Pontryagin densities and Chern-Simons form

In summary, the conversation discussed pontryagin densities and their definition in even dimension spaces, specifically focusing on a 4 dimensional space-time and the formula for arbitrary and larger groups. It was also mentioned that there are only two independent Pontrjagin classes in 4 dimensions: the Euler class and the Hirzebruch signature. The speaker also advised the asker to provide more context and references in their question for better answers.
  • #1
shereen1
51
1
Dear All
It is known that pontryagin densities are defined in even dimension space, let's say i am concerned with 4 dim space time. We also have a certain group G. What is the formula of pontryagin densities for arbitrary group? Larger group?
 
  • #3
There are only two independent (combinations of) Pontrjagin classes in 4 dimensions: the Euler class and the Hirzebruch signature.

I'm not sure what the rest of your question is asking.
 
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  • #4
Shereen,

if you expect decent answers you should spend more time on your opening posts, giving references and contexts. I think I gave that advice before.
 
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  • #5
haushofer said:
Shereen,

if you expect decent answers you should spend more time on your opening posts, giving references and contexts. I think I gave that advice before.

sure i will.
Thank you
 

1. What are Pontryagin densities and Chern-Simons form?

Pontryagin densities and Chern-Simons form are mathematical concepts used in the field of theoretical physics. They are related to the study of topological properties of spacetime and are used to describe the behavior of certain physical fields.

2. What is the significance of Pontryagin densities and Chern-Simons form in physics?

Pontryagin densities and Chern-Simons form play a crucial role in various areas of physics, including gauge theories, quantum field theory, and string theory. They have been used to study important phenomena such as topological insulators and quantum Hall effect.

3. How are Pontryagin densities and Chern-Simons form related?

Pontryagin densities and Chern-Simons form are closely connected as the latter can be expressed as an integral of the former. In other words, Chern-Simons form is a type of Pontryagin density. They both involve the use of differential forms and are used to study the topology of spacetime.

4. What are some applications of Pontryagin densities and Chern-Simons form?

Aside from their use in theoretical physics, Pontryagin densities and Chern-Simons form have found applications in other fields such as differential geometry, topology, and mathematical physics. They have also been used in the study of knot theory and in the construction of topological invariants.

5. Are there any current research developments involving Pontryagin densities and Chern-Simons form?

Yes, there is ongoing research in the field of topological field theories, which heavily rely on the use of Pontryagin densities and Chern-Simons form. Additionally, there are efforts to extend their applications to other areas such as condensed matter physics and quantum computing.

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