756 research outputs found
Branes, Quantization and Fuzzy Spheres
We propose generalized quantization axioms for Nambu-Poisson manifolds, which
allow for a geometric interpretation of n-Lie algebras and their enveloping
algebras. We illustrate these axioms by describing extensions of
Berezin-Toeplitz quantization to produce various examples of quantum spaces of
relevance to the dynamics of M-branes, such as fuzzy spheres in diverse
dimensions. We briefly describe preliminary steps towards making the notion of
quantized 2-plectic manifolds rigorous by extending the groupoid approach to
quantization of symplectic manifolds.Comment: 18 pages; Based on Review Talk at the Workshop on "Noncommutative
Field Theory and Gravity", Corfu Summer Institute on Elementary Particles and
Physics, September 8-12, 2010, Corfu, Greece; to be published in Proceedings
of Scienc
Lie 2-algebra models
In this paper, we begin the study of zero-dimensional field theories with
fields taking values in a semistrict Lie 2-algebra. These theories contain the
IKKT matrix model and various M-brane related models as special cases. They
feature solutions that can be interpreted as quantized 2-plectic manifolds. In
particular, we find solutions corresponding to quantizations of R^3, S^3 and a
five-dimensional Hpp-wave. Moreover, by expanding a certain class of Lie
2-algebra models around the solution corresponding to quantized R^3, we obtain
higher BF-theory on this quantized space.Comment: 47 pages, presentation improved, version published in JHE
The Universal Coefticient Theorem in the Category of Fuzzy Soft Modules
This paper begins with the basic concepts of chain comlexes of fuzzy soft modules. Later, we introduce short exact sequence of fuzzy soft modules and prove that split short exact sequence of fuzzy soft chain complex. Naturally, we want to investigate whether or not the universal coefficient theorems are satisfied in category of fuzzy soft chain complexes. However, in the proof of these theorems in the category of chain complexes, exact sequence of homology modules of chain complexes is used. Generally, sequence of fuzzy soft homology modules is not exact in fuzzy chain complexes. Therefore in this study, we construct exact sequence of fuzzy soft homology modules under some conditions. Universal coefficients theorem is proven by making use of this idea
Simplicial Multivalued Maps and the Witness Complex for Dynamical Analysis of Time Series
Topology based analysis of time-series data from dynamical systems is
powerful: it potentially allows for computer-based proofs of the existence of
various classes of regular and chaotic invariant sets for high-dimensional
dynamics. Standard methods are based on a cubical discretization of the
dynamics and use the time series to construct an outer approximation of the
underlying dynamical system. The resulting multivalued map can be used to
compute the Conley index of isolated invariant sets of cubes. In this paper we
introduce a discretization that uses instead a simplicial complex constructed
from a witness-landmark relationship. The goal is to obtain a natural
discretization that is more tightly connected with the invariant density of the
time series itself. The time-ordering of the data also directly leads to a map
on this simplicial complex that we call the witness map. We obtain conditions
under which this witness map gives an outer approximation of the dynamics, and
thus can be used to compute the Conley index of isolated invariant sets. The
method is illustrated by a simple example using data from the classical H\'enon
map.Comment: laTeX, 9 figures, 32 page
Inverse System in The Category of Intuitionistic Fuzzy Soft Modules
This paper begins with the basic concepts of soft module. Later, we introduce inverse system in the category of intutionistic fuzzy soft modules and prove that its limit exists in this category. Generally, limit of inverse system of exact sequences of intutionistic fuzzy soft modules is not exact. Then we define the notion which is first derived functor of the inverse limit functor. Finally, using methods of homology algebra, we prove that the inverse system limit of exact sequence of intutionistic fuzzy soft modules is exact
Fuzzy geometry
The concept of fuzzy space is due independently to
Poincaré and Zeeman. (Poincaré
used the term "physical continuum", Zeeman the term
"tolerance space". I have reluctantly introduced a
third expression since my attempts to generate a
vocabulary from either of these have all proved
impossibly unwieldy.) Both were led to it by the
nature of our perception of space, and both adapted to
it tools current in topology. Unfortunately, neither
examined the application of these tools in complete
detail, and as a result the argument from analogy
was somewhat over-extended by both. The resemblances
to topology are strong; the differences are sometimes
glaring and sometimes subtle. In the latter case the
difficulties produced by a topologically-conditioned
intuition can be severe obstacles to progress.
(Certainly, having been reared mathematically as a
topologist I have found it necessary to distrust any
conclusion whose proof is not painfully precise. )
For this reason many of the proofs in this paper are
set out in somewhat more detail than would be natural
in a more established field. For this reason also I
have here not only set out the positive results I
have so far obtained in the subject but, for the
benefit of topologists, elaborated on the failures of
analogy with topology where a more succinct exposition
would have ignored them as dead ends (e.g., in Chap. I, §2)
The Non-Abelian Self-Dual String and the (2,0)-Theory
We argue that the relevant higher gauge group for the non-abelian
generalization of the self-dual string equation is the string 2-group. We then
derive the corresponding equations of motion and discuss their properties. The
underlying geometric picture is a string structure, i.e. a categorified
principal bundle with connection whose structure 2-group is the string 2-group.
We readily write down the explicit elementary solution to our equations, which
is the categorified analogue of the 't Hooft-Polyakov monopole. Our solution
passes all the relevant consistency checks; in particular, it is globally
defined on and approaches the abelian self-dual string of charge
one at infinity. We note that our equations also arise as the BPS equations in
a recently proposed six-dimensional superconformal field theory and we show
that with our choice of higher gauge structure, the action of this theory can
be reduced to four-dimensional supersymmetric Yang-Mills theory.Comment: v3: 1+42 pages, presentation improved, typos fixed, published versio
-Algebra Models and Higher Chern-Simons Theories
We continue our study of zero-dimensional field theories in which the fields
take values in a strong homotopy Lie algebra. In a first part, we review in
detail how higher Chern-Simons theories arise in the AKSZ-formalism. These
theories form a universal starting point for the construction of
-algebra models. We then show how to describe superconformal field
theories and how to perform dimensional reductions in this context. In a second
part, we demonstrate that Nambu-Poisson and multisymplectic manifolds are
closely related via their Heisenberg algebras. As a byproduct of our
discussion, we find central Lie -algebra extensions of .
Finally, we study a number of -algebra models which are physically
interesting and which exhibit quantized multisymplectic manifolds as vacuum
solutions.Comment: 44 pages, minor corrections, published versio
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