AI Starts to Sift Through String Theory's Near-Endless Possibilities
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Welcome to the Quanta Science Podcast. Each episode, we bring you stories about developments in science and mathematics. I'm Susan Vallett. Using machine learning, string theorists are finally showing how microscopic configurations of extra dimensions translate into sets of elementary particles, though not yet those of our universe. That's next.

It's season three of The Joy of Why, and I still have a lot of questions. Like, what is this thing we call time? Why does altruism exist? And where is Jana Levin? I'm here, astrophysicist and co-host, ready for anything. That's right. I'm bringing in the A-team. So brace yourselves. Get ready to learn. I'm Jana Levin. I'm Steve Strogatz. And this is... Quantum Magazine's podcast, The Joy of Why. New episodes drop every other Thursday.

String theory captured the hearts and minds of many physicists decades ago because of a beautiful simplicity. The theory goes like this: Zoom in far enough on a patch of space, and you won't see a menagerie of particles or jittery quantum fields. There will only be identical strands of energy, vibrating and merging and separating.

By the late 1980s, physicists found that these strings can cavort in just a handful of ways. That raised the possibility that physicists could trace the path from dancing strings to the elementary particles of our world. The deepest rumblings of the strings would produce gravitons. Those are hypothetical particles believed to form the gravitational fabric of spacetime.

Other vibrations would give rise to electrons, quarks, and neutrinos. String theory was dubbed a theory of everything. Anthony Ashmore is a string theorist at Sorbonne University in Paris. People thought it was just a matter of time until you could compute everything there was to know about these things, and then just compute predictions in your particle physics theory directly from string theory. But as physicists studied string theory, they uncovered a hideous complexity.

When they zoomed out from the austere world of strings, every step toward our rich world of particles and forces introduced an exploding number of possibilities. For mathematical consistency, strings need to wriggle through 10-dimensional space-time. But our world has four dimensions, leading string theorists to conclude that the missing six dimensions are coiled into microscopic shapes resembling loofahs.

These imperceptible 6D shapes come in trillions upon trillions of varieties. On those loofahs, strings merge into the familiar ripples of quantum fields, and the formation of these fields could also come about in a bunch of different ways. So our universe would consist of the aspects of the fields that spill out from the loofahs into our giant four-dimensional world.

String theorists sought to determine whether the loofahs and fields of string theory can underlie the portfolio of elementary particles found in the real universe, but there are an overwhelming number of possibilities to consider: 10 to the 500, according to one tally. And the bigger problem is no one could figure out how to zoom out from a specific configuration of dimensions and strings to see what macro world of particles would emerge.

Laura Anderson is a physicist at Virginia Tech who's spent much of her career trying to link strings with particles. We don't know the rules for writing down 4D, our universe type string theory, because we don't know how to take the theory from 10 dimensions down to the four dimensions we'd see and actually say what it means to be a consistent string theory in four dimensions.

And that's been a major obstacle to making contact with real world phenomena. So, you know, you might say, does string theory make unique predictions yet? Is it really physics? And the jury's still out because the technology of how you calculate that, how you make the prediction, how you decide. I mean, string theory may be effectively ruled out already if we could just finish the computations and say, what do we get?

Of course, I'm sweeping loads of other computational subtleties under the rug, but a huge part is you just can't work out the theory. Now, a fresh generation of researchers has brought a new tool to bear on the old problem. They're using neural networks, the computer program's powering advances in artificial intelligence.

In recent months, two teams of physicists and computer scientists have used neural networks to calculate precisely for the first time what sort of macroscopic world would emerge from a specific microscopic world of strings. This long-sought milestone reinvigorates a quest that largely stalled decades ago: the effort to determine whether string theory can actually describe our world.

Here's Anderson again. We're not at saying these are the rules for the theory of our universe yet, but it's a big step in the right direction. The crucial feature that determines what macro world emerges from string theory is the arrangement of the six small spatial dimensions.

The simplest such arrangements are intricate 6D shapes called Calabi-Yau manifolds, the objects that resemble loofahs. They're named after the late Eugenio Calabi, the mathematician who conjectured their existence in the 1950s, and Xingtong Yao, who in the 1970s set out to prove Calabi wrong but ended up doing the opposite.

Calabi-Yau manifolds are 6D spaces with two characteristics that make them attractive to physicists. First, they can host quantum fields with a symmetry known as supersymmetry. Supersymmetric fields are much simpler to study than more irregular fields. Experiments at the Large Hadron Collider have shown that the macroscopic laws of physics are not supersymmetric.

