3,156 research outputs found
The geometry of manifolds and the perception of space
This essay discusses the development of key geometric ideas in the 19th
century which led to the formulation of the concept of an abstract manifold
(which was not necessarily tied to an ambient Euclidean space) by Hermann Weyl
in 1913. This notion of manifold and the geometric ideas which could be
formulated and utilized in such a setting (measuring a distance between points,
curvature and other geometric concepts) was an essential ingredient in
Einstein's gravitational theory of space-time from 1916 and has played
important roles in numerous other theories of nature ever since.Comment: arXiv admin note: substantial text overlap with arXiv:1301.064
Logic and Topology for Knowledge, Knowability, and Belief - Extended Abstract
In recent work, Stalnaker proposes a logical framework in which belief is
realized as a weakened form of knowledge. Building on Stalnaker's core
insights, and using frameworks developed by Bjorndahl and Baltag et al., we
employ topological tools to refine and, we argue, improve on this analysis. The
structure of topological subset spaces allows for a natural distinction between
what is known and (roughly speaking) what is knowable; we argue that the
foundational axioms of Stalnaker's system rely intuitively on both of these
notions. More precisely, we argue that the plausibility of the principles
Stalnaker proposes relating knowledge and belief relies on a subtle
equivocation between an "evidence-in-hand" conception of knowledge and a weaker
"evidence-out-there" notion of what could come to be known. Our analysis leads
to a trimodal logic of knowledge, knowability, and belief interpreted in
topological subset spaces in which belief is definable in terms of knowledge
and knowability. We provide a sound and complete axiomatization for this logic
as well as its uni-modal belief fragment. We then consider weaker logics that
preserve suitable translations of Stalnaker's postulates, yet do not allow for
any reduction of belief. We propose novel topological semantics for these
irreducible notions of belief, generalizing our previous semantics, and provide
sound and complete axiomatizations for the corresponding logics.Comment: In Proceedings TARK 2017, arXiv:1707.08250. The full version of this
paper, including the longer proofs, is at arXiv:1612.0205
Graded Lagrangians, exotic topological D-branes and enhanced triangulated categories
I point out that (BPS saturated) A-type D-branes in superstring
compactifications on Calabi-Yau threefolds correspond to {\em graded} special
Lagrangian submanifolds, a particular case of the graded Lagrangian
submanifolds considered by M. Kontsevich and P. Seidel. Combining this with the
categorical formulation of cubic string field theory in the presence of
D-branes, I consider a collection of {\em topological} D-branes wrapped over
the same Lagrangian cycle and {\em derive} its string field action from first
principles. The result is a {\em -graded} version of super-Chern-Simons
field theory living on the Lagrangian cycle, whose relevant string field is a
degree one superconnection in a -graded superbundle, in the sense
previously considered in mathematical work of J. M. Bismutt and J. Lott. This
gives a refined (and modified) version of a proposal previously made by C.
Vafa. I analyze the vacuum deformations of this theory and relate them to
topological D-brane composite formation, by using the general formalism
developed in a previous paper. This allows me to identify a large class of
topological D-brane composites (generalized, or `exotic' topological D-branes)
which do not admit a traditional description. Among these are objects which
correspond to the `covariantly constant sequences of flat bundles' considered
by Bismut and Lott, as well as more general structures, which are related to
the enhanced triangulated categories of Bondal and Kapranov. I also give a
rough sketch of the relation between this construction and the large radius
limit of a certain version of the `derived category of Fukaya's category'.Comment: 31 pages, 4 figures, minor typos corrected; v3: changed to JHEP styl
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