String Compactifications and Moonshine

String theory is a theory of quantum gravity that holds the promise of unifying the standard model of particle physics and general relativity. It requires the existence of extra spatial dimensions that need to be compactified on special spaces in order to be compatible with our four-dimensional world. We are in the process of uncovering very deep connections, commonly referred to as moonshine, that link the most widely used spaces in string compactifications, namely Calabi-Yau manifolds, to particular functions and also to so called sporadic groups. A few years ago it was shown that the four dimensional K3 manifold is related to a particular sporadic group. Recently with my collaborators I discovered that this implies that the numbers of certain two-dimensional surfaces inside a particular class of six dimensional Calabi-Yau manifolds are related to the same sporadic group. These discoveries seem to be the tip of the iceberg: We are in the process of understanding how certain eight dimensional spaces are also related to several other sporadic groups in a variety of ways. These breakthrough discoveries connect string theory and several different areas of mathematics: number theory, group theory and geometry. One unique goal of this project is to systematically work out further connections between string theory and pure mathematics to the benefit of both. One method that can lead to such new discoveries is the use of so called string dualities. These dualities mean that there are a variety of different string theories that, when compactified on different spaces, give rise to the same physics. If one can establish a connection between one of these string compactifications and a sporadic group, then such a connection immediately follows for the other dual compactifications as well. Any such newly discovered connection will very likely also have a big impact on other subfields of string theory, since they often involve Calabi-Yau manifolds that have been key components in all string compactifications for decades. One first example, relevant for string phenomenology, is that, as I have shown, certain semi-realistic descriptions of our world within string theory have particles whose interactions are controlled by a function that is closely connected to a sporadic group. The proposed highly innovated and interdisciplinary research project will search for new moonshine phenomena and study the implications of all different moonshine phenomena for other subfields of string theory.

String theory is a theory of quantum gravity that holds the promise of unifying the standard model of particle physics and general relativity. It requires the existence of extra spatial dimensions that need to be compactified on special spaces in order to be compatible with our four-dimensional world. We are in the process of uncovering very deep connections, commonly referred to as moonshine, that link the most widely used spaces in string compactifications, namely Calabi-Yau manifolds, to particular functions and also to so called sporadic groups. This research project, which was carried out at the Technical University (TU) Vienna, has studied moonshine phenomena for three years. In particular, we discovered fascinating new connections between the mathematical Mathieu groups and the above mentioned Calabi-Yau manifolds. These Calabi-Yau manifolds were extensively studied at the TU Vienna during the last decades, so that we could build on this knowledge and work with local researchers. To derive these new connections, we used so-called dualities in string theory. An interesting description of this research project appeared after an interview with the principal investigator in German in "Die Presse" in print as well as online. The article can be accessed via the following link https://www.diepresse.com/4885731/monstergruppen-berechnen-den-mondschein. In another interesting research project that was carried out during this grant, we discovered a mathematically very fascinating connection between black holes and special functions that play a role in moonshine phenomena. This connection helps us to better understand black holes and we hope, in the near future, to make further progress along this line. During this project we had joined publications with researchers at universities in Löwen, London and Dublin. We also organized a weeklong workshop at the Erwin-Schrodinger-Institute in Vienna. During this workshop 40 researchers from all over the world visited Vienna. The research results obtained during this stand-alone project will/did result in the awarding of three Phds. Furthermore, the postdoc that was working on this project as well as the principal investigator obtained professorship positions during this research project. This shows the high quality and relevance of this research project and proves that it was a great success.