less joking title:

# You Didn’t Think We’d Stop at Elliptics, Did You?

When calculating scattering amplitudes, I like to work with polylogarithms. They’re a very well-understood type of mathematical function, and thus pretty easy to work with.

Even for our favorite theory of N=4 super Yang-Mills, though, they’re not the whole story. You need other types of functions to represent amplitudes, elliptic polylogarithms that are only just beginning to be properly understood. We had our own modest contribution to that topic last year.

You can think of the difference between these functions in terms of more and more complicated curves. Polylogarithms just need circles or spheres, elliptic polylogarithms can be described with a torus.

A torus is far from the most complicated curve you can think of, though.

String theorists have done a lot of research into complicated curves, in particular ones with a property called **Calabi-Yau**. They were looking for ways to curl up six or seven extra dimensions, to get down to the four we experience. They wanted to find ways of curling that preserved some supersymmetry, in the hope that they could use it to predict new particles, and it turned out that Calabi-Yau was the condition they needed.

That hope, for the most part, didn’t pan out. There were too many Calabi-Yaus to check, and the LHC hasn’t seen any supersymmetric particles. Today, “string phenomenologists”, who try to use string theory to predict new particles, are a relatively small branch of the field.

This research did, however, have lasting impact: due to string theorists’ interest, there are huge databases of Calabi-Yau curves, and fruitful dialogues with mathematicians about classifying them.

This has proven quite convenient for us, as we happen to have some Calabi-Yaus to classify.

We call Feynman diagrams like the one above “traintrack integrals”. With two loops, it’s the elliptic integral we calculated last year. With three, though, you need a type of Calabi-Yau curve called a K3. With four loops, it looks like you start needing Calabi-Yau three-folds, the type of space used to compactify string theory to four dimensions.

“We” in this case is myself, Jacob Bourjaily, Andrew McLeod, Matthias Wilhelm, and Yang-Hui He, a Calabi-Yau expert we brought on to help us classify these things. Our new paper investigates these integrals, and the more and more complicated curves needed to compute them.

Calabi-Yaus had been seen in amplitudes before, in diagrams called “sunrise” or “banana” integrals. Our example shows that they should occur much more broadly. “Traintrack” integrals appear in our favorite N=4 super Yang-Mills theory, but they also appear in theories involving just scalar fields, like the Higgs boson. For enough loops and particles, we’re going to need more and more complicated functions, not just the polylogarithms and elliptic polylogarithms that people understand.

(And to be clear, no, nobody needs to do this calculation for Higgs bosons in practice. This diagram would calculate the result of two Higgs bosons colliding and producing ten or more Higgs bosons, all at energies so high you can ignore their mass, which is…not exactly relevant for current collider phenomenology. Still, the title proved too tempting to resist.)

Is there a way to understand traintrack integrals like we understand polylogarithms? What kinds of Calabi-Yaus do they pick out, in the vast space of these curves? We’d love to find out. For the moment, we just wanted to remind all the people excited about elliptic polylogarithms that there’s quite a bit more strangeness to find, even if we don’t leave the tracks.

ThorstenThis is pretty interesting! Is it the first time calabi yaus appear in standard field theory amplitudes not connected to string theory ? Is this a further sign of a connection between field and string theory, where calabi yaus have appeared before?

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4gravitonsandagradstudentPost authorIt’s not the first time, though I think it might be the first time they’ve been seen in a massless theory, or in four dimensions. (The only other example I’m aware of involves massive scalar particles in two dimensions, they’re the “banana” graphs I mentioned.)

The context in which Calabi-Yaus appear in string theory is pretty different, so I don’t think there’s likely to be a really direct connection. Maybe an indirect one, if something about the Calabi-Yau “flatness” condition matters to amplitudes.

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ThorstenCool! Is there any other area besides string theory/field theory in physics where Calabi-Yau s play a role in some context?

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4gravitonsandagradstudentPost authorProbably! I don’t know one off the top of my head though.

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Madeleine BirchfieldI though Calabi-Yau manifolds are from algebraic geometry, though. Isn’t this simply an example of an application of algebraic geometry in quantum field theory?

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4gravitonsPost authorThe motivation to study those specific manifolds came from physics (at least if I remember the history right). But you’re right on a more broad-strokes level, especially because there are probably amplitudes that involve other varieties that don’t correspond to Calabi-Yaus.

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Madeleine BirchfieldIt just seems to me that so many of the ‘contributions’ string theory claim to have made to other areas of physics is in reality contributions string theory made to pure and computational mathematics, which then in turn is applied to other areas of physics.

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4gravitonsPost authorThat’s the path in this case certainly, but I can’t think of any other clear examples. The knot theory applications are usually claimed as a contribution to mathematics not physics, various AdS/CMT, AdS/QCD, etc. things were never of interest to mathematics, and likewise the main contributions to amplitudes (double-copy, various techniques that come from N=4 sYM) didn’t go through mathematics first.

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