But the nature of the "microworld" beyond the standard model remains unknown. Most string theorists work under the assumption that the universe at that scale is supersymmetric, with some citing physical motivations for believing so, while others do so out of mathematical necessity.

Second, Calabi-Yau manifolds are "reachy flat." According to Albert Einstein's general theory of relativity, the presence of matter or energy bends spacetime, causing so-called "reachy curvature." Calabi-Yau manifolds lack this kind of curvature, though they can and do curve in other ways unrelated to their matter and energy contents.

To understand Ricci flatness, consider a donut, which is a low-dimensional Collabi-Yau manifold. You can unroll a donut and represent it on a flat screen. If you move off the right side, you're teleported to the left side and likewise with the top and bottom.

Rows of colorful loofah-like shapes line up on a black background. They grow increasingly complicated from one row to the next, with greater numbers of holes and curves. The general game plan for string theory, then, boils down to searching for the specific manifold that would describe the microstructure of spacetime in our universe.

One way to search is by picking a plausible 6D donut and working out whether it matches the particles we see. The first step is to work out the right class of 6D donuts.

Countable features of Calabi-Yau manifolds, such as the number of holes they have, determine the countable features of our world, such as how many distinct matter particles exist. So researchers start by searching for Calabi-Yau manifolds with the right assortment of countable features to explain the known particles.

Researchers have made steady progress on this step. In particular, over the last couple of years, a United Kingdom-based collaboration has refined the art of doughnut selection to a science. The group used insight gathered from an assortment of computational techniques in 2019 and 2020.

The researchers identified a handful of formulas that spit out classes of Collabi-Yau manifolds, producing what they call broad brush versions of the standard model, containing the right number of matter particles. These theories tend to produce long-distance forces we don't see. Still, with these tools, the UK physicists have mostly automated what were once daunting calculations.

Andre Constantin is a physicist at the University of Oxford who led the discovery of the formulas. He says the efficacy of these models is absolutely staggering. These formulas reduce the time needed for the analysis of string theory models from several months of computational efforts to a split second.

The second step is harder. String theorists aim to narrow the search beyond the class of Collabi-Yau's and identify one particular manifold. They seek to specify exactly how big it is and the precise location of every curve and dimple. These geometric details were supposed to determine all the remaining features of the macro world, including precisely how strongly particles interact and exactly what their masses are.

Completing this second step requires knowing the manifolds metric, a function that can take in any two points on the shape and tell you the distance between them. A familiar metric is the Pythagorean theorem, which encodes the geometry of a 2D plane. But as you move to higher dimensional, curvy spacetimes, metrics become richer and more complicated descriptions of the geometry.

Physicists solved Einstein's equations to get the metric for a single rotating black hole in our 4D world. But 6D spaces have been out of their league. Toby Wiseman is a physicist at Imperial College London. One of the things you learn as you carry on in physics, even like in undergrad, is that simple and beautiful equations often just don't have nice solutions. Not solutions you can write down with pen and paper, at least.

So it's one of the saddest things as a physicist that you come across. All of these beautiful examples you learn in high school where you do your maths and you integrate things and fancy fiddle around and you get some fancy answer out. They're all very special cases and almost nothing can be integrated in real life. So unfortunately, the power of mathematics, clever as it is, is quite limited when it comes to actually writing down solutions to equations. It's one of the things we struggle with as physicists.

And this is a case like that. As a postdoc at Harvard University in the early 2000s, Wiseman heard whispers of the mythical metrics of Calabi-Yau manifolds. Yao's proof that these functions exist helped win him the Fields Medal, the top prize in mathematics. But no one had ever calculated one.

At the time, Wiseman was using computers to approximate the metric of spacetime surrounding exotic black holes. He speculated that perhaps computers could also solve for the metrics of Calabi-Yau spacetimes. Everyone had known for quite a long time that Calabi-Yau metrics are known to exist, but no one had ever written one down. And I thought, well, OK, but probably maybe you can just shove it in a computer and find it. And everyone said, oh, no, you couldn't possibly do that.

So myself and a guy called Matt Hedrick, a very brilliant guy, stringed through it. We sat down and we showed it could be done. Wiseman and Hedrick, who works at Brandeis University, knew that a Calabi-Yau metric had to solve Einstein's equations for empty space. A metric satisfying this condition guaranteed that a spacetime was Ricci-flat.

Wiseman and Hedrick picked four dimensions as a proving ground. They leveraged a numerical technique sometimes taught in high school calculus classes. With that, they showed in 2005 that a 4D Calabi-Yau metric could indeed be approximated. It might not be perfectly flat at every point, but it came extremely close, like a donut with a few imperceptible dents.

