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The concept of generalised (in the sense of Colombeau) connection on a principal fibre bundle is introduced. This definition is then used to extend results concerning the geometry of principal fibre bundles to those that only have a generalised connection. Some applications to singular solutions of Yang-Mills theory are given.
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This paper lays the foundations for a nonlinear theory of differential geometry that is developed in a subsequent paper which is based on Colombeau algebras of tensor distributions on manifolds. We adopt a new approach and construct a global theory of algebras of generalised functions on manifolds based on the concept of smoothing operators. This produces a generalisation of previous theories in a form which is suitable for applications to differential geometry. The generalised Lie derivative is introduced and shown to commute with the embedding of distributions. It is also shown that the covariant derivative of a generalised scalar field commutes with this embedding at the level of association.
We introduce a notion of measuring scales for quantum abelian gauge systems. At each measuring scale a finite dimensional affine space stores information about the evaluation of the curvature on a discrete family of surfaces. Affine maps from the spaces assigned to finer scales to those assigned to coarser scales play the role of coarse graining maps. This structure induces a continuum limit space which contains information regarding curvature evaluation on all piecewise linear surfaces with boundary. The evaluation of holonomies along loops is also encoded in the spaces introduced here; thus, our framework is closely related to loop quantization and it allows us to discuss effective theories in a sensible way. We develop basic elements of measure theory on the introduced spaces which are essential for the applicability of the framework to the construction of quantum abelian gauge theories.
For any bounded domain $Omega$ in $mathbb C^m,$ let ${mathrm B}_1(Omega)$ denote the Cowen-Douglas class of commuting $m$-tuples of bounded linear operators. For an $m$-tuple $boldsymbol T$ in the Cowen-Douglas class ${mathrm B}_1(Omega),$ let $N_{boldsymbol T}(w)$ denote the restriction of $boldsymbol T$ to the subspace ${cap_{i,j=1}^mker(T_i-w_iI)(T_j-w_jI)}.$ This commuting $m$-tuple $N_{boldsymbol T}(w)$ of $m+1$ dimensional operators induces a homomorphism $rho_{_{!N_{boldsymbol T}(w)}}$ of the polynomial ring $P[z_1, ..., z_m],$ namely, $rho_{_{!N_{boldsymbol T}(w)}}(p) = pbig (N_{boldsymbol T}(w) big),, pin P[z_1, ..., z_m].$ We study the contractivity and complete contractivity of the homomorphism $rho_{_{!N_{boldsymbol T}(w)}}.$ Starting from the homomorphism $rho_{_{!N_{boldsymbol T}(w)}},$ we construct a natural class of homomorphism $rho_{_{!N^{(lambda)}(w)}}, lambda>0,$ and relate the properties of $rho_{_{!N^{(lambda)}(w)}}$ to that of $rho_{_{!N_{boldsymbol T}(w)}}.$ Explicit examples arising from the multiplication operators on the Bergman space of $Omega$ are investigated in detail. Finally, it is shown that contractive properties of $rho_{_{!N_{boldsymbol T}(w)}}$ is equivalent to an inequality for the curvature of the Cowen-Douglas bundle $E_{boldsymbol T}$.
What remains of a geometrical notion like that of a principal bundle when the base space is not a manifold but a coarse graining of it, like the poset formed by a base for the topology ordered under inclusion? Motivated by finding a geometrical framework for developing gauge theories in algebraic quantum field theory, we give, in the present paper, a first answer to this question. The notions of transition function, connection form and curvature form find a nice description in terms of cohomology, in general non-Abelian, of a poset with values in a group $G$. Interpreting a 1--cocycle as a principal bundle, a connection turns out to be a 1--cochain associated in a suitable way with this 1--cocycle; the curvature of a connection turns out to be its 2--coboundary. We show the existence of nonflat connections, and relate flat connections to homomorphisms of the fundamental group of the poset into $G$. We discuss holonomy and prove an analogue of the Ambrose-Singer theorem.
We obtain Gabor frame characterisations of modulation spaces defined via a class of translation-modulation invariant Banach spaces of distributions that was recently introduced in $[10]$. We show that these spaces admit an atomic decomposition through Gabor expansions and that they are characterised by summability properties of their Gabor coefficients. Furthermore, we construct a large space of admissible windows. This generalises several fundamental results for the classical modulation spaces $ M^{p,q}_{w}$. Due to the absence of solidity assumptions on the Banach spaces defining these modulation spaces, the methods used for the spaces $M^{p,q}_{w}$ (or, more generally, in coorbit space theory) fail in our setting and we develop here a new approach based on the twisted convolution.
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