In this thesis we study conformal invariance of two-dimensional physical systems, and its connections to the Virasoro algebra. We first discuss how symmetries relate to Lie algebras, and how the...Show moreIn this thesis we study conformal invariance of two-dimensional physical systems, and its connections to the Virasoro algebra. We first discuss how symmetries relate to Lie algebras, and how the Virasoro algebra corresponds to conformal symmetries. We then study the theory of the free open bosonic string. Starting with the Nambu-Goto action, we first solve the equations of motion of the classical string using the light-cone gauge. From there, we construct the quantum theory and show it is indeed a representation of the Virasoro algebra. We then look more directly into representations of the Virasoro algebra. We define lowest weight representations and Verma modules, and discuss reducibility of the latter.Show less
In this thesis, we will investigate the transformation of electromagnetic fields under conformal maps. When a conformal map is applied to such a field, the resulting field is again a valid...Show moreIn this thesis, we will investigate the transformation of electromagnetic fields under conformal maps. When a conformal map is applied to such a field, the resulting field is again a valid electromagnetic field. Even when the conformal map is complex, i.e. it mixes real and complex points of space, the resulting field is valid. To better understand complex conformal maps, we introduce Dirac spinors and Twistor space. Using these concepts, we find a nicer expression for a — possibly complex — conformal transformation. This could ease the calculation of the transformed electromagnetic field.Show less
In this thesis we will try to find explicit examples and characterisations of the quantum moduli algebras on ribbon graphs with one vertex. First, we will study the classical case of group gauge...Show moreIn this thesis we will try to find explicit examples and characterisations of the quantum moduli algebras on ribbon graphs with one vertex. First, we will study the classical case of group gauge theory, in which we identify the moduli algebra with the function algebra on the moduli space of flat connections. Secondly, the group gauge theory case will be extended to the group algebra case, for which we show that the quantum moduli algebra is isomorphic to the moduli algebra in the group gauge theory case. Thirdly, we will give a general construction of how to obtain quantum moduli algebras of semisimple finite-dimensional Hopf algebras, and we will identify this construction with the construction of the quantum moduli algebra in the group algebra case. Fourthly, we will be examining the situation in which our Hopf algebra is the Drinfel'd double of a group algebra. After giving some examples, we will show that the quantum moduli algebra in the case of the Drinfel'd double is isomorphic to the quantum moduli algebra in the group algebra case.Show less
In this thesis we present Dijkgraaf-Witten theory. We start by considering a two-dimensional topological quantum field theory that can be used to prove Mednykh's formula along the way. Subsequently...Show moreIn this thesis we present Dijkgraaf-Witten theory. We start by considering a two-dimensional topological quantum field theory that can be used to prove Mednykh's formula along the way. Subsequently, we define the Dijkgraaf-Witten invariant as a partition function where we assign a specified weight to the 3-simplices of a compact, oriented and triangulated 3-manifold with boundary. The partition function depends on how we assign elements of a finite, discrete group G to all the oriented edges of the manifold. We prove that, whenever the triangulation of the boundary is fixed, the invariant does not depend on the triangulation of the manifold. Finally, we define a similar invariant where we model the weight of the 3-simplices to mimic the action of Chern-Simons theory. We demonstrate that by demanding invariance, we obtain the Dijkgraaf-Witten invariant.Show less
The main subject of this thesis is a reformulation of Einstein’s equation. In this reformulation, the variable is not a metric, but a connection on a vector bundle. Nevertheless, we can associate a...Show moreThe main subject of this thesis is a reformulation of Einstein’s equation. In this reformulation, the variable is not a metric, but a connection on a vector bundle. Nevertheless, we can associate a Riemannian metric to a connection. This allows us to relate the new formulation to the usual formulation, i.e. this allows us to argue that the new formulation is in fact a reformulation of Einstein’s equation. Since the physically significant metrics are of Lorentzian signature, we also consider modifying the new formulation in an attempt to make it suitable for Lorentzian metrics.Show less
In this thesis, we introduce the language of smooth manifolds, which is the natural setting for general relativity, and show how the restricted Lorentz group is related to the complex special...Show moreIn this thesis, we introduce the language of smooth manifolds, which is the natural setting for general relativity, and show how the restricted Lorentz group is related to the complex special linear group in two dimensions, and argue how this relation shows that spinors come up naturally in general relativity. We then consider fibre bundles and how they come up in general relativity, and how they are necessary to define what a spin structure is, and examine under which assumptions it exists. We conclude with a proper definition of Einstein’s field equation.Show less
