group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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.
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 $X//G \to \mathbf{B} U(1)$, can be explicitly described as a function $\lambda: G\times X \to U(1)$ such that
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:
The Liouville cocycle with central charge $c\in \mathbb{R}$ is the function
defined by
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, 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 $c\neq 0$, one says that the conformal field theory has a conformal anomaly.