On the Geometry of Infinite-Dimensional Grassmannian Manifolds and Gauge Theory

This PhD Thesis is about certain infinite-dimensional Grassmannian manifolds that arise naturally in geometry, representation theory and mathematical physics. From the physics point of view one encounters these infinite-dimensional manifolds when trying to understand the second quantization of fermions. The many particle Hilbert space of the second quantized fermions is called the fermionic Fock space. A typical element of the fermionic Fock space can be thought to be a linear combination of the configurations m particles and n anti-particles . Geometrically the fermionic Fock space can be constructed as holomorphic sections of a certain (dual)determinant line bundle lying over the so called restricted Grassmannian manifold, which is a typical example of an infinite-dimensional Grassmannian manifold one encounters in QFT. The construction should be compared with its well-known finite-dimensional analogue, where one realizes an exterior power of a finite-dimensional vector space as the space of holomorphic sections of a determinant line bundle lying over a finite-dimensional Grassmannian manifold. The connection with infinite-dimensional representation theory stems from the fact that the restricted Grassmannian manifold is an infinite-dimensional homogeneous (Kähler) manifold, i.e. it is of the form G/H where G is a certain infinite-dimensional Lie group and H its subgroup. A central extension of G acts on the total space of the dual determinant line bundle and also on the space its holomorphic sections; thus G admits a (projective) representation on the fermionic Fock space. This construction also induces the so called basic representation for loop groups (of compact groups), which in turn are vitally important in string theory / conformal field theory. The Thesis consists of three chapters: the first chapter is an introduction to the backround material and the other two chapters are individually written research articles. The first article deals in a new way with the well-known question in Yang-Mills theory, when can one lift the action of the gauge transformation group on the space of connection one forms to the total space of the Fock bundle in a compatible way with the second quantized Dirac operator. In general there is an obstruction to this (called the Mickelsson-Faddeev anomaly) and various geometric interpretations for this anomaly, using such things as group extensions and bundle gerbes, have been given earlier. In this work we give a new geometric interpretation for the Faddeev-Mickelsson anomaly in terms of differentiable gerbes (certain sheaves of categories) and central extensions of Lie groupoids. The second research article deals with the question how to define a Dirac-like operator on the restricted Grassmannian manifold, which is an infinite-dimensional space and hence not in the landscape of standard Dirac operator theory. The construction relies heavily on infinite-dimensional representation theory and one of the most technically demanding challenges is to be able to introduce proper normal orderings for certain infinite sums of operators in such a way that all divergences will disappear and the infinite sum will make sense as a well-defined operator acting on a suitable Hilbert space of spinors. This research article was motivated by a more extensive ongoing project to construct twisted K-theory classes in Yang-Mills theory via a Dirac-like operator on the restricted Grassmannian manifold.

Characterisation of properties of coniferous wood tracheids by x-ray diffraction, laser scattering and microscopy

In recent years there has been growing interest in selecting suitable wood raw material to increase end product quality and to increase the efficiency of industrial processes. Genetic background and growing conditions are known to affect properties of growing trees, but only a few parameters reflecting wood quality, such as volume and density can be measured on an industrial scale. Therefore research on cellular level structures of trees grown in different conditions is needed to increase understanding of the growth process of trees leading to desired wood properties. In this work the cellular and cell wall structures of wood were studied. Parameters, such as the mean microfibril angle (MFA), the spiral grain angles, the fibre length, the tracheid cell wall thickness and the cross-sectional shape of the tracheid, were determined as a function of distance from the pith towards the bark and mutual dependencies of these parameters were discussed. Samples from fast-grown trees, which belong to a same clone, grown in fertile soil and also from fertilised trees were measured. It was found that in fast-grown trees the mean MFA decreased more gradually from the pith to the bark than in reference stems. In fast-grown samples cells were shorter, more thin-walled and their cross-sections were rounder than in slower-grown reference trees. Increased growth rate was found to cause an increase in spiral grain variation both within and between annual rings. Furthermore, methods for determination of the mean MFA using x-ray diffraction were evaluated. Several experimental arrangements including the synchrotron radiation based microdiffraction were compared. For evaluation of the data analysis procedures a general form for diffraction conditions in terms of angles describing the fibre orientation and the shape of the cell was derived. The effects of these parameters on the obtained microfibril angles were discussed. The use of symmetrical transmission geometry and tangentially cut samples gave the most reliable MFA values.