Around the same time, Simon Donaldson, a prominent mathematician also at Imperial, was also studying Calabi-Yau metrics for mathematical reasons. He soon worked up another algorithm for approximating metrics. String theorists, including Anderson, started trying to calculate specific metrics in these ways, but their procedures took a long time and produced overly bumpy doughnuts. That would mess up attempts to make precise particle predictions.

So attempts to complete step two died out for nearly a decade. But as researchers focused on step one and on solving other problems in string theory, a powerful new technology for approximating functions swept computer science: neural networks. These adjust huge grids of numbers until their values can stand in for some unknown function.

Neural networks found functions that could identify objects in images, translate speech into other languages, and even master humanity's most complicated board games. In 2016, the AlphaGo algorithm, created by the artificial intelligence company DeepMind, bested a top human Go player. That's when physicist Fabian Moulet took notice. I thought if this thing can...

outperform the world champion in Go and chess and whatever, maybe can outperform mathematicians or physicists like me at least. Roulet, who is now at Northwestern, and his collaborators took up the old problem of approximating Calabi-Yau metrics. Anderson and others also revitalized their early attempts to overcome step two. The physicists found that neural networks provided the speed and flexibility that earlier techniques had lacked.

The algorithms were able to guess a metric, check the curvature at many thousands of points in 6D space, and repeatedly adjust the guess until the curvature vanished all over the manifold. All the researchers had to do was tweak freely available machine learning packages. By 2020, multiple groups had released custom packages for computing Calabi-Yau metrics.

With the ability to obtain metrics, physicists could finally contemplate the finer features of the large-scale universes corresponding to each manifold. Here's Roulet again. And so we constructed this and we then used it, of course, the first thing I did after I had it, I actually calculated masses of particles.

In 2021, Roulet and Ashmore cranked out the masses of exotic heavy particles that depend only on the curves of the Calabi-Yau. But these hypothetical particles would be far too massive to detect. To calculate the masses of familiar particles like electrons, a goal string theorists have chased for decades, the machine learners would have to do more.

Lightweight matter particles acquire their masses through interactions with the Higgs field, a field of energy that extends throughout space. The more a given particle takes notice of the Higgs field, the heavier it is. How strongly each particle interacts with the Higgs is labeled by a quantity called its Yukawa coupling. And in string theory, Yukawa couplings depend on two things:

One is the metric of the Calabi-Yau manifold, which is like the shape of the doughnut. The other is the way quantum fields, arising as collections of strings, spread out over the manifold. These quantum fields are a bit like sprinkles. Their arrangement is related to the doughnut's shape, but also somewhat independent.

Roulet and other physicists had released software packages that could get the doughnut shape. The last step was to get the sprinkles, and neural networks proved capable of that task too. Two teams put all the pieces together earlier this year. An international collaboration led by Challenger Mishra of the University of Cambridge first used a homegrown neural network to calculate the metric, the geometry of the doughnut itself.

They then harnessed additional original algorithms to compute the way the quantum fields overlap as they curve around the manifold, like the donut sprinkles. Importantly, they worked in a context where the geometry of the fields and that of the manifold are tightly linked, a setup in which the Yukawa couplings could be calculated in an alternative way, although this had never been done before.

When the group calculated the couplings in both manners, the results matched. Moreover, the couplings they found hinted at a separation between particle masses, a mysterious feature of the standard model. Here's Mishra. People have been wanting to do this, you know, even from before I was born in the 80s. But ultimately, now is the time where we can exploit machine learning to do this.

And the other important factor, I think, is what this new machine learning approach is doing is it is galvanizing the community of string theory researchers. A group led by string theory veterans Bert Ovrut of the University of Pennsylvania and Andre Lucas of Oxford also started with Roulet's metric calculating software. Lucas knew it well. He'd helped develop it.

Building on that foundation, they added an array of 11 neural networks to handle the different types of sprinkles. These networks allowed them to calculate an assortment of fields that could take on a richer variety of shapes, creating a more realistic setting that can't be studied with any other techniques.

This army of machines learned the metric and the arrangement of the fields, calculated the Yukawa couplings, and spit out the masses of three types of quarks. It did all this for six differently shaped Calabi-Yau manifolds.

physicist Lara Anderson. I think it's exciting. This is the first time anybody has been able to calculate them to that degree of accuracy. None of those Calabi-Yau's underlies our universe because two of the quarks have identical masses, while the six varieties in our world come in three tiers of masses. Rather, the results represent a proof of principle that machine learning algorithms can take physics from a Calabi-Yau manifold all the way to specific particle masses.

