Doctor of Philosophy
van Suijlekom, Walter D.
We investigate the metric nature of spectral triples in two ways.
Given an oriented Riemannian embedding i:X->Y of codimension 1 we construct a family of unbounded KK-cycles i!(epsilon), each of which represents the shriek class of i in KK-theory. These unbounded KK-cycles are further equipped with connections, allowing for the explicit computation of the products of i! with the spectral triple of Y at the unbounded level. In the limit epsilon to 0 the product of these unbounded KK-cycles with the canonical spectral triple for Y admits an asymptotic expansion. The divergent part of this expansion is known and universal, the constant term in the expansion gives the canonical spectral triple for X. Furthermore, the curvature of these unbounded KK-cycles converges to the square of the mean curvature of X in Y as epsilon goes to 0.
We define a random matrix ensemble for the Dirac operator on the (0,1) fuzzy geometry incorporating both the geometric and fermionic aspects of the spectral action. This yields a unitarily invariant, single-matrix multi-trace model. We generalize Coulomb-gas techniques for finding the spectral density of single-trace models to multi-trace models and apply these to our model of a fermionic fuzzy geometry. The resulting Fredholm integral equation for the spectral density is analyzed numerically and the effect of various parameters on the spectral density is investigated.
Summary for Lay Audience
Descriptions of quantum mechanics often use matrices. These are square arrays of numbers that can be added, subtracted and multiplied similar to normal numbers with one major difference: if A and B are matrices A times B and B times A may give different results. In technical terms this is called noncommutativity.
On the other hand we have general relativity, in which space and time are described using geometry. The theory of noncommutative geometry describes geometry using matrices (or their infinite dimensional generalizations). This makes noncommutative geometry a natural language to try and unite the quantum mechanical with the world of general relativity. In this thesis we explore two aspects of noncommutative geometry.
In the first part of this thesis we contribute to one of the open questions in noncommutative geometry: How to describe maps between geometries? For these maps you can think of a map in an atlas, which describes one geometry (the world) as a part of another (the page). We construct a description in noncommutative geometry language of maps that come from hypersurfaces. An example of a hypersurface is the shell of a ball in 3D space.
The second part of the thesis concerns a separate project where we consider a specific class of noncommutative geometries, called the (0,1) fuzzy geometries. We are interested in this class of geometries since they provide a toy model of quantum gravity, where reality is a superposition of many different geometries of this (0,1) type.
Fuzzy geometries have many useful qualities for us. They have, for example, a minimum resolution built in. If you look very closely, details become fuzzy. Such a minimum scale, called the Planck length, is predicted by the combination of general relativity and quantum mechanics.
We focus on the fuzzy geometries of type (0,1) since for this type the situation further simplifies to allow us to find exact solutions. The main addition we make to this model of quantum gravity on fuzzy geometries is the inclusion of particles living on space time. We compute in particular how these particles affect the superposition of geometries.
Verhoeven, Luuk S., "Geometry in spectral triples: Immersions and fermionic fuzzy geometries" (2023). Electronic Thesis and Dissertation Repository. 9561.
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