A Introduction to topological field theory?

[Kris-L]

Hi!

I have been looking at differential forms, manifolds and de Rham cohomology. Now I'm trying to figure out the connection from cohomology and equations of motions and topological field theory. Problem is that I am only looking at abelian field theories and I only find introductions into non-abelian or/and Lie groups.

Do you have any tips with a easy introduction from the math into the physics?

 

[azntoon]

Thanks in advance!The best way to understand the connection between cohomology and equations of motion is to look at the action functional. The action functional is a mathematical expression that describes the dynamics of a physical system. It is an integral over a certain space of fields that describes how the system evolves over time. In the case of abelian field theories, the action functional is a complex-valued integral over the space of fields. This integral can be written as the sum of two components: one that depends only on the field values (the "kinetic" term) and one that depends on the derivatives of the field (the "potential" term). The differential form associated with the action functional is known as the Lagrangian. It is the integral of the product of the Lagrangian density (which is a function of the fields and their derivatives) and the volume element of the space of fields. The equations of motion can be derived from the Lagrangian by taking the variational derivatives of the action functional with respect to the fields. These equations of motion are then used to determine the dynamics of the system.The de Rham cohomology is then used to study the topological properties of the system. This involves looking at the symmetries of the action functional and the differentials of the action functional. These symmetries and differentials can be used to understand the conserved quantities of the system and the topological features such as the number of solutions to the equations of motion. It is also possible to use the de Rham cohomology to study the symmetries of the action functional in more detail. This can be used to derive the Ward identities that describe how the equations of motion for a system change under certain transformations. It can also be used to investigate the renormalization group structure of the system. Finally, there is a connection between de Rham cohomology and topological field theory. Topological field theory is a type of quantum field theory which has no local degrees of freedom. Instead, it is characterized by global features such as topological invariants and topological terms in the action functional. The de Rham cohomology can be used to explore these global features and to understand how they are related to the renormalization group structure of the system.