TITLE: Why is a Topological Field Theory equivalent to a Frobenius algebra?
QUESTION [15 upvotes]: 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 models 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.

REPLY [13 votes]: Arun Debray's answer is good, with one addendum: 2d TQFT is the same thing as a commutative Frobenius algebra.  But to truly understand what's going on, one also needs some pictures.  A commutative Frobenius algebra can be defined using pictures that just happen to look exactly like the relations holding between 2-dimensional cobordisms.  The best place to learn about this is here:

Joachim Kock, Frobenius Algebras and 2d Topological Quantum Field Theories, Cambridge U. Press, Cambridge, 2004.

But briefly, a Frobenius algebra has a multiplication and unit obeying associativity and the left and right unit laws:

and also a comultiplication and counit obeying coassociativity and the left and right counit laws:

where the multiplication and comultiplication obey the Frobenius laws:

A Frobenius algebra is commutative if switching two things and then multiplying them is the same as multiplying them:

Furthermore, a Frobenius algebra is commutative iff it is cocommutative, meaning this:

The ultimate statement of this fact is that $2\mathrm{Cob}$, the category with compact oriented 2-dimensional cobordisms as morphisms, is the free symmetric monoidal category on a commutative Frobenius monoid object.  The story gets much more interesting in 3 dimensions, and for that I recommend this paper:

Bruce Bartlett, Christopher L. Douglas, Christopher J. Schommer-Pries and Jamie Vicary, Extended 3-dimensional bordism as the theory of modular objects.