How can a physicist understand a 2-dimensional topological field theory as a Frobenius algebra? Are there some explicit examples in order to understand this relation?
The definition (e.g. on Wikipedia) of the Frobenius algebra is quite clear, it is a finite dimensional associative algebra equipped with a special non-degenerate bilinear form. If this algebra is represented by $n\times n$ matrices then the bilinear form is the trace.
Now, I fail to see how this is a TFT.
In specific two very famous topological field theories in 2-dimensions (that physicists work with a lot) are the topological A-model and the topological B-model (a la Witten). These model are related via mirror symmetry. How are they defined as a Frobenius algebra and what are their differences as a Frobenius algebra. And natural question to ask is wether mirror symmetry in the TFT side relates somehow the two Frobenius algebras on the algebra side.
So I would like to understand intuitively why a Frobenius algebra and a TFT are the same thing and the detail I ask in specific about the A-model and the B-model which are of my main interest.
Michael Atiyah has already written an article that does this well. Here's the citation, should the link ever rot:
There is also this, the very first google hit in my search, that also appears to get the job done, but it seems to have more detail. It might be good to keep the errata to the book that it comes from handy too.
Finally, here is another article to use.