14,397 research outputs found
Algebraic description of spacetime foam
A mathematical formalism for treating spacetime topology as a quantum
observable is provided. We describe spacetime foam entirely in algebraic terms.
To implement the correspondence principle we express the classical spacetime
manifold of general relativity and the commutative coordinates of its events by
means of appropriate limit constructions.Comment: 34 pages, LaTeX2e, the section concerning classical spacetimes in the
limit essentially correcte
Labelled transition systems as a Stone space
A fully abstract and universal domain model for modal transition systems and
refinement is shown to be a maximal-points space model for the bisimulation
quotient of labelled transition systems over a finite set of events. In this
domain model we prove that this quotient is a Stone space whose compact,
zero-dimensional, and ultra-metrizable Hausdorff topology measures the degree
of bisimilarity such that image-finite labelled transition systems are dense.
Using this compactness we show that the set of labelled transition systems that
refine a modal transition system, its ''set of implementations'', is compact
and derive a compactness theorem for Hennessy-Milner logic on such
implementation sets. These results extend to systems that also have partially
specified state propositions, unify existing denotational, operational, and
metric semantics on partial processes, render robust consistency measures for
modal transition systems, and yield an abstract interpretation of compact sets
of labelled transition systems as Scott-closed sets of modal transition
systems.Comment: Changes since v2: Metadata updat
Chiral expansion and Macdonald deformation of two-dimensional Yang-Mills theory
We derive the analog of the large Gross-Taylor holomorphic string
expansion for the refinement of -deformed Yang-Mills theory on a
compact oriented Riemann surface. The derivation combines Schur-Weyl duality
for quantum groups with the Etingof-Kirillov theory of generalized quantum
characters which are related to Macdonald polynomials. In the unrefined limit
we reproduce the chiral expansion of -deformed Yang-Mills theory derived by
de Haro, Ramgoolam and Torrielli. In the classical limit , the expansion
defines a new -deformation of Hurwitz theory wherein the refined
partition function is a generating function for certain parameterized Euler
characters, which reduce in the unrefined limit to the orbifold Euler
characteristics of Hurwitz spaces of holomorphic maps. We discuss the
geometrical meaning of our expansions in relation to quantum spectral curves
and -ensembles of matrix models arising in refined topological string
theory.Comment: 45 pages; v2: References adde
Differential KO-theory: constructions, computations, and applications
We provide a systematic and detailed treatment of differential refinements of
KO-theory. We explain how various flavors capture geometric aspects in
different but related ways, highlighting the utility of each. While general
axiomatics exist, no explicit constructions seem to have appeared before. This
fills a gap in the literature in which K-theory is usually worked out leaving
KO-theory essentially untouched, with only scattered partial information in
print. We compare to the complex case, highlighting which constructions follow
analogously and which are much more subtle. We construct a pushforward and
differential refinements of genera, leading to a Riemann-Roch theorem for
-theory. We also construct the corresponding
Atiyah-Hirzebruch spectral sequence (AHSS) and explicitly identify the
differentials, including ones which mix geometric and topological data. This
allows us to completely characterize the image of the Pontrjagin character.
Then we illustrate with examples and applications, including higher tangential
structures, Adams operations, and a differential Wu formula.Comment: 105 pages, very minor changes, comments welcom
Orientifolds and the Refined Topological String
We study refined topological string theory in the presence of orientifolds by
counting second-quantized BPS states in M-theory. This leads us to propose a
new integrality condition for both refined and unrefined topological strings
when orientifolds are present. We define the SO(2N) refined Chern-Simons theory
which computes refined open string amplitudes for branes wrapping Seifert
three-manifolds. We use the SO(2N) refined Chern-Simons theory to compute new
invariants of torus knots that generalize the Kauffman polynomials. At large N,
the SO(2N) refined Chern-Simons theory on the three-sphere is dual to refined
topological strings on an orientifold of the resolved conifold, generalizing
the Gopakumar-Sinha-Vafa duality. Finally, we use the (2,0) theory to define
and solve refined Chern-Simons theory for all ADE gauge groups
The categorical limit of a sequence of dynamical systems
Modeling a sequence of design steps, or a sequence of parameter settings,
yields a sequence of dynamical systems. In many cases, such a sequence is
intended to approximate a certain limit case. However, formally defining that
limit turns out to be subject to ambiguity. Depending on the interpretation of
the sequence, i.e. depending on how the behaviors of the systems in the
sequence are related, it may vary what the limit should be. Topologies, and in
particular metrics, define limits uniquely, if they exist. Thus they select one
interpretation implicitly and leave no room for other interpretations. In this
paper, we define limits using category theory, and use the mentioned relations
between system behaviors explicitly. This resolves the problem of ambiguity in
a more controlled way. We introduce a category of prefix orders on executions
and partial history preserving maps between them to describe both discrete and
continuous branching time dynamics. We prove that in this category all
projective limits exist, and illustrate how ambiguity in the definition of
limits is resolved using an example. Moreover, we show how various problems
with known topological approaches are now resolved, and how the construction of
projective limits enables us to approximate continuous time dynamics as a
sequence of discrete time systems.Comment: In Proceedings EXPRESS/SOS 2013, arXiv:1307.690
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