234 research outputs found
Non-Abelian Vortices, Super-Yang-Mills Theory and Spin(7)-Instantons
We consider a complex vector bundle E endowed with a connection A over the
eight-dimensional manifold R^2 x G/H, where G/H = SU(3)/U(1)xU(1) is a
homogeneous space provided with a never integrable almost complex structure and
a family of SU(3)-structures. We establish an equivalence between G-invariant
solutions A of the Spin(7)-instanton equations on R^2 x G/H and general
solutions of non-Abelian coupled vortex equations on R^2. These vortices are
BPS solitons in a d=4 gauge theory obtained from N=1 supersymmetric Yang-Mills
theory in ten dimensions compactified on the coset space G/H with an
SU(3)-structure. The novelty of the obtained vortex equations lies in the fact
that Higgs fields, defining morphisms of vector bundles over R^2, are not
holomorphic in the generic case. Finally, we introduce BPS vortex equations in
N=4 super Yang-Mills theory and show that they have the same feature.Comment: 14 pages; v2: typos fixed, published versio
Hamiltonian 2-forms in Kahler geometry, III Extremal metrics and stability
This paper concerns the explicit construction of extremal Kaehler metrics on
total spaces of projective bundles, which have been studied in many places. We
present a unified approach, motivated by the theory of hamiltonian 2-forms (as
introduced and studied in previous papers in the series) but this paper is
largely independent of that theory.
We obtain a characterization, on a large family of projective bundles, of
those `admissible' Kaehler classes (i.e., the ones compatible with the bundle
structure in a way we make precise) which contain an extremal Kaehler metric.
In many cases, such as on geometrically ruled surfaces, every Kaehler class is
admissible. In particular, our results complete the classification of extremal
Kaehler metrics on geometrically ruled surfaces, answering several
long-standing questions.
We also find that our characterization agrees with a notion of K-stability
for admissible Kaehler classes. Our examples and nonexistence results therefore
provide a fertile testing ground for the rapidly developing theory of stability
for projective varieties, and we discuss some of the ramifications. In
particular we obtain examples of projective varieties which are destabilized by
a non-algebraic degeneration.Comment: 40 pages, sequel to math.DG/0401320 and math.DG/0202280, but largely
self-contained; partially replaces and extends math.DG/050151
Instantons and Yang-Mills Flows on Coset Spaces
We consider the Yang-Mills flow equations on a reductive coset space G/H and
the Yang-Mills equations on the manifold R x G/H. On nonsymmetric coset spaces
G/H one can introduce geometric fluxes identified with the torsion of the spin
connection. The condition of G-equivariance imposed on the gauge fields reduces
the Yang-Mills equations to phi^4-kink equations on R. Depending on the
boundary conditions and torsion, we obtain solutions to the Yang-Mills
equations describing instantons, chains of instanton-anti-instanton pairs or
modifications of gauge bundles. For Lorentzian signature on R x G/H, dyon-type
configurations are constructed as well. We also present explicit solutions to
the Yang-Mills flow equations and compare them with the Yang-Mills solutions on
R x G/H.Comment: 1+12 page
Morse theory of the moment map for representations of quivers
The results of this paper concern the Morse theory of the norm-square of the
moment map on the space of representations of a quiver. We show that the
gradient flow of this function converges, and that the Morse stratification
induced by the gradient flow co-incides with the Harder-Narasimhan
stratification from algebraic geometry. Moreover, the limit of the gradient
flow is isomorphic to the graded object of the
Harder-Narasimhan-Jordan-H\"older filtration associated to the initial
conditions for the flow. With a view towards applications to Nakajima quiver
varieties we construct explicit local co-ordinates around the Morse strata and
(under a technical hypothesis on the stability parameter) describe the negative
normal space to the critical sets. Finally, we observe that the usual Kirwan
surjectivity theorems in rational cohomology and integral K-theory carry over
to this non-compact setting, and that these theorems generalize to certain
equivariant contexts.Comment: 48 pages, small revisions from previous version based on referee's
comments. To appear in Geometriae Dedicat
Stability and BPS branes
We define the concept of Pi-stability, a generalization of mu-stability of
vector bundles, and argue that it characterizes N=1 supersymmetric brane
configurations and BPS states in very general string theory compactifications
with N=2 supersymmetry in four dimensions.Comment: harvmac, 18 p
A framework for the classification and prioritization of arrival and departure routes in Multi-Airport Systems Terminal Manoeuvring Areas
