HELP Why CS is topological? Why BF is topological?

[astros]

Hello,
I know that the CS Lagrangian is a topological invariant, in the sense that it does not depend on the connection we choose. OK, but a TFT is a field theory whose Lagrangian and all other observables do not depend on the metric, a connection in general is not uniquely defined by a metric! Then how can I see CS theory as topological? Another problem, CS Lagrangian is a 3-forme, normally; a Lagrangian must be a scalar!? After that, can I see that CS is topological by showing that the number of constraints is equal to the number of degrees of freedom? If yes HOW! The same for BF! Please help me RANI N’NAGER!

 

[astros]

nobody!

 

[Entropia]

Hello RANI N’NAGER,

Thank you for your question. I understand your confusion about how Chern-Simons (CS) and BF theories can be considered topological. Let me try to explain it in a simple way.

Firstly, a topological field theory (TFT) is a field theory whose observables do not depend on the metric or the connection. This means that the theory is independent of the geometry of the underlying spacetime. In CS and BF theories, the Lagrangian is a topological invariant, meaning that it does not change under small deformations of the underlying manifold. This is why these theories are considered topological.

Now, let's address your concerns about the CS Lagrangian being a 3-form and not a scalar. In TFTs, we are interested in topological invariants, not physical quantities. Therefore, the Lagrangian does not have to be a scalar, as we are not concerned with the actual value of the Lagrangian but rather its topological properties. Additionally, the number of constraints in a TFT is indeed equal to the number of degrees of freedom, which is another characteristic of topological theories.

I hope this helps to clarify why CS and BF theories are considered topological. If you have any further questions, please don't hesitate to ask. Best of luck in your studies!