Principal Investigator: Katrin A. Wendland
It has become a successful practice in mathematics to take advantage of lessons learnt from physics. In this proposal, three such lessons are singled out, which should be of interest in their own right but also contribute to the proposer's long term research aims, a complete understanding of conformal field theories (CFTs) on K3 surfaces and beyond. The first part of the project is devoted to classically unknown properties of elliptically fibered Calabi-Yau 3-folds, arising from the conjectured heterotic-type IIA string-string duality. We will combine an algebraic geometric and an index theoretic approach, which is novel. A new index theorem and novel insights into the mathematics of the duality are aimed for. A recent conjecture of John McKay is also tested, relating Dynkin data to sporadic groups. The second part of the project deals with so-called TERP structures, which underlie the geometry of moduli spaces of supersymmetric quantum field theories. Here, D-branes and their stability conditions shall be studied using techniques from geometric analysis and integrable systems rather than the standard approaches. CFTs on K3 surfaces are studied in the third part of the project, where for a powerful new family of examples various techniques for the study of boundary CFTs and vertex operator algebras shall be tested. A generalization of previously obtained results about CFTs on K3 surfaces to certain Borcea-Voisin 3-folds is also planned. This research will underpin and generalize the PI's previous results, confirming and extending an applicable dictionary between geometry and CFT.
String theory is as yet the most promising candidate for a physical model of the real world. It describes particles and forces of nature using one-dimensional extended objects, the strings, whose behavior naturally depends on the ambient geometry. Vice versa, one can view these strings as testing the ambient geometry, whose properties can be read off from the behavior of the strings. This bi-directionality is crucial, since string theory on the other hand uses the language of quantum field theory which in many ways is richer than classical geometry. Hence string theory provides a natural bridge between two fundamental, seemingly unrelated concepts: quantum field theory and geometry. The proposed research aims to make use of this bridge. In particular, it is proposed to use classically unknown predictions coming from string theory to gain a deeper understanding of various geometric structures, including particular Calabi-Yau 3-folds and so-called tt* geometry in general.
