I can't seem to find a good source for the motivation for defining the big quantum cohomology ring with its quantum product. Collecting the Gromov-Witten invariants in a generating function seems like a sensible thing to do (and is supposedly related to free energy in the topological $A$-model), but I don't see where the quantum product comes from.
I vaguely understand that this is supposed to come from quantum field theory/string theory, but the only connection to that effect that I'm aware of is the equivalence between 2D TQFTs and Frobenius algebras and the identity $\langle a*b, c\rangle = \langle a,b,c\rangle$ (which comes from diffeomorphism invariance); setting the three point functions to be Gromov-Witten invariants for (as an example) $\mathbb{P}^r$ produces a TQFT with quantum product $\ast$ defined by the above identity and linearity: $$h^i * h^j = \sum_{e+f =r} \langle h^i,h^j,h^e \rangle h^f$$ where $h \in H^\ast(\mathbb{P}^r)$ is the hyperplane class. Examining the nonvanishing GW invariants, this produces the ring structure $\mathbb{Z}[h]/(h^{r+1}-1)$ which is the $q=1$ limit of the small quantum cohomology ring, or, equivalently, the $\mathbf{x} =0$ limit of the big quantum cohomology ring. Is the leap from this to either of the quantum cohomology rings ad-hoc, or is there an underlying explanation? Is there a physical model underlying these constructions that is more complicated than a TQFT?
Edit: $\mathbb{Z}[h]/(h^{r+1} -1)$ is precisely the deformation of the ordinary cohomology ring described in Witten's paper 2D Gravity and Intersection Theory on Moduli Space (up to choice of coefficients) as the "quantum cohomology ring", and he says that the big quantum cohomology ring generalizes this, but it's opaque to me how one would come up with the definition $\mathcal{O}_\alpha \mathcal{O}_\beta = f_{\alpha \beta}^\gamma \mathcal{O}_\gamma$ where $f_{\alpha \beta \gamma} = \frac{\partial^3 F}{\partial y^\alpha \partial y^\beta \partial y^\gamma}$ and where $F$ is the Gromov-Witten potential (this is precisely the quantum product). He writes that the above equality is equivalent to a certain overdetermined set of PDEs, which might be related to the boundary divisor identities (e.g the three boundary points on $\overline{M}_{0,4} = \mathbb{P}^1$ are linearly equivalent, and pulling these boundary divisors back produces nontrivial equivalences on the spaces $\overline{M}_{0,n}(\mathbb{P}^r,d)$).