845 research outputs found
Diffeomorphisms and families of Fourier-Mukai transforms in mirror symmetry
Assuming the standard framework of mirror symmetry, a conjecture is
formulated describing how the diffeomorphism group of a Calabi-Yau manifold Y
should act by families of Fourier-Mukai transforms over the complex moduli
space of the mirror X. The conjecture generalizes a proposal of Kontsevich
relating monodromy transformations and self-equivalences. Supporting evidence
is given in the case of elliptic curves, lattice-polarized K3 surfaces and
Calabi-Yau threefolds. A relation to the global Torelli problem is discussed.Comment: Approx. 20 pages LaTeX. One reference adde
On the classification of Kahler-Ricci solitons on Gorenstein del Pezzo surfaces
We give a classification of all pairs (X,v) of Gorenstein del Pezzo surfaces
X and vector fields v which are K-stable in the sense of Berman-Nystrom and
therefore are expected to admit a Kahler-Ricci solition. Moreover, we provide
some new examples of Fano threefolds admitting a Kahler-Ricci soliton.Comment: 21 pages, ancillary files containing calculations in SageMath; minor
correction
On the spectrum of the Page and the Chen-LeBrun-Weber metrics
We give bounds on the first non-zero eigenvalue of the scalar Laplacian for
both the Page and the Chen-LeBrun-Weber Einstein metrics. One notable feature
is that these bounds are obtained without explicit knowledge of the metrics or
numerical approximation to them. Our method also allows the calculation of the
invariant part of the spectrum for both metrics. We go on to discuss an
application of these bounds to the linear stability of the metrics. We also
give numerical evidence to suggest that the bounds for both metrics are
extremely close to the actual eigenvalue.Comment: 15 pages, v2 substantially rewritten, section on linear stability
added; v3 updated to reflect referee's comments, v4 final version to appear
in Ann. Glob. Anal. Geo
Morse homology for the heat flow
We use the heat flow on the loop space of a closed Riemannian manifold to
construct an algebraic chain complex. The chain groups are generated by
perturbed closed geodesics. The boundary operator is defined in the spirit of
Floer theory by counting, modulo time shift, heat flow trajectories that
converge asymptotically to nondegenerate closed geodesics of Morse index
difference one.Comment: 89 pages, 3 figure
Instantons and Killing spinors
We investigate instantons on manifolds with Killing spinors and their cones.
Examples of manifolds with Killing spinors include nearly Kaehler 6-manifolds,
nearly parallel G_2-manifolds in dimension 7, Sasaki-Einstein manifolds, and
3-Sasakian manifolds. We construct a connection on the tangent bundle over
these manifolds which solves the instanton equation, and also show that the
instanton equation implies the Yang-Mills equation, despite the presence of
torsion. We then construct instantons on the cones over these manifolds, and
lift them to solutions of heterotic supergravity. Amongst our solutions are new
instantons on even-dimensional Euclidean spaces, as well as the well-known
BPST, quaternionic and octonionic instantons.Comment: 40 pages, 2 figures v2: author email addresses and affiliations adde
Torus equivariant K-stability
It is conjectured that to test the K-polystability of a polarised variety it is enough to consider test-configurations which are equivariant with respect to a torus in the automorphism group. We prove partial results towards this conjecture. We also show that it would give a new proof of the K-polystability of constant scalar curvature polarised manifolds
The role of ultrasound simulators in education: An investigation into sonography student experiences and clinical mentor perceptions
Introduction: Simulation as an effective pedagogy is gaining momentum at all levels of healthcare education. Limited research has been undertaken on the role of simulated learning in healthcare, and further evaluation is needed to explore the quality of learning opportunities offered, and their effectiveness in the preparation of students for clinical practice. This study was undertaken to explore ways of integrating simulation into sonography training to enhance clinical preparation.Research method: A qualitative study was undertaken, using interviews to investigate the experiences of a group of sonography students after interacting with an ultrasound simulator. The perceptions of their clinical mentors on the effectiveness of this equipment to support the education and development of sonographers, were also explored.Findings: The findings confirm that ultrasound simulators provide learning opportunities in an unpressurised environment, which reduces stress for the student and potential harm to patients. Busy clinical departments acknowledge the advantages of opportunities for students to acquire basic psychomotor skills in a classroom setting, thereby avoiding the inevitable reduction in patient throughput which results from clinical training. The limitations of simulation equipment to support the development of the full range of clinical skills required by sonographers, were highlighted, and suggestions made for more effective integration of simulation into the teaching and learning process. Conclusion: Ultrasound simulators have a role in sonography education, but continued research needs to be undertaken in order to develop appropriate strategies to support students, educators, and mentors to effectively integrate this methodology
Approximate Hermitian-Yang-Mills structures and semistability for Higgs bundles. II: Higgs sheaves and admissible structures
We study the basic properties of Higgs sheaves over compact K\"ahler
manifolds and we establish some results concerning the notion of semistability;
in particular, we show that any extension of semistable Higgs sheaves with
equal slopes is semistable. Then, we use the flattening theorem to construct a
regularization of any torsion-free Higgs sheaf and we show that it is in fact a
Higgs bundle. Using this, we prove that any Hermitian metric on a
regularization of a torsion-free Higgs sheaf induces an admissible structure on
the Higgs sheaf. Finally, using admissible structures we proved some properties
of semistable Higgs sheaves.Comment: 18 pages; some typos correcte
Some Curvature Problems in Semi-Riemannian Geometry
In this survey article we review several results on the curvature of
semi-Riemannian metrics which are motivated by the positive mass theorem. The
main themes are estimates of the Riemann tensor of an asymptotically flat
manifold and the construction of Lorentzian metrics which satisfy the dominant
energy condition.Comment: 25 pages, LaTeX, 4 figure
Kahler-Einstein metrics emerging from free fermions and statistical mechanics
We propose a statistical mechanical derivation of Kahler-Einstein metrics,
i.e. solutions to Einstein's vacuum field equations in Euclidean signature
(with a cosmological constant) on a compact Kahler manifold X. The microscopic
theory is given by a canonical free fermion gas on X whose one-particle states
are pluricanonical holomorphic sections on X (coinciding with higher spin
states in the case of a Riemann surface). A heuristic, but hopefully physically
illuminating, argument for the convergence in the thermodynamical (large N)
limit is given, based on a recent mathematically rigorous result about
exponentially small fluctuations of Slater determinants. Relations to effective
bosonization and the Yau-Tian-Donaldson program in Kahler geometry are pointed
out. The precise mathematical details will be investigated elsewhere.Comment: v1: 22 pages v2: 25 pages. The relation to quantum gravity has been
further developed by working over the moduli space of all complex structures.
Relations to Donaldson's program pointed out. References adde
- …