© 2015 American Institute of Aeronautics and Astronautics Inc, AIAA. All right reserved.Typically major cities (London, New York, Tokyo) are served by several airports effectively creating a Multi-Airport System or Metroplex. The operations of the Metroplex airports are highly dependent on one another, which renders their efficient management difficult. This paper proposes a framework for the prioritization of arrival and departure routes in Multi-Airport Systems Terminal Manoeuvring Areas. The framework consists of three components. The first component presents a new procedure for clustering arrival and departure flights into dynamic routes based on their temporal and spatial distributions through the identification of the important traffic flow patterns throughout the day of operations. The second component is a novel Analytic Hierarchy Process model for the prioritization of the dynamic routes, accounting for a set of quantitative and qualitative characteristics important for Multi-Airport Systems operations. The third component is a priority-based model for the facility location of the optimal terminal waypoints (fixes), which accounts for the derived priorities of each dynamic route, while meeting the required separation distances. The proposed Analytic Hierarchy Process model characteristics are validated by subject matter experts. The developed framework is applied to the London Metroplex case study
The spectrum of BPS branes on a noncompact Calabi-Yau
We begin the study of the spectrum of BPS branes and its variation on lines
of marginal stability on O_P^2(-3), a Calabi-Yau ALE space asymptotic to
C^3/Z_3. We show how to get the complete spectrum near the large volume limit
and near the orbifold point, and find a striking similarity between the
descriptions of holomorphic bundles and BPS branes in these two limits. We use
these results to develop a general picture of the spectrum. We also suggest a
generalization of some of the ideas to the quintic Calabi-Yau.Comment: harvmac, 45 pp. (v2: added references
Reducible connections and non-local symmetries of the self-dual Yang-Mills equations
We construct the most general reducible connection that satisfies the
self-dual Yang-Mills equations on a simply connected, open subset of flat
. We show how all such connections lie in the orbit of the flat
connection on under the action of non-local symmetries of the
self-dual Yang-Mills equations. Such connections fit naturally inside a larger
class of solutions to the self-dual Yang-Mills equations that are analogous to
harmonic maps of finite type.Comment: AMSLatex, 15 pages, no figures. Corrected in line with the referee's
comments. In particular, restriction to simply-connected open sets now
explicitly stated. Version to appear in Communications in Mathematical
Physic
Crystal Melting and Toric Calabi-Yau Manifolds
We construct a statistical model of crystal melting to count BPS bound states
of D0 and D2 branes on a single D6 brane wrapping an arbitrary toric Calabi-Yau
threefold. The three-dimensional crystalline structure is determined by the
quiver diagram and the brane tiling which characterize the low energy effective
theory of D branes. The crystal is composed of atoms of different colors, each
of which corresponds to a node of the quiver diagram, and the chemical bond is
dictated by the arrows of the quiver diagram. BPS states are constructed by
removing atoms from the crystal. This generalizes the earlier results on the
BPS state counting to an arbitrary non-compact toric Calabi-Yau manifold. We
point out that a proper understanding of the relation between the topological
string theory and the crystal melting involves the wall crossing in the
Donaldson-Thomas theory.Comment: 28 pages, 9 figures; v2: section 5 removed to simplify discussion on
black hole
Infinitesimal Gribov copies in gauge-fixed topological Yang-Mills theories
We study the Gribov problem in four-dimensional topological Yang-Mills
theories following the Baulieu-Singer approach in the (anti-)self-dual Landau
gauges. This is a gauge-fixed approach that allows to recover the topological
spectrum, as first constructed by Witten, by means of an equivariant (or
constrained) BRST cohomology. As standard gauge-fixed Yang-Mills theories
suffer from the gauge copy (Gribov) ambiguity, one might wonder if and how this
has repercussions for this analysis. The resolution of the small
(infinitesimal) gauge copies, in general, affects the dynamics of the
underlying theory. In particular, treating the Gribov problem for the standard
Landau gauge condition in non-topological Yang-Mills theories strongly affects
the dynamics of the theory in the infrared. In the current paper, although the
theory is investigated with the same gauge condition, the effects of the copies
turn out to be completely different. In other words: in both cases, the copies
are there, but the effects are very different. As suggested by the tree-level
exactness of the topological model in this gauge choice, the Gribov copies are
shown to be inoffensive at the quantum level. To be more precise, following
Gribov, we discuss the path integral restriction to the Gribov horizon. The
associated gap equation, which fixes the so-called Gribov parameter, is however
shown to only possess a trivial solution, making the restriction obsolete. We
relate this to the absence of radiative corrections in both gauge and ghost
sectors. We give further evidence by employing the renormalization group which
shows that, for this kind of topological model, the gap equation indeed forbids
the introduction of a massive Gribov parameter.Comment: 21 pages. Final version accepted for publication in Physics Letters
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