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.

- Peskin Schroeder
*Quantum Field Theory*
- Zuber
*Quantum Field Theory*

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

For any Feynman graph $G$ with generic kinematics $q,m$, there is a canonical way to associate to a convergent integral
$$ I_G(q,m) = \int \bigg[ \frac{P(\alpha_E) \Omega_G }{\Psi_G^A \Xi_G(q,m)^B } \bigg] $$

- an object $\text{mot}_G$ in $\mathcal{H}(S)$, where $S$ is a zariski open [set] in the space of kinematics
- a de Rham class $[\omega_G]$ in the generic fiber of $(\text{mot}_G)_{dR} $
- a Betti class $[\sigma_G]$ in a certain (Euclidean) fiber of $(\text{mot}_G)^\hacek_B$

such that the [Feynman diagram] integral is the period
$$ \sigma_G\big( c(\omega_G) \big) = I_G(q,m)$$

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

- the space you're integrating over is part of
~~Betti~~ singular cohomology (this is the one from topology)
- the integrand is part of the deRham cohomology (this is the one from calculus)
- By some sort of Poincaré duality there's a way to match the Betti and deRham cohomolog to get a
**number**

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.