From the sound of it, you are reading Costello's book.
In point particle QFT the stable graphs you are referring to can be thought of as the paths of particles through spacetime the vertices are where the particles interact.
Now in the Deligne-Mumford case you can think of taking the stable graph and "thickening" it and replacing the particles with little loops of string to obtain a Riemann surface. The interactions then can be thought of as corresponding to the joining and splitting of these little loops of string.
It happens to be the case that in bosonic string theory one is in this second "thickened" case and the symmetries of the theory are such that in doing the path integral one ends up integrating over the Deligne-Mumford spaces of such Riemann surfaces. Thus, the Deligne-Mumford stratification is of use there. Furthermore, you can relate Deligne-Mumford spaces of such Riemann surfaces to point particle QFT by simply taking the limit of infinite string tension.
Let $d=\deg S$. If $d\ge 2$ and $S$ is non-degenerate near infinity, in the sense that $|\nabla S(x)|^2 \ge c|x|^{2d-2}$ for $|x|\ge R$ for some positive $R$ and $c$ then the integral converges in a sense close to what you propose. Namely,
$$\lim_{r\rightarrow \infty} \int_{\mathbb R^n} \phi(|x|/r) e^{iS(x)} dx$$ exists where $\phi$ is any sufficiently smooth function with $\phi(t)\equiv 1$ for $0\le t\le 1$ and $\phi (t)\equiv 0$ for $t\ge 2$.
To prove this use integration by parts. Fix $r < r'$ and let
$$I_{r,r'} = \int_{\mathbb R^n} \phi_{r,r'}(x) e^{i S(x)}dx,$$where $\phi_{r,r'}(x)=\phi(|x|/r)-\phi(|x|/r')$. Assume $r >R$. Multiply and divide by $|\nabla S(x)|^2$, noting that
$$|\nabla S(x)|^2e^{i S(x)}= -i\nabla S(x)\cdot \nabla e^{i S(x)}.$$
So we have, after IBP,
$$I_{r,r'} = -i \int_{\mathbb R^n} \left [ \nabla \cdot \left ( \frac{\phi_{r,r'}(x)}{|\nabla S(x)|^2} \nabla S(x) \right ) \right ] e^{i S(x)} d x.$$
The integrand is supported in the region $r \le |x| \le 2 r'$, and you can check that it is bounded in magnitude by some constant times $|x|^{1-d}$. (It is useful to note that an $m$-th derivative of $S$ is bounded from above by $|x|^{d-m}$ since it is a polynomial of degree $d-2$.) If $d$ is large enough this may be enough to control the integral. If not repeat the procedure as many times as necessary to produce a factor that is integrable and you can use to show that $I_{r,r'}$ is small for $r,r'$ sufficiently large. Basically, each time you integrate by parts after multiplying and dividing by $|\nabla S(x)|^2$ you produce a an extra factor of size $|x|^{1-d}$.
All of the above can be extended to integrals with $S$ not necessarily a polynomial, but sufficiently smooth in a neighborhood of infinity and with derivatives that satisfy suitable estimates. The real difficulty with such integrals is not proving that they exist, but estimating their size. Here stationary phase is useful, when applicable, but I don't know of much else.
Best Answer
If I understand correctly, Quantum Field Theory was successful in so far as it was predictive of experimental results. Somehow these badly divergent integrals, when combined correctly product an answer aligning with the result you see in the particle accelerator.
Having gone through a quantum field theory class, I was warned not to worry about being too rigorous as long as I can get the correct result. And the course moved so quickly there wasn't enough time to reflect whether I had seen these tools in my math courses. And I left the course confused and dissatisfied.
There are many types of math and many types of physics. So, I think a great question to ask to is which math and which physics are being related in a given paper.
A few names are mentioned again and again in the paper. Francis Brown, so here is one if his:
Certainly there is earlier literature. Dirk Kreimer is mentioned so I pull one out of a hat:
None of the Hopf algebra structures or Tannakian categories discussed here ever appear in a QFT course. Real QFT homework consists of pages and pages of integerals followed by more integrals and you are never told which matematical theory can formalize this.
Even worse, when you try to formalize these computations, you drown in rigor and entirely lose the spirit of the original computation.
Using the Feynman Rules we can put together diagrams which contribute to the scattering cross sections of various quantum fields theories. In particular $\phi^3$ and $\phi^4$ theory I am seeing a lot. It doesn't matter because they all use the same diagrams. Here's an integral:
$$ \left[ \prod_{k=1}^L \int \frac{d^4 p_k}{p_k^2 (p_k + p)^2}\right]\prod_{m=0}^L \frac{1}{(p_{m+1}-p_m)^2} $$
and later on in that paper we obtain the value of a slightly different diagram: $$ \int [\dots] \, d^4p_k = p^2 \left( \frac{\pi}{p}\right)^{2L} \binom{2L}{L} \zeta(2L-1) $$ for the ladder with $L$ rungs. And this is nothing short of PHENOMENAL.
I believe part of the miracle is, that although we could write down the contributing integrals, we had no idea what the $\int$ evaluated to, even in these simple cases. In class, I thought we had covered this but I guess not.
And I left out the domain of integration - which loosely involves conservation of momentum - but there may be other factors as well.
And there are many facets to these integrals that we are just beginning to find out.
Here are the sources I have looked up. And I recognize that moduli space, it is the moduli space of $n$ marked points on the sphere. Or as I like to think of them as polyhedra (hopefully that is accurate).
The second diagram counts how many Feynman diagrams - which is like a zillion.
Having explained a tiny bit why there is a question at all, we learn that Feynman diagrams evaluate to multiple zeta values or other periods. As I learn a bit more, maybe I can explain why the first examples of motives. QFT students doing their homework.
Here is theorem 0.1 in Brown - page 1
Unless your Deligne or Kontsevich you probably have no idea what that means. I read it was: in the process of doing Feynman integrals we have done a lot of other things:
a motive is a thing that appears in many cohomology theories
Bettisingular cohomology (this is the one from topology)I believe Brown's kinematics is what happens when you evaluate the Feynman integral over momentum space $dp$ rather than position space $dx$.
He also does something rather strange getting these cohomology theories over the fractions $\mathbb{Q}$ rather than $\mathbb{C}$. And the space of kinematic data (e.g. from convervation of momentum) he says define a scheme rather than a variety. Complicating things further, but separating it from what a day-to-day high energy physics graduate student might call a "Feynman diagram"
At least now we see what the motives are and why they are appearing. If you are an expert in motives I direct you to the relevant papers and textbooks. Lastly... a great discussion on Motives (in French)
the last one one of my favorite resources.