x)=\lambda(h, g x)\lambda(g,x).

Now fix a Riemann surface $\Sigma$ and take as $X$ the space of Riemannian metrics on $\Sigma$, and as $G$ the additive group of real-valued smooth functions on $\Sigma$, acting on metrics by conformal rescaling:

$\left(f,{g}_{ij}\right)↦{e}^{f}{g}_{ij}\phantom{\rule{thinmathspace}{0ex}}.$(f,g_{i j})\mapsto e^f g_{i j} \,.

The Liouville cocycle with central charge $c\in ℝ$ is the function

$\lambda :{C}^{\infty }\left(\Sigma ,ℝ\right)×\mathrm{Met}\left(\Sigma \right)\to U\left(1\right)$\lambda: C^\infty(\Sigma,\mathbb{R})\times Met(\Sigma) \to U(1)

defined by

$\lambda \left(f,g\right)=\mathrm{exp}\left(\frac{ic}{2}{\int }_{\Sigma }\left(df\wedge {*}_{g}df+4f{R}_{g}d{\mu }_{g}\right)\right),$\lambda(f,g)=exp(\frac{i c}{2} \int_\Sigma(d f\wedge *_g d f +4 f R_g d \mu_g)),

where ${*}_{g}$ 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.

## 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 $c$; when $c\ne 0$, 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)