On the Semi-Classical Vacuum Structure of the Electroweak Interaction

It is shown that in the semi-classical approximation of the electroweak sector of the Standard Model the moduli space of vacua can be identified with the first de Rham cohomology group of space-time. This gives a slightly different physical interpretation of the occurrence of the well-known Ahoronov-Bohm effect. Moreover, when charge conjugation is taken into account, the existence of a non-trivial ground state of the Higgs boson is shown to be equivalent to the triviality of the electroweak gauge bundle. As a consequence, the gauge bundle of the electromagnetic interaction must also be trivial. Though derived at ``tree level'' the results presented here may also have some consequences for quantizing, e. g., electromagnetism on an arbitrary curved space-time.Comment: 26 pages, no figure

(Fermionic)Mass Meets (Intrinsic)Curvature

Using the notion of vacuum pairs we show how the (square of the) mass matrix of the fermions can be considered geometrically as curvature. This curvature together with the curvature of space-time, defines the total curvature of the Clifford module bundle representing a ``free'' fermion within the geometrical setup of spontaneously broken Yang-Mills-Higgs gauge theories. The geometrical frame discussed here gives rise to a natural class of Lagrangian densities. It is shown that the geometry of the Clifford module bundle representing a free fermion is described by a canonical spectral invariant Lagrangian density.Comment: 14 page

The functional of super Riemann surfaces -- a "semi-classical" survey

This article provides a brief discussion of the functional of super Riemann surfaces from the point of view of classical (i.e. not "super-) differential geometry. The discussion is based on symmetry considerations and aims to clarify the "borderline" between classical and super differential geometry with respect to the distinguished functional that generalizes the action of harmonic maps and is expected to play a basic role in the discussion of "super Teichm\"uller space". The discussion is also motivated by the fact that a geometrical understanding of the functional of super Riemann surfaces from the point of view of super geometry seems to provide serious issues to treat the functional analytically

On the Determinant of One-Dimensional Elliptic Boundary Value Problems

We discuss the $\zeta-$regularized determinant of elliptic boundary value problems on a line segment. Our framework is applicable for separated and non-separated boundary conditions.Comment: LaTeX, 18 page

Applications of hypercomplex automorphic forms in Yang-Mills gauge theories

In this paper we show how hypercomplex function theoretical objects can be used to construct explicitly self-dual SU(2)-Yang-Mills instanton solutions on certain classes of conformally flat 4-manifolds. We use a hypercomplex argument principle to establish a natural link between the fundamental solutions of $D \Delta f = 0$ and the second Chern class of the SU(2) principal bundles over these manifolds. The considered base manifolds of the bundles are not simply-connected, in general. Actually, this paper summarizes an extension of the corresponding results of G\"ursey and Tze on a hyper-complex analytical description of SU(2) instantons. Furthermore, it provides an application of the recently introduced new classes of hypercomplex-analytic automorphic forms

Symmetries and conservation laws of a nonlinear sigma model with gravitino

We show that the action functional of the nonlinear sigma model with gravitino considered in a previous article [18] is invariant under rescaled conformal transformations, super Weyl transformations and diffeomorphisms. We give a careful geometric explanation how a variation of the metric leads to the corresponding variation of the spinors. In particular cases and despite using only commutative variables, the functional possesses a degenerate super symmetry. The corresponding conservation laws lead to a geometric interpretation of the energy-momentum tensor and supercurrent as holomorphic sections of appropriate bundles.Comment: 27 page

Regularity of Solutions of the Nonlinear Sigma Model with Gravitino

We propose a geometric setup to study analytic aspects of a variant of the super symmetric two-dimensional nonlinear sigma model. This functional extends the functional of Dirac-harmonic maps by gravitino fields. The system of Euler--Lagrange equations of the two-dimensional nonlinear sigma model with gravitino is calculated explicitly. The gravitino terms pose additional analytic difficulties to show smoothness of its weak solutions which are overcome using Rivi\`ere's regularity theory and Riesz potential theory.Comment: 24 pages. This is a revised version, with some typos corrected. To appear in Commun. Math. Phy

(Bosonic)Mass Meets (Extrinsic)Curvature

In this paper we discuss the mechanism of spontaneous symmetry breaking from the point view of vacuum pairs, considered as ground states of a Yang-Mills-Higgs gauge theory. We treat a vacuum as a section in an appropriate bundle that is naturally associated with a minimum of a (general) Higgs potential. Such a vacuum spontaneously breaks the underlying gauge symmetry if the invariance group of the vacuum is a proper subgroup of the gauge group. We show that each choice of a vacuum admits to geometrically interpret the bosonic mass matrices as ``normal'' sections. The spectrum of these sections turns out to be constant over the manifold and independent of the chosen vacuum. Since the mass matrices commute with the invariance group of the chosen vacuum one may decompose the Hermitian vector bundles which correspond to the bosons in the eigenbundles of the bosonic mass matrices. This decomposition is the geometrical analogue of the physical notion of a ``particle multplet''. In this sense the basic notion of a ``free particle'' also makes sense within the geometrical context of a gauge theory, provided the gauge symmetry is spontaneously broken by some vacuum. We also discuss the Higgs-Kibble mechanism (``Higgs Dinner'') from a geometrical point of view. It turns out that the ``unitary gauge'', usually encountered in the context of discussing the Higgs Dinner, is of purely geometrical origin. In particular, we discuss rotationally symmetric Higgs potentials and give a necessary and sufficient condition for the unitary gauge to exist. As a specific example we discuss in some detail the electroweak sector of the standard model of particle physics in this context.Comment: 26 page

Perturbative Description of the Fermionic Projector: Normalization, Causality and Furry's Theorem

The causal perturbation expansion of the fermionic projector is performed with a contour integral method. Different normalization conditions are analyzed. It is shown that the corresponding light-cone expansions are causal in the sense that they only involve bounded line integrals. For the resulting loop diagrams we prove a generalized Furry theorem.Comment: 34 pages, LaTeX, 2 ancillary files, minor improvements (published version