A new puzzle piece for string theory research: Study proves 4-graviton scattering conjecture

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String theory aims to explain all fundamental forces and particles in the universe—essentially, how the world operates on the smallest scales. Though it has not yet been experimentally verified, work in string theory has already led to significant advancements in mathematics and theoretical physics.

Dr. Ksenia Fedosova, a researcher at the Mathematics Münster Cluster of Excellence at the University of Münster has, along with two co-authors, added a new piece to this puzzle: They have proven a conjecture related to so-called 4-graviton scattering, which physicists have proposed for certain equations. The results have been published in the Proceedings of the National Academy of Sciences.

Gravitons are hypothetical particles responsible for gravity. "The 4-graviton scattering can be thought of as two gravitons moving freely through space until they interact in a 'black box' and then emerge as two gravitons," explains Fedosova, providing the physical background for her work. "The goal is to determine the probability of what happens in this black box."

This scattering probability is described by a function that depends on information about all four gravitons involved. "While the exact form of this function is not known, we can approximate this scattering amplitude for specific types of interactions within the black box, as long as the energies involved in the process are relatively low."

To calculate this approximation, its dependency on another variable must also be considered, namely the so-called string coupling constant, which describes the strength of interactions between strings. "In our research setup, its domain of definition connects string theory and number theory," explains Fedosova.

The string coupling constant is represented by the shape of a torus or, topologically, a donut—which in this case is used to compactify invisible dimensions. For number theorists, the string coupling constant, or torus, is represented by a point on a well-known modular surface. The latter is a curved 2-dimensional surface with two conical and one cusp singularity used in mathematics and physics to analyze specific number patterns and geometric structures.

This is how functions defined on a modular surface arise in the context of string theory. Fedosova, Prof. Dr. Kim Klinger-Logan and Dr. Danylo Radchenko investigated these functions, which must satisfy certain partial differential equations, and found the correct homogeneous part of some functions that appear in 4-graviton scattering. The homogeneous part is frequently used in mathematics to understand the fundamental structure or behavior of a function.

"To simplify the process, we solved the partial differential equations on an 'unfolded' version of the modular surface and then investigated whether it was possible to 'fold' the solution back," says the mathematician. For this, Fedosova and her collaborators needed to evaluate infinite sums that involve the so-called divisor functions.

The first example of these sums was found by physicists, and based on numerical evaluations, it was conjectured that they vanish. The research team discovered further examples of such sums. "Interestingly, however, other sums did not necessarily vanish as physicists had expected. Our results suggest that there should be a better choice for a starting partial differential equation than the one currently considered by physicists."

More information: Ksenia Fedosova et al, Convolution identities for divisor sums and modular forms, Proceedings of the National Academy of Sciences (2024). DOI: 10.1073/pnas.2322320121

Journal information: Proceedings of the National Academy of Sciences

Provided by University of Münster