Geometric Structures in Field and String Theory

Harvard University, Cambridge MA

Investigators

Abstract

This research is concerned with the study of geometric structures occurring in field and string theories using methods from mathematics as well as from physics. The intertwine of research in both fields has led to many surprising connections and new ideas. A rich source of insights is the study of the deformation of the geometrical structures associated to physical quantities. This has led to an understanding of physical dualities and has uncovered a rich variety of new mathematical structures. The change of certain characteristics of the underlying mathematical description known as wall crossing is a very powerful feature of many mathematical deformation problems. These questions are best studied using mirror symmetry which combines techniques from various fields of mathematics. The projects proposed here intend to derive new mathematical ideas, structures and tools associated to physical deformation problems shedding thus new light both on the physical theories as well as on the interconnections between different mathematical structures. A class of objects which is suitable for studying wall crossing phenomena are supersymmetric BPS (Bogomol'nyi-Prasad-Sommerfield) objects. These have played a prominent role in understanding field theories, dualities and black hole micro-state counting problems. They will be studied in the context of string theory compactifications where they are associated to a special set of objects, the D-branes, which are studied using tools from topological string theory and mirror symmetry. These allow one to systematically study the dependence of these on the moduli of the theory, leading to powerful equations and important insights. Understanding BPS spectra and their jumping is furthermore crucial to unravel their deep role in the description of physical theories. By connecting ideas from theoretical physics and very diverse areas of mathematics such as differential and algebraic geometry, representation theory and number theory, these projects are pushing the boundaries of knowledge in these fields. The PI is a pioneer and leader in the field of geometry and mathematical aspects of string theory. Broader Impacts: Lying at the intersection of cutting edge research in both physics and mathematics, the project seeks to enhance the interactions and communication between the two communities. An integral part is to train postdocs and graduate students to be comfortable using methods and ideas from various fields and to conduct advanced cross disciplinary research. The PI is also actively bringing fundamental science and research to the public through various talks, presentations and publications.