Relation between QFT and algebraic geometry

Question

So, I've read this article who mentioned that physicists in LHC, while calculating the Feynman diagrams in one of their experiments, noticed a strange pattern: the numbers emerging from Feynman's diagrams were the same of periods in algebraic geometry.

Original source : Quanta Magazine , which seems well worth reading.

Over the last decade physicists and mathematicians have been exploring a surprising correspondence that has the potential to breathe new life into the venerable Feynman diagram and generate far-reaching insights in both fields. It has to do with the strange fact that the values calculated from Feynman diagrams seem to exactly match some of the most important numbers that crop up in a branch of mathematics known as algebraic geometry. These values are called “periods of motives,” and there’s no obvious reason why the same numbers should appear in both settings. Indeed, it’s as strange as it would be if every time you measured a cup of rice, you observed that the number of grains was prime.

So, my question is:

If there is, actually, this relation between algebraic geometry and quantum mechanics, what would be the greatest implications and how would it be helpful for physics?

Answer 1

The Quanta magazine article you link references "Knots and Numbers in $\varphi^4$ Theory to 7 Loops and Beyond" by Broadhurst and Kreimer from 1995 which does not even mention the word "period" anywhere in it, but nevertheless draws connections between the numbers appearing in amplitudes via Feynman diagram computations and combinatorial objects, the Catalan numbers.

However, the connection between scattering amplitudes and algebro-geometric objects has indeed been an active area of research in the last decades. From the amplituhedron showing the equivalence of certain amplitudes with volumes of Graßmannians by Arkani-Hamed over "Motivic Multiple Zeta Values and Superstring Amplitudes" by Schlotterer and Stieberger to "Periods and motives in the spectral action of Robertson-Walker spacetimes" by Fathizadeh and Marcolli (semi-random examples off the top of my head) various amplitudes in physics have been related to purely mathematical objects.

The "greatest implication" of these discoveries is that if the connection between them could be shown more generally than just for the examples already investigated, the connection would obviate the need to compute Feynman diagrams, and we could instead just compute the relevant algebro-geometric invariants equivalent to the amplitudes, which is often a rather "easy" computation (both conceptually and numerically) when compared to the integrals of Feynman diagrams.

What exactly the implications for our theory of quantum fields would be depends on what exactly the final result of these connections may be. In the most striking case, it may well be that it would point the way to a more algebraic instead of analytic conception of spacetime and quantum fields on it, but to say anything concrete about this would be mere unfounded speculation on my part.