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.

Folding and Selective Exit of Reporter Proteins from the Yeast Endoplasmic Reticulum

The present study analyses the traffic of Hsp150 fusion proteins through the endoplasmic reticulum (ER) of yeast cells, from their post-translational translocation and folding to their exit from the ER via a selective COPI-independent pathway. The reporter proteins used in the present work are: Hsp150p, an O-glycosylated natural secretory protein of Saccharomyces cerevisiae, as well as fusion proteins consisting of a fragment of Hsp150 that facilitates in the yeast ER proper folding of heterologous proteins fused to it. It is thought that newly synthesized polypeptides are kept in an unfolded form by cytosolic chaperones to facilitate the post-translational translocation across the ER membrane. However, beta-lactamase, fused to the Hsp150 fragment, folds in the cytosol into bioactive conformation. Irreversible binding of benzylpenicillin locked beta-lactamase into a globular conformation, and prevented the translocation of the fusion protein. This indicates that under normal conditions the beta-lactamase portion unfolds for translocation. Cytosolic machinery must be responsible for the unfolding. The unfolding is a prerequisite for translocation through the Sec61 channel into the lumen of the ER, where the polypeptide is again folded into a bioactive and secretion-competent conformation. Lhs1p is a member of the Hsp70 family, which functions in the conformational repair of misfolded proteins in the yeast ER. It contains Hsp70 motifs, thus it has been thought to be an ATPase, like other Hsp70 members. In order to understand its activity, authentic Lhs1p and its recombinant forms expressed in E. coli, were purified. However, no ATPase activity of Lhs1p could be detected. Nor could physical interaction between Lhs1p and activators of the ER Hsp70 chaperone Kar2p, such as the J-domain proteins Sec63p, Scj1p, and Jem1p and the nucleotide exchange factor Sil1p, be demonstrated. The domain structure of Lhs1p was modelled, and found to consist of an ATPase-like domain, a domain resembling the peptide-binding domain (PBD) of Hsp70 proteins, and a C-terminal extension. Crosslinking experiments showed that Lhs1p and Kar2p interact. The interacting domains were the C-terminal extension of Lhs1p and the ATPase domain of Kar2p, and this interaction was independent of ATPase activity of Kar2p. A model is presented where the C-terminal part of Lhs1p forms a Bag-like 3 helices bundle that might serve in the nucleotide exchange function for Kar2p in translocation and folding of secretory proteins in the ER. Exit of secretory proteins in COPII-coated vesicles is believed to be dependent of retrograde transport from the Golgi to the ER in COPI-coated vesicles. It is thought that receptors escaping to the Golgi must be recycled back to the ER exit sites to recruit cargo proteins. We found that Hsp150 leaves the ER even in the absence of functional COPI-traffic from the Golgi to the ER. Thus, an alternative, COPI-independent ER exit pathway must exists, and Hsp150 is recruited to this route. The region containing the signature guiding Hsp150 to this alternative pathway was mapped.

Effects of forestry extension on the use of allowable cut in non-industrial private forests.

XVIII IUFRO World Congress, Ljubljana 1986.