It depends on your point of view.
As a physicist, a solid foundation on QFT is the first thing you need. Then you need to familiar yourself with supersymmetric QFT in various dimensions. From there, you can follow all sorts of topological twists to get to topological field theories.
To...
People usually interpret the geometric modulus of a Calabi-Yau internal space as fields in the effective field theory, after compactification of a string theory on some Calabi-Yau space, without a good explanation of this interpretation. If you have a good explanation, please share with us.
String theory - THE largest and deepest area combining both physics and mathematics. Algebraic geometry, algebraic topology, differential geometry... all got used in string theory.
There is also this field called "physical mathematics", such as Topological Field Theory, in which you can use...
For any free p-brane
S_p=-T_p\int \sqrt{-det(G_{ab})}d^{p+1}\sigma
where G_{ab} is the induced metric on the world volume of the p-brane. The idea is that this action gives you the volume of the p-brane world volume. The p=1 case gives you the Nambu-Goto action for strings. Read GSW or...
D-branes in string theory are defined as subspaces of spacetime on which open strings can end. The idea is that the open string spectrum can be described by fields living on the world volume of D-branes. Therefore, in string theory, D-brane actions are constructed using open string modes...
You are right. But still they have the sam cohomology as the cylinder, which is the same as the cohomology of the circle. In this way, I got the result that the Klein bottle and the torus have the same cohomology. I do not know if this is correct, but I think I am doing the right thing.
Yes, I used tow cylinders to cover it. The result I got this way shows that Klein bottle has the same cohomology as the torus. I am not sure whether this is correct.