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Dr. Ksenia Fedosova from the Cluster of Excellence Mathematics Münster, along with an international research team, has proven a conjecture in string theory that physicists had proposed regarding certain equations. Their findings were published in the journal Proceedings of the National Academy of Sciences (PNAS).
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 proposed for certain equations. The results have been published in the journal "Proceedings of the National Academy of Sciences" (PNAS).
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 Ksenia 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 Ksenia Fedosova. The string coupling constant is represented by a 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 analyse specific number patterns and geometric structures.
This is how functions defined on a modular surface arise in the context of string theory. Ksenia 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," the mathematician explains their approach. For this, Ksenia 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."
Dr. Ksenia Fedosova
Mathematisches Institut | University of Münster
E-Mail: kfedosov@uni-muenster.de
Ksenia Fedosova, Kim Klinger-Logan, Danylo Radchenko (2024): Convolution identities for divisor sums and modular forms. Proceedings of the National Academy of Sciences (PNAS), Vol. 121, No. 44, DOI: https://doi.org/10.1073/pnas.2322320121
Dr. Ksenia Fedosova presents an example of the equations from the work published in PNAS.
Victoria Liesche
Uni MS - Victoria Liesche
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