Higher differential geometry and non-integral Kac-Moody extensions
Date
2024
Authors
McCusker, James Anthony
Editors
Advisors
Stevenson, Daniel
Vozzo, Raymond
Vozzo, Raymond
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Thesis
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Abstract
This thesis reviews the theory of higher differential geometry, in particular that of bundle gerbes and principal 2-bundles, providing the necessary background for groupoids, 2- groups, crossed-modules, and principal G-bundles. Along the way, we give a novel result generalising an equivalence between connections on principal bundles and splittings of a vector bundle sequence to the bundle gerbe setting, as well as filling a gap in the literature with a discussion of adjoint representations for Lie 2-groups. The main result of our work, however, is the construction of a strict symmetric Lie 2-group extension of the loop group ΩG for a compact, simple and simply connected Lie group G for all real levels k, generalising the well-known integral level Kac-Moody extension. We further show that this construction gives rise to a generalisation of the basic bundle gerbe and admits a principal 2-bundle connection and curvature which, in the integer level case, reduces to the usual connection and curvature. These results enrich the area of higher differential geometry by providing explicit calculations and constructions, examples of which are sparse in the existing literature, and by generalising a familiar and important class of examples.
School/Discipline
School of Mathematical and Computer Sciences
Dissertation Note
Thesis (MPhil) -- University of Adelaide, School of Mathematical and Computer Sciences, 2024
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