Chern–Simons theory

The Chern - Simons functional is in differential geometry, topology, and mathematical physics of meaning. In mathematics, it is used to define the Chern - Simons invariant of connections on principal bundles over 3-manifolds. Originally introduced by Chern and Simons theory of secondary characteristic classes, it had at least two unexpected applications, one being Witten's Position in the quantum field theory with a physical geometrical interpretation of the Jones polynomial and, secondly, the interpretation of the Chern - Simons invariant flat bundle as a complex- version of the hyperbolic volume.

Definition

Be a simply connected Lie group and a 3-dimensional, closed, orientable manifold. Under these conditions, every principal bundle is trivialisierbar, so it has a cut.

For a connection

Is defined by its Chern - Simons action functional

This definition depends a priori on the choice of a cut off for a gauge transformation

But applies

The Maurer - Cartan form.

One thus gets a well-defined modulo value

Properties

Let be a closed, orientable 3-manifold and. We call on with the ( infinite-dimensional ) manifold of all connections on principal bundles.

Then is smooth and has the following properties:

• ( Functoriality )

• ( Additivity )

• (Extension of the structure group)

Flat relationships

It is

The curvature form of the connection referred to. The critical points of the Chern - Simons functionals are therefore currently the flat connections. In particular, the Chern - Simons functional constant flat on the connected components of the moduli space correlations is on.

Set of Yoshida

It is a closed, orientable hyperbolic 3-manifold and their holonomy. Then for the associated flat bundle

The Riemannian Chern - Simons invariant of the Levi -Civita connection denoted.

The image of the fundamental class of the representation defines a homology class

In the extended Bloch group and the Rogers Dilogarithm

Maps to. This provides an explicit formula for the Chern - Simons invariant and an alternative proof of the theorem by Yoshida.

Algorithm for flat bundles

It is a flat bundle over a closed, orientable 3-manifold with holonomy. Then the Rogers Dilogarithm makes up down where the canonical homomorphism called. The value of can be calculated from the Ptolemaic coordinates of the representation to a triangulation of. ( This approach also works for 3-manifolds with boundary, as long as the restriction of is unipotent on the fundamental groups of the edge. ), This algorithm is implemented in Ptolemy modules as part of the software Snappy.

Generalization

In arbitrary dimensions, one can use Chern - Simons forms for the definition of secondary characteristic classes.