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Resurgence and Brane-quantization in Topological Quantum Field Theory

Carlsberg Foundation Reintegration Fellowships

What

This project is about topological quantum field theory (TQFT), which is an interdisciplinary field between mathematics and physics. More concretely, it is about the mathematical theories of the Reshetikhin-Turaev Topological Quantum Field Theory, Quantum-Chern-Simons theory, Symplectic and Algebraic Geometry and the new theories from physics called resurgence and brane-quantization. The aim of the project is to further the development of the theories of brane-quantization and resurgence, to synthesize them, and to use them in Topological Quantum Field Theory to prove new mathematical results.

Why

From a broad viewpoint the project addresses two foundational problems in physics. 1) The first is the use of Feynman-path-integrals in physics. This have been a great success, giving some of the most accurately verified predictions in all of science. However; there is no sound mathematical foundation for their use. The heuristics is understood though, and this is via so-called perturbation theory. Resurgence is a theory which applies to perturbation theory. 2) The second is the problem of quantization and concerns the passage from classical mechanicals to quantum theory. Brane-quantization is a new proposal for this. Many important mathematical results in geometry are expected to follow this project. Historically, mathematics have greatly benefitted from new ideas from physics.

How

My prior research is about the so-called Reshetikhin-Turaev Topological Quantum Field Theory (TQFT), which is a theory in which the quantization process as well as the Feynman-path-integrals are mathematically well-understood. The latter are known as Quantum Invariants. Part of my work have been to mathematically prove conjectures about these quantum invariants, which are motivated from the resurgence theory. Continuing this line of research will further our understanding of resurgence. Simultaneously, I will work on developing the mathematical framework for brane-quantization and employing it to TQFT. Central to the quantization process in TQFT is the study of moduli spaces over surfaces, which allow for the use of powerful techniques from algebraic and symplectic geometry.

SSR

Research in TQFT is potentially of immense importance to society. TQFT have already found applications in quantum computing, and possibly in the future resurgence will contribute to this, as resurgence gives computational recursive schemes. The long term societal benefits from a precise mathematical geometric understanding of quantum field theory and quantization is potentially of even bigger significance - and hopefully this project will contribute hereto.