105 research outputs found
Supersymmetry, homology with twisted coefficients and n-dimensional knots
Let be any natural number. Let be any -dimensional knot in
. We define a supersymmetric quantum system for with the following
properties. We firstly construct a set of functional spaces (spaces of
fermionic \{resp. bosonic\} states) and a set of operators (supersymmetric
infinitesimal transformations) in an explicit way. Thus we obtain a set of the
Witten indexes for . Our Witten indexes are topological invariants for
-dimensional knots. Our Witten indexes are not zero in general. If is
equivalent to the trivial knot, all of our Witten indexes are zero. Our Witten
indexes restrict the Alexander polynomials of -knots. If one of our Witten
indexes for an -knot is nonzero, then one of the Alexander polynomials
of is nontrivial. Our Witten indexes are connected with homology with
twisted coefficients. Roughly speaking, our Witten indexes have path integral
representation by using a usual manner of supersymmetric theory.Comment: 10pages, no figure
Dyson processes on the octonion algebra
We consider Brownian motion on symmetric matrices of octonions, and study the
law of the spectrum. Due to the fact that the octonion algebra is
nonassociative, the dimension of the matrices plays a special role. We provide
two specific models on octonions, which give some indication of the relation
between the multiplicity of eigenvalues and the exponent in the law of the
spectrum
Exotic Differentiable Structures and General Relativity
We review recent developments in differential topology with special concern
for their possible significance to physical theories, especially general
relativity. In particular we are concerned here with the discovery of the
existence of non-standard (``fake'' or ``exotic'') differentiable structures on
topologically simple manifolds such as , \R and
Because of the technical difficulties involved in the smooth case, we begin
with an easily understood toy example looking at the role which the choice of
complex structures plays in the formulation of two-dimensional vacuum
electrostatics. We then briefly review the mathematical formalisms involved
with differentiable structures on topological manifolds, diffeomorphisms and
their significance for physics. We summarize the important work of Milnor,
Freedman, Donaldson, and others in developing exotic differentiable structures
on well known topological manifolds. Finally, we discuss some of the geometric
implications of these results and propose some conjectures on possible physical
implications of these new manifolds which have never before been considered as
physical models.Comment: 11 pages, LaTe
The anomaly line bundle of the self-dual field theory
In this work, we determine explicitly the anomaly line bundle of the abelian
self-dual field theory over the space of metrics modulo diffeomorphisms,
including its torsion part. Inspired by the work of Belov and Moore, we propose
a non-covariant action principle for a pair of Euclidean self-dual fields on a
generic oriented Riemannian manifold. The corresponding path integral allows to
study the global properties of the partition function over the space of metrics
modulo diffeomorphisms. We show that the anomaly bundle for a pair of self-dual
fields differs from the determinant bundle of the Dirac operator coupled to
chiral spinors by a flat bundle that is not trivial if the underlying manifold
has middle-degree cohomology, and whose holonomies are determined explicitly.
We briefly sketch the relevance of this result for the computation of the
global gravitational anomaly of the self-dual field theory, that will appear in
another paper.Comment: 41 pages. v2: A few typos corrected. Version accepted for publication
in CM
The Seven-sphere and its Kac-Moody Algebra
We investigate the seven-sphere as a group-like manifold and its extension to
a Kac-Moody-like algebra. Covariance properties and tensorial composition of
spinors under are defined. The relation to Malcev algebras is
established. The consequences for octonionic projective spaces are examined.
Current algebras are formulated and their anomalies are derived, and shown to
be unique (even regarding numerical coefficients) up to redefinitions of the
currents. Nilpotency of the BRST operator is consistent with one particular
expression in the class of (field-dependent) anomalies. A Sugawara construction
is given.Comment: 22 pages. Macropackages used: phyzzx, epsf. Three epsf figure files
appende
Compactification, topology change and surgery theory
We study the process of compactification as a topology change. It is shown
how the mediating spacetime topology, or cobordism, may be simplified through
surgery. Within the causal Lorentzian approach to quantum gravity, it is shown
that any topology change in dimensions may be achieved via a causally
continuous cobordism. This extends the known result for 4 dimensions.
Therefore, there is no selection rule for compactification at the level of
causal continuity. Theorems from surgery theory and handle theory are seen to
be very relevant for understanding topology change in higher dimensions.
Compactification via parallelisable cobordisms is particularly amenable to
study with these tools.Comment: 1+19 pages. LaTeX. 9 associated eps files. Discussion of disconnected
case adde
Cosmological Creation of D-branes and anti-D-branes
We argue that the early universe may be described by an initial state of
space-filling branes and anti-branes. At high temperature this system is
stable. At low temperature tachyons appear and lead to a phase transition,
dynamics, and the creation of D-branes. These branes are cosmologically
produced in a generic fashion by the Kibble mechanism. From an entropic point
of view, the formation of lower dimensional branes is preferred and
brane-worlds are exponentially more likely to form than higher dimensional
branes. Virtually any brane configuration can be created from such phase
transitions by adjusting the tachyon profile. A lower bound on the number
defects produced is: one D-brane per Hubble volume.Comment: 30 pages, 5 eps figures; v2 more references added; v3 section 4
slightly improve
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