Physical Mathematics: String Theory, Quantization and Geometry

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Physics has influenced the development of mathematics in many different ways. It is well known that many areas of mathematics have been developed to provide a language to formulate physical theories. But it is perhaps not as well known that the intricate mathematical consistency required of physical theories often uncover new, unexpected structures in mathematics. In particular, dualities in quantum field theory and string theory often give rise to deep, fascinating connections between areas of mathematics that are a priori unrelated. My research program focuses on this intriguing interaction between mathematics and physics, sometimes known as "physical mathematics".******One particular example of such interaction is the so-called "topological recursion", which originated as a solution to the calculation of physical observables in some particular quantum field theory. Because of dualities in string theory and quantum field theory, it is now clear that this recursive structure is a unifying theme in many areas of geometry. The underlying mathematical reason for the ubiquity of this structure appears to lie in the process of quantization. In fact, this recursive structure appears to be connected to quantization in two different ways: through so-called "quantum Airy structures", and via "quantum curves". One of the goals of my research program is to shed light on the connections between these two quantization processes, the topological recursion, and its various applications in geometry, knot theory, and the theory of modular forms. In particular, I propose a novel generalization of this quantum Airy structures, which implies fascinating new connections between algebra, geometry and physics, and opens up many new research questions. I also propose to study the question of whether it is possible impose an extra symmetry, known as supersymmetry, on these quantization processes. What would then be the geometric meaning of the objects calculated by such a supersymmetric topological recursion?******In physics it is often the case that observables of a given theory have strong invariance properties. For instance, they should not depend on a choice of coordinate system used to describe a physical phenomenon. Those invariance properties generally follow from physical consistency, but are often far from obvious mathematically. Another aspect of my research program consists in studying one such invariance requirement that arose from our study of particular D-brane states in string theory. Invariance of these states suggests a new construction in the theory of modular forms, which is quite general and elegant. I intend to complete this construction and study its properties and consequences, both mathematically and physically.**