4,352 research outputs found
Quantifier elimination in C*-algebras
The only C*-algebras that admit elimination of quantifiers in continuous
logic are , Cantor space and
. We also prove that the theory of C*-algebras does not have
model companion and show that the theory of is not
-axiomatizable for any .Comment: More improvements and bug fixes. To appear in IMR
General Properties of Noncommutative Field Theories
In this paper we study general properties of noncommutative field theories
obtained from the Seiberg-Witten limit of string theories in the presence of an
external B-field. We analyze the extension of the Wightman axioms to this
context and explore their consequences, in particular we present a proof of the
CPT theorem for theories with space-space noncommutativity. We analyze as well
questions associated to the spin-statistics connections, and show that
noncommutative N=4, U(1) gauge theory can be softly broken to N=0 satisfying
the axioms and providing an example where the Wilsonian low energy effective
action can be constructed without UV/IR problems, after a judicious choice of
soft breaking parameters is made. We also assess the phenomenological prospects
of such a theory, which are in fact rather negative.Comment: 39 pages. LaTeX. 4 figures. Typos corrected. Comments and references
added. To appear in Nuclear Physics
Supersymmetric QCD and noncommutative geometry
We derive supersymmetric quantum chromodynamics from a noncommutative
manifold, using the spectral action principle of Chamseddine and Connes. After
a review of the Einstein-Yang-Mills system in noncommutative geometry, we
establish in full detail that it possesses supersymmetry. This noncommutative
model is then extended to give a theory of quarks, squarks, gluons and gluinos
by constructing a suitable noncommutative spin manifold (i.e. a spectral
triple). The particles are found at their natural place in a spectral triple:
the quarks and gluinos as fermions in the Hilbert space, the gluons and squarks
as bosons as the inner fluctuations of a (generalized) Dirac operator by the
algebra of matrix-valued functions on a manifold. The spectral action principle
applied to this spectral triple gives the Lagrangian of supersymmetric QCD,
including soft supersymmetry breaking mass terms for the squarks. We find that
these results are in good agreement with the physics literature
Lie Groupoids and Lie algebroids in physics and noncommutative geometry
The aim of this review paper is to explain the relevance of Lie groupoids and
Lie algebroids to both physicists and noncommutative geometers. Groupoids
generalize groups, spaces, group actions, and equivalence relations. This last
aspect dominates in noncommutative geometry, where groupoids provide the basic
tool to desingularize pathological quotient spaces. In physics, however, the
main role of groupoids is to provide a unified description of internal and
external symmetries. What is shared by noncommutative geometry and physics is
the importance of Connes's idea of associating a C*-algebra C*(G) to a Lie
groupoid G: in noncommutative geometry C*(G) replaces a given singular quotient
space by an appropriate noncommutative space, whereas in physics it gives the
algebra of observables of a quantum system whose symmetries are encoded by G.
Moreover, Connes's map G -> C*(G) has a classical analogue G -> A*(G) in
symplectic geometry due to Weinstein, which defines the Poisson manifold of the
corresponding classical system as the dual of the so-called Lie algebroid A(G)
of the Lie groupoid G, an object generalizing both Lie algebras and tangent
bundles. This will also lead into symplectic groupoids and the conjectural
functoriality of quantization.Comment: 39 pages; to appear in special issue of J. Geom. Phy
Zero Modes and the Atiyah-Singer Index in Noncommutative Instantons
We study the bosonic and fermionic zero modes in noncommutative instanton
backgrounds based on the ADHM construction. In k instanton background in U(N)
gauge theory, we show how to explicitly construct 4Nk (2Nk) bosonic (fermionic)
zero modes in the adjoint representation and 2k (k) bosonic (fermionic) zero
modes in the fundamental representation from the ADHM construction. The number
of fermionic zero modes is also shown to be exactly equal to the Atiyah-Singer
index of the Dirac operator in the noncommutative instanton background. We
point out that (super)conformal zero modes in non-BPS instantons are affected
by the noncommutativity. The role of Lorentz symmetry breaking by the
noncommutativity is also briefly discussed to figure out the structure of U(1)
instantons.Comment: v3: 24 pages, Latex, corrected typos, references added, to appear in
Phys. Rev.
The Dual Gromov-Hausdorff Propinquity
Motivated by the quest for an analogue of the Gromov-Hausdorff distance in
noncommutative geometry which is well-behaved with respect to C*-algebraic
structures, we propose a complete metric on the class of Leibniz quantum
compact metric spaces, named the dual Gromov-Hausdorff propinquity. This metric
resolves several important issues raised by recent research in noncommutative
metric geometry: it makes *-isomorphism a necessary condition for distance
zero, it is well-adapted to Leibniz seminorms, and --- very importantly --- is
complete, unlike the quantum propinquity which we introduced earlier. Thus our
new metric provides a natural tool for noncommutative metric geometry, designed
to allow for the generalizations of techniques from metric geometry to
C*-algebra theory.Comment: 42 pages in elsarticle 3p format. This third version has many small
typos corrections and small clarifications included. Intended form for
publicatio
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