Exotic Calabi-Yaus from Supersymmetric Gauge Theory

Calabi-Yau spaces play a crucial role in string theory and mathematics. One of the great questions in modern theoretical physics is whether there exists a theory which unifies all the fundamental forces of nature. Currently the only candidate for such a Theory of Everything is string theory. Gravity and the forces governing interactions of elementary particles can be treated within the same theoretical framework, thus making possible unification. Roughly speaking, closed strings are responsible for gravity, while open strings and their boundary conditions, called D-branes, account for particle interactions. String theory requires a ten-dimensional space-time, not the four dimensions we see around us. This issue can be resolved by string compactification which means that the extra six dimensions are curled up and so small that they have not been observed yet. Further conditions lead to the mathematical requirement that the extra dimensions are so-called Calabi-Yau spaces. The choice of Calabi-Yau has great influence on what we observe in our four-dimensional world. It determines the properties of the elementary particles as well as how they interact with other particles. Triggered by the insights from string theory, Calabi-Yaus have also become an active research area in mathematics. While many examples of Calabi-Yau spaces are known, most of them belong to a very specific class. Moreover, Calabi-Yaus come with free parameters called moduli. For explicit calculations, one typically has to restrict them to specific values where the theoretical tools at hand are applicable. Naturally, focusing on such specific examples and specific loci in the moduli space potentially leads to a huge bias concerning the perception of typical properties of Calabi-Yaus and string compactifications. This project to be carried out at the Mathematical Physics Group at the University of Vienna is concerned with studying new types of Calabi-Yaus, their moduli spaces and D-branes away from known territory. We will combine new methods from string theory, quantum field theory and mathematics to tackle this challenging problem. In particular we make use of the fact that Calabi-Yaus can be described in terms of certain supersymmetric theories. We will study the following questions in concrete examples: What are the properties of exotic Calabi-Yaus which do not belong the the class of standard examples? What can we learn about the "obscure" regions of the moduli space? What are the properties of D-branes of exotic CYs in different regions of the moduli space? We expect to uncover interesting physics and mathematics lurking in the dark corners of string compactifications. Our approach follows the ideas of "physical mathematics" which uses physics methods to find new mathematical structures, which in turn are relevant for physics applications.

The main goal of the project was to study extra dimensions in string theory, with particular focus on exotic configurations that are not sufficiently understood. String theory is a theory that unifies all known fundamental forces by replacing point particles with strings. Self-consistency of string theory leads to the necessity of introducing extra dimensions. String theory requires a ten-dimensional space-time, not the four we see around us. This issue can be resolved by string compactification, which means that the extra six dimensions are curled up and so small that they have not been observed yet. Further conditions lead to the mathematical requirement that the extra dimensions are Calabi-Yau spaces. The choice of Calabi-Yau has great influence on what we observe in our four-dimensional world. It determines the properties of the elementary particles and how they interact. Triggered by the insights from string theory, Calabi-Yaus have also become an active research area in mathematics. While many examples of Calabi-Yaus are known, most of them belong to a very specific class. Moreover, such spaces come with free parameters called moduli. For explicit calculations, one typically has to restrict them to specific values where the theoretical tools at hand are applicable. Naturally, focusing on such specific examples and specific loci in the moduli space potentially leads to a huge bias concerning the perception of typical properties of Calabi-Yaus and string compactifications. A goal of the project was to remove these constraints and to study more general configurations. The most important tool was a supersymmetric gauge theory in two dimensions, called gauged linear sigma model. It provides a framework to construct new Calabi-Yaus and to study the properties of the associated moduli spaces by standard methods. It also provides tools to compute physical quantities in string compactifications. These methods were successfully applied in the course of the project. One important result was the construction of new types of Calabi-Yaus beyond known classes. In particular, gauged linear sigma model techniques were used to construct Calabi-Yaus with specific mathematical properties that could not be realised previously. A further result was the discovery of certain universal mathematical structures in physical quantities arising in Calabi-Yau compactifications that are valid in all areas of the moduli space. Furthermore these techniques and results were used to confirm and extend the so-called swampland conjectures. These conjectures are concerned with the fundamental question of what are defining properties of consistent theories of quantum gravity, one of which is string theory. The results received considerable attention in the international research community and have been presented at several international research conferences. The outcomes were published in high-profile international research journals. The members of the research group will continue their research in this area.