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Effects of surface roughness on the paramagnetic response of small unconventional superconductors
We theoretically study effects of surface roughness on the magnetic response
of small unconventional superconductors by solving the Eilenberger equation for
the quassiclassical Green function and the Maxwell equation for the vector
potential simultaneously and self-consistently. The paramagnetic phase of
spin-singlet -wave superconducting disks is drastically suppressed by the
surface roughness, whereas that of spin-triplet -wave disks is robust even
in the presence of the roughness. Such difference derives from the orbital
symmetry of paramagnetic odd-frequency Cooper pairs appearing at the surface of
disks. The orbital part of the paramagnetic pairing correlation is -wave
symmetry in the -wave disks, whereas it is -wave symmetry in the -wave
ones. Calculating the free-energy, we also confirm that the paramagnetic state
is more stable than the normal state, which indicates a possibility of
detecting the paramagnetic effect in experiments. Indeed our results are
consistent with an experimental finding on high- thin films.Comment: 11 pages, 10 figure
Borcherds symmetries in M-theory
It is well known but rather mysterious that root spaces of the Lie
groups appear in the second integral cohomology of regular, complex, compact,
del Pezzo surfaces. The corresponding groups act on the scalar fields (0-forms)
of toroidal compactifications of M theory. Their Borel subgroups are actually
subgroups of supergroups of finite dimension over the Grassmann algebra of
differential forms on spacetime that have been shown to preserve the
self-duality equation obeyed by all bosonic form-fields of the theory. We show
here that the corresponding duality superalgebras are nothing but Borcherds
superalgebras truncated by the above choice of Grassmann coefficients. The full
Borcherds' root lattices are the second integral cohomology of the del Pezzo
surfaces. Our choice of simple roots uses the anti-canonical form and its known
orthogonal complement. Another result is the determination of del Pezzo
surfaces associated to other string and field theory models. Dimensional
reduction on corresponds to blow-up of points in general position
with respect to each other. All theories of the Magic triangle that reduce to
the sigma model in three dimensions correspond to singular del Pezzo
surfaces with (normal) singularity at a point. The case of type I and
heterotic theories if one drops their gauge sector corresponds to non-normal
(singular along a curve) del Pezzo's. We comment on previous encounters with
Borcherds algebras at the end of the paper.Comment: 30 pages. Besides expository improvements, we exclude by hand real
fermionic simple roots when they would naively aris
New Complete Non-compact Spin(7) Manifolds
We construct new explicit metrics on complete non-compact Riemannian
8-manifolds with holonomy Spin(7). One manifold, which we denote by A_8, is
topologically R^8 and another, which we denote by B_8, is the bundle of chiral
spinors over . Unlike the previously-known complete non-compact metric of
Spin(7) holonomy, which was also defined on the bundle of chiral spinors over
S^4, our new metrics are asymptotically locally conical (ALC): near infinity
they approach a circle bundle with fibres of constant length over a cone whose
base is the squashed Einstein metric on CP^3. We construct the
covariantly-constant spinor and calibrating 4-form. We also obtain an
L^2-normalisable harmonic 4-form for the A_8 manifold, and two such 4-forms (of
opposite dualities) for the B_8 manifold. We use the metrics to construct new
supersymmetric brane solutions in M-theory and string theory. In particular, we
construct resolved fractional M2-branes involving the use of the L^2 harmonic
4-forms, and show that for each manifold there is a supersymmetric example. An
intriguing feature of the new A_8 and B_8 Spin(7) metrics is that they are
actually the same local solution, with the two different complete manifolds
corresponding to taking the radial coordinate to be either positive or
negative. We make a comparison with the Taub-NUT and Taub-BOLT metrics, which
by contrast do not have special holonomy. In an appendix we construct the
general solution of our first-order equations for Spin(7) holonomy, and obtain
further regular metrics that are complete on manifolds B^+_8 and B^-_8 similar
to B_8.Comment: Latex, 29 pages. Appendix obtaining general solution of first-order
equations and additional complete Spin(7) manifolds adde
Force dipoles and stable local defects on fluid vesicles
An exact description is provided of an almost spherical fluid vesicle with a
fixed area and a fixed enclosed volume locally deformed by external normal
forces bringing two nearby points on the surface together symmetrically. The
conformal invariance of the two-dimensional bending energy is used to identify
the distribution of energy as well as the stress established in the vesicle.
While these states are local minima of the energy, this energy is degenerate;
there is a zero mode in the energy fluctuation spectrum, associated with area
and volume preserving conformal transformations, which breaks the symmetry
between the two points. The volume constraint fixes the distance , measured
along the surface, between the two points; if it is relaxed, a second zero mode
appears, reflecting the independence of the energy on ; in the absence of
this constraint a pathway opens for the membrane to slip out of the defect.
Logarithmic curvature singularities in the surface geometry at the points of
contact signal the presence of external forces. The magnitude of these forces
varies inversely with and so diverges as the points merge; the
corresponding torques vanish in these defects. The geometry behaves near each
of the singularities as a biharmonic monopole, in the region between them as a
surface of constant mean curvature, and in distant regions as a biharmonic
quadrupole. Comparison of the distribution of stress with the quadratic
approximation in the height functions points to shortcomings of the latter
representation. Radial tension is accompanied by lateral compression, both near
the singularities and far away, with a crossover from tension to compression
occurring in the region between them.Comment: 26 pages, 10 figure
Extremal Transitions in Heterotic String Theory
In this paper we study extremal transitions between heterotic string
compactifications, i.e., transitions between pairs (M,V) where M is a
Calabi-Yau manifold and V a gauge bundle. Bundle transitions are described
using language recently espoused by Friedman, Morgan, Witten. In addition,
partly as a check on our methods, we also study how small instantons are
described in the same language, and also describe the sheaves corresponding to
small instantons.Comment: 26 pages, LaTex, 3 figures, references adde
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