Positivity relations on a locale

This paper analyses the notion of a positivity relationof Formal Topology from the point of view of the theory of Locales. It is shown that a positivity relation on a locale corresponds to a suitable class of points of its lower powerlocale. In particular, closed subtopologies associated to the positivity relation correspond to overt (that is, with open domain) weakly closed sublocales. Finally, some connection is revealed between positivity relations and localic suplattices (these are algebras for the powerlocale monad)

Supersymmetric Canonical Commutation Relations

We present unitarily represented supersymmetric canonical commutation relations which are subsequently used to canonically quantize massive and massless chiral,antichiral and vector fields. The massless fields, especially the vector one, show new facets which do not appear in the non superymmetric case. Our tool is the supersymmetric positivity induced by the Hilbert-Krein structure of the superspace.Comment: 14 page

Quantum Spacetime: a Disambiguation

We review an approach to non-commutative geometry, where models are constructed by quantisation of the coordinates. In particular we focus on the full DFR model and its irreducible components; the (arbitrary) restriction to a particular irreducible component is often referred to as the "canonical quantum spacetime". The aim is to distinguish and compare the approaches under various points of view, including motivations, prescriptions for quantisation, the choice of mathematical objects and concepts, approaches to dynamics and to covariance.Comment: special issue of SIGMA "Noncommutative Spaces and Fields

Embedding locales and formal topologies into positive topologies

A positive topology is a set equipped with two particular relations between elements and subsets of that set: a convergent cover relation and a positivity relation. A set equipped with a convergent cover relation is a predicative counterpart of a locale, where the given set plays the role of a set of generators, typically a base, and the cover encodes the relations between generators. A positivity relation enriches the structure of a locale; among other things, it is a tool to study some particular subobjects, namely the overt weakly closed sublocales. We relate the category of locales to that of positive topologies and we show that the former is a re\ufb02ective subcategory of the latter. We then generalize such a result to the (opposite of the) category of suplattices, which we present by means of (not necessarily convergent) cover relations. Finally, we show that the category of positive topologies also generalizes that of formal topologies, that is, overt locales

Supersymmetric Distributions, Hilbert Spaces of Supersymmetric Functions and Quantum Fields

The recently investigated Hilbert-Krein and other positivity structures of the superspace are considered in the framework of superdistributions. These tools are applied to problems raised by the rigorous supersymmetric quantum field theory.Comment: 24 page

Fundamentals of Quantum Gravity

The outline of a recent approach to quantum gravity is presented. Novel ingredients include: (1) Affine kinematical variables; (2) Affine coherent states; (3) Projection operator approach toward quantum constraints; (4) Continuous-time regularized functional integral representation without/with constraints; and (5) Hard core picture of nonrenormalizability. The ``diagonal representation'' for operator representations, introduced by Sudarshan into quantum optics, arises naturally within this program.Comment: 15 pages, conference proceeding

Cohomological Donaldson-Thomas theory

This review gives an introduction to cohomological Donaldson-Thomas theory: the study of a cohomology theory on moduli spaces of sheaves on Calabi-Yau threefolds, and of complexes in 3-Calabi-Yau categories, categorifying their numerical DT invariant. Local and global aspects of the theory are both covered, including representations of quivers with potential. We will discuss the construction of the DT sheaf, a nontrivial topological coefficient system on such a moduli space, along with some cohomology computations. The Cohomological Hall Algebra, an algebra structure on cohomological DT spaces, will also be introduced. The review closes with some recent appearances, and extensions, of the cohomological DT story in the theory of knot invariants, of cluster algebras, and elsewhere.Comment: 33 pages, some references adde