nLab
Liouville cocycle

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 $U(1)$-cocycle on the action groupoid $X//G$, i.e. a functor $X//G\to BU(1)$, can be explicitly described as a function $\lambda :G\times X\to U(1)$ such that

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

$$(f,g_{ij})\mapsto e^f{g}_{ij}.$$(f,g_{i j})\mapsto e^f g_{i j} \,.

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

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

defined by

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

where ${*}_g$ 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.