3,843 research outputs found
A functional-analytic theory of vertex (operator) algebras, I
This paper is the first in a series of papers developing a
functional-analytic theory of vertex (operator) algebras and their
representations. For an arbitrary Z-graded finitely-generated vertex algebra
(V, Y, 1) satisfying the standard grading-restriction axioms, a locally convex
topological completion H of V is constructed. By the geometric interpretation
of vertex (operator) algebras, there is a canonical linear map from the tensor
product of V and V to the algebraic completion of V realizing linearly the
conformal equivalence class of a genus-zero Riemann surface with analytically
parametrized boundary obtained by deleting two ordered disjoint disks from the
unit disk and by giving the obvious parametrizations to the boundary
components. We extend such a linear map to a linear map from the completed
tensor product of H and H to H, and prove the continuity of the extension. For
any finitely-generated C-graded V-module (W, Y_W) satisfying the standard
grading-restriction axioms, the same method also gives a topological completion
H^W of W and gives the continuous extensions from the completed tensor product
of H and H^W to H^W of the linear maps from the tensor product of V and W to
the algenbraic completion of W realizing linearly the above conformal
equivalence classes of the genus-zero Riemann surfaces with analytically
parametrized boundaries.Comment: LaTeX file. 31 pages, 1 figur
Topology on cohomology of local fields
Arithmetic duality theorems over a local field are delicate to prove if
. In this case, the proofs often exploit topologies
carried by the cohomology groups for commutative finite type
-group schemes . These "\v{C}ech topologies", defined using \v{C}ech
cohomology, are impractical due to the lack of proofs of their basic
properties, such as continuity of connecting maps in long exact sequences. We
propose another way to topologize : in the key case ,
identify with the set of isomorphism classes of objects of the
groupoid of -points of the classifying stack and invoke
Moret-Bailly's general method of topologizing -points of locally of finite
type -algebraic stacks. Geometric arguments prove that these "classifying
stack topologies" enjoy the properties expected from the \v{C}ech topologies.
With this as the key input, we prove that the \v{C}ech and the classifying
stack topologies actually agree. The expected properties of the \v{C}ech
topologies follow, which streamlines a number of arithmetic duality proofs
given elsewhere.Comment: 36 pages; final version, to appear in Forum of Mathematics, Sigm
Finitary Topos for Locally Finite, Causal and Quantal Vacuum Einstein Gravity
Previous work on applications of Abstract Differential Geometry (ADG) to
discrete Lorentzian quantum gravity is brought to its categorical climax by
organizing the curved finitary spacetime sheaves of quantum causal sets
involved therein, on which a finitary (:locally finite), singularity-free,
background manifold independent and geometrically prequantized version of the
gravitational vacuum Einstein field equations were seen to hold, into a topos
structure. This topos is seen to be a finitary instance of both an elementary
and a Grothendieck topos, generalizing in a differential geometric setting, as
befits ADG, Sorkin's finitary substitutes of continuous spacetime topologies.
The paper closes with a thorough discussion of four future routes we could take
in order to further develop our topos-theoretic perspective on ADG-gravity
along certain categorical trends in current quantum gravity research.Comment: 49 pages, latest updated version (errata corrected, references
polished) Submitted to the International Journal of Theoretical Physic
Twisted convolution and Moyal star product of generalized functions
We consider nuclear function spaces on which the Weyl-Heisenberg group acts
continuously and study the basic properties of the twisted convolution product
of the functions with the dual space elements. The final theorem characterizes
the corresponding algebra of convolution multipliers and shows that it contains
all sufficiently rapidly decreasing functionals in the dual space.
Consequently, we obtain a general description of the Moyal multiplier algebra
of the Fourier-transformed space. The results extend the Weyl symbol calculus
beyond the traditional framework of tempered distributions.Comment: LaTeX, 16 pages, no figure
Extraordinary dimension theories generated by complexes
We study the extraordinary dimension function dim_{L} introduced by
\v{S}\v{c}epin. An axiomatic characterization of this dimension function is
obtained. We also introduce inductive dimensions ind_{L} and Ind_{L} and prove
that for separable metrizable spaces all three coincide. Several results such
as characterization of dim_{L} in terms of partitions and in terms of mappings
into -dimensional cubes are presented. We also prove the converse of the
Dranishnikov-Uspenskij theorem on dimension-raising maps
Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds
We present a perturbative construction of interacting quantum field theories
on any smooth globally hyperbolic manifold. We develop a purely local version
of the Stueckelberg-Bogoliubov-Epstein-Glaser method of renormalization using
techniques from microlocal analysis. As byproducts, we describe a perturbative
construction of local algebras of observables, present a new definition of Wick
polynomials as operator-valued distributions on a natural domain, and we find a
general method for the extension of distributions which were defined on the
complement of some surfaces.Comment: 38 pages, LaTeX with AMSLaTeX style option, Micro.tex macrofil
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