String theory is a ten-dimensional theory that aims to combine the known laws of nature into a mathematically consistent framework. Our current understanding is that string theory describes interactions between multi-dimensional objects called membranes. The ultimate motivation for my project is to understand the theories that describe these membranes; this will help clarify the dynamics of string theory, and hopefully in the long run help shed light on whether string theory correctly describes the universe we live in.
One of the most powerful concepts in theoretical physics is that of symmetry. Thanks to symmetry we can explain virtually all phenomena observed at particle accelerators in terms of a handful of mathematical equations. Even mundane phenomena like water boiling are governed by an important symmetry: conformal symmetry. The same conformal symmetry also governs the behavior of membranes in string theory. Additionally, the theories of membranes also display supersymmetry, a symmetry which relates to each other the two fundamental types of particles found in nature: bosons and fermions. Theories that enjoy both conformal symmetry and supersymmetry are known as superconformal field theories, or SCFTs.
Because of the large amount of symmetry, SCFTs are described by very rigid mathematical structures. A consequence of this rigid structure is that SCFTs can only exist in six dimensions or less. Six-dimensional (6d) SCFTs are in some ways the most fundamental: for example, starting from a 6d SCFT one can obtain infinitely many superconformal theories in lower dimensions. It is even possible that all lower-dimensional theories can be obtained starting from 6d SCFTs.
At the same time, 6d SCFTs display unusual features. For example, a key role in their dynamics is played by two-dimensional, string-like objects. Remarkably, from the properties of these strings (in particular from a function known as the ‘elliptic genus’ of the strings) one can reconstruct very detailed information about the 6d theory, which is encoded in a function called the ‘6d superconformal index’. These strings have previously not been well understood, and are the subject of this research project.
The first objective of the project has been to develop a systematic understanding of the strings in arbitrary 6d SCFTs, and devise approaches to compute their elliptic genus from first principles.
The second objective has been to apply these results to a different field in string theory: topological string theory. Topological string theory is important from various perspectives: first, it is an area of string theory where very detailed computations can be made, which enable us to tackle fundamental problems in string theory. For example, it has been used to understand the internal structure of (supersymmetric) black holes, which is one of the most mysterious aspects of quantum gravity. Second, topological string theory has very deep connections to a number of active areas of research in mathematics, including geometry of so-called Calabi-Yau spaces, and knot theory; for this reason, progress in topological string theory often also leads to advances in the realm of pure mathematics.