You remember physicist Andrei Konstantin from earlier. He's a member of the group based at Oxford. Until now, and until all these techniques about computing Calabiau metrics have been developed, any such calculations would have been just unthinkable. The message is that these calculations are possible now, and

There are a lot of models on which such methods can be applied. The neural networks choke on donuts with more than a handful of holes, and researchers would eventually like to study manifolds with hundreds. And so far, the researchers have considered only rather simple quantum fields. String theorist Anthony Ashmore says we'll need more to go all the way to the standard model. It's a more involved problem, and you probably need a more sophisticated neural network.

Bigger challenges loom on the horizon. Attempting to find our particle physics in the solutions of string theory, if it's in there at all, is a numbers game. The more sprinkle-laden doughnuts you can check, the more likely you are to find a match. After decades of effort, string theorists can finally check doughnuts and compare them with reality, the masses and couplings of the elementary particles we observe.

But even the most optimistic theorists recognize that the odds of finding a match by blind luck are cosmically low.

the number of Calabi-Yau doughnuts alone may be infinite. Here's Roulet again. Or you need to learn how to game the system. If you know how to rig the lottery, so if you know how to sort of select which numbers to pick that will be winning, you win. One approach is to check thousands of Calabi-Yau manifolds and try to suss out any patterns that could steer the search. For instance, by stretching and squeezing the manifolds in different ways, physicists might develop an intuitive sense of what shapes lead to what particles.

Here's Ash Moore again. What you really hope actually is that you have some really strong reason for looking at particular models that you think are particularly hopeful. Maybe you stumble into the right kind of model that gives all will.

Or generally, you want to learn something about these Calabi-Yau's that you didn't understand before. You want to gain some general intuition for what kinds of models give the right kind of physics in our world. Lucas and colleagues at Oxford plan to start that exploration, prodding their most promising doughnuts and fiddling more with the sprinkles as they try to find a manifold that produces a realistic population of quarks.

Konstantin believes that they will find a manifold reproducing the masses of the rest of the known particles in a matter of years. But other string theorists think it's premature to start scrutinizing individual manifolds. Tomas van Riet of KU Leuven is a string theorist pursuing the Swampland Research Program.

It seeks to identify features shared by all mathematically consistent string theory solutions, such as the extreme weakness of gravity relative to the other forces. Van Riet and his colleagues aspire to rule out broad swaths of string solutions, that is, possible universes, before they even start to think about specific donuts and sprinkles. We need to take a step back. We need to zoom out. We need to think about

the underlying principles, the patterns. And now what you're asking is a detail. And I think it's good that people do this machine learning business because I'm sure we will need it at some point. And then the technology is there. Plenty of physicists have moved on from string theory to pursue other theories of quantum gravity. And the recent machine learning developments are unlikely to bring them back.

Renata Lowell, a physicist at Radboud University in the Netherlands, says to truly impress, string theorists will need to predict and confirm new physical phenomena beyond the standard model. She says it's a needle in a haystack search, and while she's not sure what we'll learn from it, there should be some interesting new physical predictions.

New predictions are indeed the ultimate goal of many of the machine learners. They hope that string theory will prove rather rigid, in the sense that donuts matching our universe will have commonalities. For instance, these donuts might all contain a kind of novel particle that could serve as a target for experiments. For now, though, that's purely aspirational, and it might not pan out.

Nima Arkani-Hamed, a theoretical physicist at the Institute for Advanced Study in Princeton, New Jersey, points out that the track record for qualitatively correct statements about the universe is really garbage.

Ultimately, the question of what string theory predicts remains open. Now that string theorists are leveraging the power of neural networks to connect the 60 microworlds of strings with the 40 macroworlds of particles, they stand a better chance of someday answering it. Or in the words of physicist Lara Anderson, Without doubt, there are loads of string theories that have nothing to do with the unique.

I could say that definitively. But the question is, are there any that have something to do with it? That's the interesting question. And the answer might be no, but I think it's really interesting to try and push the theory to decide. ♪

Arlene Santana helped with this episode. I'm Susan Vallett. For more on this story, read Charlie Wood's full article, "AI Starts to Sift Through String Theories Near Endless Possibilities," on our website, quantamagazine.org. Explore math mysteries in the quanta book, "The Prime Number Conspiracy," published by the MIT Press, available now at amazon.com, barnesandnoble.com, or your local bookstore.