93 research outputs found
Non-imprisonment conditions on spacetime
The non-imprisonment conditions on spacetimes are studied. It is proved that
the non-partial imprisonment property implies the distinction property.
Moreover, it is proved that feeble distinction, a property which stays between
weak distinction and causality, implies non-total imprisonment. As a result the
non-imprisonment conditions can be included in the causal ladder of spacetimes.
Finally, totally imprisoned causal curves are studied in detail, and results
concerning the existence and properties of minimal invariant sets are obtained.Comment: 12 pages, 2 figures. v2: improved results on totally imprisoned
curves, a figure changed, some misprints fixe
Khovanov homology is an unknot-detector
We prove that a knot is the unknot if and only if its reduced Khovanov
cohomology has rank 1. The proof has two steps. We show first that there is a
spectral sequence beginning with the reduced Khovanov cohomology and abutting
to a knot homology defined using singular instantons. We then show that the
latter homology is isomorphic to the instanton Floer homology of the sutured
knot complement: an invariant that is already known to detect the unknot.Comment: 124 pages, 13 figure
Weak distinction and the optimal definition of causal continuity
Causal continuity is usually defined by imposing the conditions (i)
distinction and (ii) reflectivity. It is proved here that a new causality
property which stays between weak distinction and causality, called feeble
distinction, can actually replace distinction in the definition of causal
continuity. An intermediate proof shows that feeble distinction and future
(past) reflectivity implies past (resp. future) distinction. Some new
characterizations of weak distinction and reflectivity are given.Comment: 9 pages, 2 figures. v2: improved and expanded version. v3: a few
misprints have been corrected and a reference has been update
The causal ladder and the strength of K-causality. I
A unifying framework for the study of causal relations is presented. The
causal relations are regarded as subsets of M x M and the role of the
corresponding antisymmetry conditions in the construction of the causal ladder
is stressed. The causal hierarchy of spacetime is built from chronology up to
K-causality and new characterizations of the distinction and strong causality
properties are obtained. The closure of the causal future is not transitive, as
a consequence its repeated composition leads to an infinite causal subladder
between strong causality and K-causality - the A-causality subladder. A
spacetime example is given which proves that K-causality differs from infinite
A-causality.Comment: 16 pages, one figure. Old title: ``On the relationship between
K-causality and infinite A-causality''. Some typos fixed; small change in the
proof of lemma 4.
K-causality and degenerate spacetimes
The causal relation was introduced by Sorkin and Woolgar to extend the
standard causal analysis of spacetimes to those that are only . Most
of their results also hold true in the case of spacetimes with degeneracies. In
this paper we seek to examine explicitly in the case of Lorentzian
topology changing Morse spacetimes containing isolated degeneracies. We first
demonstrate some interesting features of this relation in globally Lorentzian
spacetimes. In particular, we show that is robust and that it coincides
with the Seifert relation when the spacetime is stably causal. Moreover, the
Hawking and Sachs characterisation of causal continuity translates into a
natural expression in terms of for general spacetimes. We then examine
in topology changing Morse spacetimes both with and without the
degeneracies and find further characterisations of causal continuity.Comment: Latex, 23 pages, 4 figure
Sheaves on fibered threefolds and quiver sheaves
This paper classifies a class of holomorphic D-branes, closely related to
framed torsion-free sheaves, on threefolds fibered in resolved ADE surfaces
over a general curve C, in terms of representations with relations of a twisted
Kronheimer--Nakajima-type quiver in the category Coh(C) of coherent sheaves on
C. For the local Calabi--Yau case C\cong\A^1 and special choice of framing, one
recovers the N=1 ADE quiver studied by Cachazo--Katz--Vafa.Comment: 13 pages, 2 figures, minor change
Proper time and Minkowski structure on causal graphs
For causal graphs we propose a definition of proper time which for small
scales is based on the concept of volume, while for large scales the usual
definition of length is applied. The scale where the change from "volume" to
"length" occurs is related to the size of a dynamical clock and defines a
natural cut-off for this type of clock. By changing the cut-off volume we may
probe the geometry of the causal graph on different scales and therey define a
continuum limit. This provides an alternative to the standard coarse graining
procedures. For regular causal lattice (like e.g. the 2-dim. light-cone
lattice) this concept can be proven to lead to a Minkowski structure. An
illustrative example of this approach is provided by the breather solutions of
the Sine-Gordon model on a 2-dimensional light-cone lattice.Comment: 15 pages, 4 figure
Hitchin Equation, Singularity, and N=2 Superconformal Field Theories
We argue that Hitchin's equation determines not only the low energy effective
theory but also describes the UV theory of four dimensional N=2 superconformal
field theories when we compactify six dimensional theory on a
punctured Riemann surface. We study the singular solution to Hitchin's equation
and the Higgs field of solutions has a simple pole at the punctures; We show
that the massless theory is associated with Higgs field whose residual is a
nilpotent element; We identify the flavor symmetry associated with the puncture
by studying the singularity of closure of the moduli space of solutions with
the appropriate boundary conditions. For the mass-deformed theory the residual
of the Higgs field is a semi-simple element, we identify the semi-simple
element by arguing that the moduli space of solutions of mass-deformed theory
must be a deformation of the closure of the moduli space of the massless
theory. We also study the Seiberg-Witten curve by identifying it as the
spectral curve of the Hitchin's system. The results are all in agreement with
Gaiotto's results derived from studying the Seiberg-Witten curve of four
dimensional quiver gauge theory.Comment: 42 pages, 20 figures, Hitchin's equation for N=2 theory is derived by
comparing different order of compactification of six dimensional theory on
T^2\times \Sigma. More discussion about flavor symmetries. Typos are
correcte
PU(2) monopoles. II: Top-level Seiberg-Witten moduli spaces and Witten's conjecture in low degrees
In this article we complete the proof---for a broad class of
four-manifolds---of Witten's conjecture that the Donaldson and Seiberg-Witten
series coincide, at least through terms of degree less than or equal to c-2,
where c is a linear combination of the Euler characteristic and signature of
the four-manifold. This article is a revision of sections 4--7 of an earlier
version, while a revision of sections 1--3 of that earlier version now appear
in a separate companion article (math.DG/0007190). Here, we use our
computations of Chern classes for the virtual normal bundles for the
Seiberg-Witten strata from the companion article (math.DG/0007190), a
comparison of all the orientations, and the PU(2) monopole cobordism to compute
pairings with the links of level-zero Seiberg-Witten moduli subspaces of the
moduli space of PU(2) monopoles. These calculations then allow us to compute
low-degree Donaldson invariants in terms of Seiberg-Witten invariants and
provide a partial verification of Witten's conjecture.Comment: Journal fur die Reine und Angewandte Mathematik, to appear; 65 pages.
Revision of sections 4-7 of version v1 (December 1997
Loop and surface operators in N=2 gauge theory and Liouville modular geometry
Recently, a duality between Liouville theory and four dimensional N=2 gauge
theory has been uncovered by some of the authors. We consider the role of
extended objects in gauge theory, surface operators and line operators, under
this correspondence. We map such objects to specific operators in Liouville
theory. We employ this connection to compute the expectation value of general
supersymmetric 't Hooft-Wilson line operators in a variety of N=2 gauge
theories.Comment: 60 pages, 11 figures; v3: further minor corrections, published
versio
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