72,922 research outputs found
Enumeration Reducibility in Closure Spaces with Applications to Logic and Algebra
In many instances in first order logic or computable algebra, classical
theorems show that many problems are undecidable for general structures, but
become decidable if some rigidity is imposed on the structure. For example, the
set of theorems in many finitely axiomatisable theories is nonrecursive, but
the set of theorems for any finitely axiomatisable complete theory is
recursive. Finitely presented groups might have an nonrecursive word problem,
but finitely presented simple groups have a recursive word problem. In this
article we introduce a topological framework based on closure spaces to show
that many of these proofs can be obtained in a similar setting. We will show in
particular that these statements can be generalized to cover arbitrary
structures, with no finite or recursive presentation/axiomatization. This
generalizes in particular work by Kuznetsov and others. Examples from first
order logic and symbolic dynamics will be discussed at length
N=2 structures in all string theories
The BRST cohomology of any topological conformal field theory admits the
structure of a Batalin--Vilkovisky algebra, and string theories are no
exception. Let us say that two topological conformal field theories are
``cohomologically equivalent'' if their BRST cohomologies are isomorphic as
Batalin--Vilkovisky algebras. What we show in this paper is that any string
theory (regardless of the matter background) is cohomologically equivalent to
some twisted N=2 superconformal field theory. We discuss three string theories
in detail: the bosonic string, the NSR string and the W_3 string. In each case
the way the cohomological equivalence is constructed can be understood as
coupling the topological conformal field theory to topological gravity. These
results lend further supporting evidence to the conjecture that _any_
topological conformal field theory is cohomologically equivalent to some
topologically twisted N=2 superconformal field theory. We end the paper with
some comments on different notions of equivalence for topological conformal
field theories and this leads to an improved conjecture.Comment: 23 pages (12 physical pages), .dvi.uu (+ some hyperlinks
Topological M-theory as Unification of Form Theories of Gravity
We introduce a notion of topological M-theory and argue that it provides a
unification of form theories of gravity in various dimensions. Its classical
solutions involve G_2 holonomy metrics on 7-manifolds, obtained from a
topological action for a 3-form gauge field introduced by Hitchin. We show that
by reductions of this 7-dimensional theory one can classically obtain
6-dimensional topological A and B models, the self-dual sector of loop quantum
gravity in 4 dimensions, and Chern-Simons gravity in 3 dimensions. We also find
that the 7-dimensional M-theory perspective sheds some light on the fact that
the topological string partition function is a wavefunction, as well as on
S-duality between the A and B models. The degrees of freedom of the A and B
models appear as conjugate variables in the 7-dimensional theory. Finally, from
the topological M-theory perspective we find hints of an intriguing holographic
link between non-supersymmetric Yang-Mills in 4 dimensions and A model
topological strings on twistor space.Comment: 65 pages, 2 figures, harvmac; v2: references added, small
corrections/clarification
Beyond Topologies, Part I
Arguments on the need, and usefulness, of going beyond the usual
Hausdorff-Kuratowski-Bourbaki, or in short, HKB concept of topology are
presented. The motivation comes, among others, from well known {\it topological
type processes}, or in short TTP-s, in the theories of Measure, Integration and
Ordered Spaces. These TTP-s, as shown by the classical characterization given
by the {\it four Moore-Smith conditions}, can {\it no longer} be incorporated
within the usual HKB topologies. One of the most successful recent ways to go
beyond HKB topologies is that developed in Beattie & Butzmann. It is shown in
this work how that extended concept of topology is a {\it particular} case of
the earlier one suggested and used by the first author in the study of
generalized solutions of large classes of nonlinear partial differential
equations
Coherent Topological Charge Structure in Models and QCD
In an effort to clarify the significance of the recent observation of
long-range topological charge coherence in QCD gauge configurations, we study
the local topological charge distributions in two-dimensional sigma
models, using the overlap Dirac operator to construct the lattice topological
charge. We find long-range sign coherence of topological charge along extended
one-dimensional structures in two-dimensional spacetime. We discuss the
connection between the long range topological structure found in and
the observed sign coherence along three-dimensional sheets in four-dimensional
QCD gauge configurations. In both cases, coherent regions of topological charge
form along membrane-like surfaces of codimension one. We show that the Monte
Carlo results, for both two-dimensional and four-dimensional gauge theory,
support a view of topological charge fluctuations suggested by Luscher and
Witten. In this framework, the observed membranes are associated with
boundaries between ``k-vacua,'' characterized by an effective local value of
which jumps by across the boundary.Comment: 26 page
Higher Structures in M-Theory
The key open problem of string theory remains its non-perturbative completion
to M-theory. A decisive hint to its inner workings comes from numerous
appearances of higher structures in the limits of M-theory that are already
understood, such as higher degree flux fields and their dualities, or the
higher algebraic structures governing closed string field theory. These are all
controlled by the higher homotopy theory of derived categories, generalised
cohomology theories, and -algebras. This is the introductory chapter
to the proceedings of the LMS/EPSRC Durham Symposium on Higher Structures in
M-Theory. We first review higher structures as well as their motivation in
string theory and beyond. Then we list the contributions in this volume,
putting them into context.Comment: 22 pages, Introductory Article to Proceedings of LMS/EPSRC Durham
Symposium Higher Structures in M-Theory, August 2018, references update
- …