What is a Chern Simons term?

What is a Chern Simons term?

Mathematical origin Moreover, the Chern–Simons term is described as the eta invariant defined by Atiyah, Patodi and Singer. The gauge invariance and the metric invariance can be viewed as the invariance under the adjoint Lie group action in the Chern–Weil theory.

What Chern Simons theory assigns to a point?

We answer the questions, “What does Chern–Simons theory assign to a point?” and “What kind of mathematical object does Chern–Simons theory assign to a point?” Our answer to the first question is representations of the based loop group.

Is Chern Simons theory a CFT?

Chern-Simons Gauge Theory and the AdS(3)/CFT(2) Correspondence. The bulk partition function of pure Chern-Simons theory on a three-manifold is a state in the space of conformal blocks of the dual boundary RCFT, and therefore transforms non-trivially under the boundary modular group.

Why do we use gauge transformation?

importance of gauge theory …of the field variables (gauge transformations) that leaves the basic physics of the quantum field unchanged. This condition, called gauge invariance, gives the theory a certain symmetry, which governs its equations.

What is the Chern character?

Traditionally, in the strict sense of the term, the Chern character is a universal characteristic class of vector bundles or equivalently of their topological K-theory classes, which is a rational combination of all Chern classes. to the smash product with the Eilenberg-MacLane spectrum over the real numbers.

How are Chern classes calculated?

For Chern class, we have this formula c(E⊕F)=c(E)c(F), where E and F are complex vector bundle over a manifold M. c(E)=1+c1(E)+⋯ is the total chern class of E.

Why are they called gauge theories?

On the other hand there is general relativity (whose quantum version is still under construction) which describes gravitational interaction. These theories are called gauge theories because their standard formulation contains unobservable and hence apparently redundant elements.

What is Lorentz and Coulomb gauge?

Lorentz gauge and Coulomb gauge are two concepts that are important in quantum chemistry. Lorentz gauge is a partial gauge fixing of the electromagnetic vector potential while Coulomb gauge is a type of gauge that is expressed in terms of instantaneous values of the fields and densities.

Why is Chern number an integer?

The integrals of the Chern-classes associated with a complex vector bundle over any manifold M are guaranteed to be integers because they give the winding numbers of the map from the manifold M into a subset of the Schubert cycles in the complex Grassmanian Gr(n,N), where n is the dimension of the complex vector space.

What is gauge theory simple explanation?

In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups).