Higher Geometry and Field Theories
Final Report Abstract
The project goal was to develop the link between higher geometric structures and mathematical approaches to physical field theories. In particular, it is an important research problem to classify such field theories, i.e. fully describe all possible field theories that can exist. My proposal was that for an important class of field theories this classification can be achieved through so-called higher-dimensional parallel transports. Parallel transport is a method for comparing data attached to any two points in a given space along any smooth path connecting the two points. They are a fundamental tool in differential geometry, describing, for instance, trajectories of freely falling observers in general relativity. In modern mathematics and physics, the necessity has emerged to also transport data along higher-dimensional objects such as surfaces which interpolate between different paths. On the field theory side, the project focussed on functorial field theories (FFTs). Like in the better understood TQFTs, data is assigned to geometric shapes of increasing dimension in very specific ways, but in FFTs one keeps track of how the data changes under smooth variations of the geometric shapes. Further, the geometric shapes may be decorated with additional data (such as bundles), and FFTs keep track of smooth changes of these decorations. During this project, unfortunately a range of obstacles emerged. It became clear that mathematical background technology had to be developed before being able to tackle several of the main research goals. This concerned, for instance, technology to handle very explicitly data with infinite levels of coherence, as well as smooth families of geometric data with infinitesimally defined higher-dimensional parallel transports (so-called connections); this had been treated at various abstract levels in the literature, but it turned out that the existing methods did not provide sufficient computational power to carry out some of the key steps in this project. Developing this technology cost time, but was in the end successful and led to results which have already been useful beyond this project. External obstacles included another group publishing a preprint and announcing a second one right at the start of this project which claimed to subsume many of the proposed results. This had a major impact on the project. Unfortunately, it proved difficult to communicate with that group and in the end meant that I had to reposition myself in the research landscape to some extent. Nevertheless, I achieved several of the main objectives and made significant progress towards others during the funding period. I will publish further important results from the project in the near future.
Publications
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An ∞-categorical localisation functor for categories of simplicial diagrams
Bunk, S.
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Higher geometric structures on manifolds and the gauge theory of Deligne cohomology
Bunk, S. & Shahbazi, C. S.
