A Frobenius manifold is a type of manifolds with extra structure.
The main examples are quantum cohomology (viewed as a space itself), GBV algebras, the ``Saito'' examples arising from singularities (e.g. I think the ADE singularity correspoding to Lie group $\mathfrak{g}$ give the manifold $\mathfrak{h}/W$, at least for $\mathfrak{sl}_n$), and the ADE examples by Dubrovin.
Maybe it's unsurprising that quantum cohomology, which is defined in terms of Gromov-Witten invariants (=maps from Riemann surfaces into a space), should have something to do with Frobenius algebras (=a $1$+$1$d TQFT), using the pants decomposition of a Riemann surface.
However, is there a more precise answer (perhaps coming from physics) motivating the notion of a Frobenius algebra?
What is the physical system they are modelling?
Finally, how should I interpret the structure connection of a Frobenius manifold?
Ideally such an answer would make clear why these seemingly unrelated examples are instances of the same phenomenon. I'm especially curious about Saito's examples about singularities, and why Hodge theory/the Brieskorn lattice appears.
