Liouville cocycle

**cohomology**
* cocycle, coboundary, coefficient
* homology
* chain, cycle, boundary
* characteristic class
* universal characteristic class
* secondary characteristic class
* differential characteristic class
* fiber sequence/long exact sequence in cohomology
* fiber ∞-bundle, principal ∞-bundle, associated ∞-bundle,
twisted ∞-bundle
* ∞-group extension
* obstruction
### Special and general types ###
* cochain cohomology
* ordinary cohomology, singular cohomology
* group cohomology, nonabelian group cohomology, Lie group cohomology
* Galois cohomology
* groupoid cohomology, nonabelian groupoid cohomology
* generalized (Eilenberg-Steenrod) cohomology
* cobordism cohomology theory
* integral cohomology
* K-theory
* elliptic cohomology, tmf
* taf
* abelian sheaf cohomology
* Deligne cohomology
* de Rham cohomology
* Dolbeault cohomology
* etale cohomology
* group of units, Picard group, Brauer group
* crystalline cohomology
* syntomic cohomology
* motivic cohomology
* cohomology of operads
* Hochschild cohomology, cyclic cohomology
* string topology
* nonabelian cohomology
* principal ∞-bundle
* universal principal ∞-bundle, groupal model for universal principal ∞-bundles
* principal bundle, Atiyah Lie groupoid
* principal 2-bundle/gerbe
* covering ∞-bundle/local system
* (∞,1)-vector bundle / (∞,n)-vector bundle
* quantum anomaly
* orientation, Spin structure, Spin^c structure, String structure, Fivebrane structure
* cohomology with constant coefficients / with a local system of coefficients
* ∞-Lie algebra cohomology
* Lie algebra cohomology, nonabelian Lie algebra cohomology, Lie algebra extensions, Gelfand-Fuks cohomology,
* bialgebra cohomology
### Special notions
* Čech cohomology
* hypercohomology
### Variants ###
* equivariant cohomology
* equivariant homotopy theory
* Bredon cohomology
* twisted cohomology
* twisted bundle
* twisted K-theory, twisted spin structure, twisted spin^c structure
* twisted differential c-structures
* twisted differential string structure, twisted differential fivebrane structure
* differential cohomology
* differential generalized (Eilenberg-Steenrod) cohomology
* differential cobordism cohomology
* Deligne cohomology
* differential K-theory
* differential elliptic cohomology
* differential cohomology in a cohesive topos
* Chern-Weil theory
* ∞-Chern-Weil theory
* relative cohomology
### Extra structure
* Hodge structure
* orientation, in generalized cohomology
### Operations ###
* cohomology operations
* cup product
* connecting homomorphism, Bockstein homomorphism
* fiber integration, transgression
* cohomology localization
### Theorems
* universal coefficient theorem
* Künneth theorem
* de Rham theorem, Poincare lemma, Stokes theorem
* Hodge theory, Hodge theorem
nonabelian Hodge theory, noncommutative Hodge theory
* Brown representability theorem
* hypercovering theorem
* Eckmann-Hilton-Fuks duality

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

$\lambda(h g,x)=\lambda(h, g x)\lambda(g,x).$

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:

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

The **Liouville cocycle** with central charge $c\in \mathbb{R}$ is the function

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

defined by

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

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, 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*.

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