Mathematicians and String Theorists Push Each Other’s Fields Forward

When asked why he wanted to climb Mount Everest, legendary British mountaineer George Mallory purportedly said, “Because it’s there.”

Mathematicians, says Lev Rozansky, are drawn to certain mathematical problems for the same reason. “Many mathematicians think there are no practical applications to some problems, but people attempt them anyway, because they’re there,” says Rozansky, a theoretical physicist and mathematician at the University of North Carolina at Chapel Hill and an investigator with the Simons Collaboration on New Structures in Low-Dimensional Topology. “But it turns out there are many places where there is, in fact, overlap with practical applications.”

One of the areas with the largest overlap is a branch of theoretical physics known as string theory. String theorists posit that if you break down matter to its smallest scales, you will eventually find that everything is composed of one-dimensional bits of vibrating energy called strings. The fundamental framework for how these strings move and interact is string theory. This realm of physics requires both an understanding of some of the furthest frontiers of fundamental physics and specialized knowledge of the mathematics at play. That’s a combination that’s largely out of reach for any one person.

“It’s a huge challenge to master several different subjects, and therefore most people tend to stay in one area,” says Sergei Gukov, a professor of theoretical physics and mathematics at the California Institute of Technology. The solution, he says, is a greater collaboration between physicists and mathematicians.

Driving that cooperation are initiatives supported by the Simons Foundation that bring together top mathematicians and physicists from around the world. For much of history, math and physics grew together thanks to polymaths such as Isaac Newton and Galileo Galilei. These days, researchers are often siloed in their respective fields, with physicists typically learning only the math necessary for their disciplines and mathematicians largely forgoing the study of physics. At some universities, math and physics departments aren’t even located in the same part of campus.

“The cultural gap between mathematicians and physicists is a problem,” Rozansky says. “But on the other hand, for those who can understand both sides, there is a great opportunity.”

That opportunity includes solving one of the greatest unsolved problems looming over the fields of mathematics and physics. Called the Yang–Mills existence and mass gap problem, this mystery is one of seven with a $1 million bounty for its solution offered by the Clay Mathematics Institute.

The famous Yang–Mills problem is one that the Simons Collaboration on Confinement and QCD Strings was established to tackle. Since 2023, members of the collaboration, comprising a mix of physicists and mathematicians, have been working together to study aspects of string theory and the strong nuclear force (one of the four fundamental forces along with gravity, electromagnetism and the weak force).

The strong force holds together strongly interacting particles such as protons and neutrons and describes the subatomic particles they’re made of, called quarks and gluons. The fundamental description of the strong force is known as quantum chromodynamics, or QCD. Physicists believe that QCD predicts that quarks never travel alone and are instead always ‘confined’ by stringlike tubes of energy into larger particles like protons and neutrons.

Princeton University’s Igor Klebanov, director of the QCD collaboration, is working on understanding quark confinement using ideas at the intersection of string theory and mathematics. Using simplified models and mathematical symmetries, Klebanov aims to explain why a single quark can never be found in isolation. This work led Klebanov, along with three other scientists in the QCD collaboration, to a serendipitous finding that a special kind of mathematical symmetry can be applied in a simplified version of Yang–Mills theory. The findings are helping theoretical physicists perform numerical calculations with greater accuracy than before.

“Cutting-edge developments in mathematics are essential for getting a better grip on confining strings and string theory,” says Ross Dempsey, a postdoctoral fellow at the Massachusetts Institute of Technology who is affiliated with the QCD collaboration and who worked on the finding with Klebanov and fellow collaboration members Silviu Pufu and Bernardo Zan. “But you also see it going the other way, where advances in string theory inspire new conjectures and new directions in mathematics.”

For example, string theory is helping expand the study of multidimensional geometry. Unlike the world we live in, with its three dimensions plus the dimension of time, string theory operates in 10 dimensions. Findings from string theory have led to new discoveries about complex geometries that operate in high dimensions. That means that even if string theory is someday ruled out as a description of physical reality, it will still have played a key role in advancing mathematics.

In another area of mathematics, a connection to string theory led to the proof of the monstrous moonshine theory (its actual name), which involved connecting wildly different areas of mathematics to understand a high-dimensional mathematical structure. The mathematician Richard Borcherds was inspired by the way string theory uses dimensions rolled up on strings into tiny spheres and doughnuts. Borcherds’ proof in the 1990s has served as a steppingstone for other researchers who have since uncovered further connections between string theory and this branch of mathematics.

“There’s enormous mutual influence between string theory and mathematics,” Dempsey says. “Any individual example would be shocking and amazing, but then there are so many of these examples that they become commonplace.”

Furthering the study of the mathematical components underlying many aspects of string theory is the Simons Collaboration on New Structures in Low-Dimensional Topology. Low-dimensional topology is the study of three- and four-dimensional manifolds, which are mathematical spaces or surfaces that appear flat on small scales but reveal complex structures, such as knots and chains, when viewed from a distance. These types of spaces are important to fields of physics including relativity, quantum mechanics and string theory. With a diverse group of researchers, the New Structures collaboration aims to bring together new perspectives that can help overcome the barriers that have long thwarted progress in the field.

“Our collaboration could not exist without mathematicians and physicists coming together,” says Gukov, a co-director of the New Structures collaboration. “I hope this collaboration will allow us to solve long-standing questions, some that have stood the test of time for many decades.”

Besides furthering discoveries at the interface of mathematics and string theory, the collaboration’s work is also advancing other areas of physics. The mathematics of manifolds, a type of modern algebra, is also relevant to the quantum realm. Last August, the collaboration discovered that a new class of mathematical theories could be used to make notoriously fragile quantum computers more stable.

“The subjects of mathematics and theoretical physics are getting more interwoven in a meaningful way,” Gukov says. “As this connection becomes more meaningful, the bridges between the subjects become more solid and interesting, and it helps bring more people from one community to the other. It’s a scenario where one plus one is actually bigger than two.”