Spectral Geometry of Fuzzy SpacesTools Druce, Paul Joseph (2020) Spectral Geometry of Fuzzy Spaces. PhD thesis, University of Nottingham.
AbstractIn this thesis, a number of tools are developed to better understand fuzzy spaces and finite noncommutative geometries in general. These tools depend only on the spectrum of the Dirac operator. Dimensional measures based on Weyl's law and heat kernel asymptotics are defined. A new dimensional measured called the spectral variance is defined as a modification of the spectral dimension to remove some of its undesirable properties. Volume measures based upon the Dixmier trace and the work of Stern are adapted to the finite setting and tested on the fuzzy spaces. The distance between two geometries is investigated by comparing the spectral zeta functions using the method of Cornelissen and Kontogeorgis. All of these tools are then used to investigate the fuzzy sphere, the fuzzy tori and the random fuzzy spaces introduced by Barrett and Glaser. The role of symmetry in the creation of fuzzy spaces is investigated using the characterisation of the Dirac operator given by Barrett. It is shown that all SU(2)-equivariant Dirac operators for type (0,3) and (1,3) fuzzy spaces produce the round metric on the sphere, despite the Dirac operators not agreeing with Grosse-Presnadjer or Barrett operators. A pathway for further research is presented along these lines.
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