137 research outputs found
An Invariant Theory of Spacelike Surfaces in the Four-dimensional Minkowski Space
We consider spacelike surfaces in the four-dimensional Minkowski space and
introduce geometrically an invariant linear map of Weingarten-type in the
tangent plane at any point of the surface under consideration. This allows us
to introduce principal lines and an invariant moving frame field. Writing
derivative formulas of Frenet-type for this frame field, we obtain eight
invariant functions. We prove a fundamental theorem of Bonnet-type, stating
that these eight invariants under some natural conditions determine the surface
up to a motion. We show that the basic geometric classes of spacelike surfaces
in the four-dimensional Minkowski space, determined by conditions on their
invariants, can be interpreted in terms of the properties of the two geometric
figures: the tangent indicatrix, and the normal curvature ellipse. We apply our
theory to a class of spacelike general rotational surfaces.Comment: 23 pages; to appear in Mediterr. J. Math., Vol. 9 (2012
An Extension of the Character Ring of sl(3) and Its Quantisation
We construct a commutative ring with identity which extends the ring of
characters of finite dimensional representations of sl(3). It is generated by
characters with values in the group ring of the extended affine
Weyl group of at . The `quantised' version at
rational level realises the fusion rules of a WZW conformal field
theory based on admissible representations of .Comment: contains two TeX files: main file using harvmac.tex, amssym.def,
amssym.tex, 35p.; file with figures using XY-pic package, 4p; v2: minor
corrections, Note adde
On reduction, quantum gauge transformations, and algebras singular vectors
The problem of describing the singular vectors of \cW_3 and \cW_3^{(2)}
Verma modules is addressed, viewing these algebras as BRST quantized
Drinfeld-Sokolov (DS) reductions of . Singular vectors of an
Verma module are mapped into \W algebra singular vectors and
are shown to differ from the latter by terms trivial in the BRST cohomology.
These maps are realized by quantum versions of the highest weight DS gauge
transformations.Comment: 9 page
AdS(3) holography for non-BPS geometries
By using the approach introduced in arXiv:2107.09677 we construct non-BPS
solutions of 6D supergravity coupled to two tensor multiplets as a
perturbation of AdS. These solutions are both regular and
asymptotically AdS, so according to the standard holographic
framework they must have a dual CFT interpretation as non-supersymmetric heavy
operators of the D1-D5 CFT. We provide quantitative evidence that such heavy
CFT operators are bound states of a large number of light BPS operators that
are mutually non-BPS.Comment: 36 pages, 2 Mathematica files containing data to reproduce our
perturbative expansions, 1 readme file summarising how to use the Mathematica
file
Kahler manifolds with quasi-constant holomorphic curvature
The aim of this paper is to classify compact Kahler manifolds with
quasi-constant holomorphic sectional curvature.Comment: 18 page
An invariant theory of marginally trapped surfaces in the four-dimensional Minkowski space
A marginally trapped surface in the four-dimensional Minkowski space is a
spacelike surface whose mean curvature vector is lightlike at each point. We
associate a geometrically determined moving frame field to such a surface and
using the derivative formulas for this frame field we obtain seven invariant
functions. Our main theorem states that these seven invariants determine the
surface up to a motion in Minkowski space.
We introduce meridian surfaces as one-parameter systems of meridians of a
rotational hypersurface in the four-dimensional Minkowski space. We find all
marginally trapped meridian surfaces.Comment: 16 page
Parafermionic algebras, their modules and cohomologies
We explore the Fock spaces of the parafermionic algebra introduced by H.S.
Green. Each parafermionic Fock space allows for a free minimal resolution by
graded modules of the graded 2-step nilpotent subalgebra of the parafermionic
creation operators. Such a free resolution is constructed with the help of a
classical Kostant's theorem computing Lie algebra cohomologies of the nilpotent
subalgebra with values in the parafermionic Fock space. The Euler-Poincar\'e
characteristics of the parafermionic Fock space free resolution yields some
interesting identities between Schur polynomials. Finally we briefly comment on
parabosonic and general parastatistics Fock spaces.Comment: 10 pages, talk presented at the International Workshop "Lie theory
and its applications in Physics" (17-23 June 2013, Varna, Bulgaria
Special biconformal changes of K\"ahler surface metrics
The term "special biconformal change" refers, basically, to the situation
where a given nontrivial real-holomorphic vector field on a complex manifold is
a gradient relative to two K\"ahler metrics, and, simultaneously, an
eigenvector of one of the metrics treated, with the aid of the other, as an
endomorphism of the tangent bundle. A special biconformal change is called
nontrivial if the two metrics are not each other's constant multiples. For
instance, according to a 1995 result of LeBrun, a nontrivial special
biconformal change exists for the conformally-Einstein K\"ahler metric on the
two-point blow-up of the complex projective plane, recently discovered by Chen,
LeBrun and Weber; the real-holomorphic vector field involved is the gradient of
its scalar curvature. The present paper establishes the existence of nontrivial
special biconformal changes for some canonical metrics on Del Pezzo surfaces,
viz. K\"ahler-Einstein metrics (when a nontrivial holomorphic vector field
exists), non-Einstein K\"ahler-Ricci solitons, and K\"ahler metrics admitting
nonconstant Killing potentials with geodesic gradients.Comment: 16 page
Vanishing Theorems and String Backgrounds
We show various vanishing theorems for the cohomology groups of compact
hermitian manifolds for which the Bismut connection has (restricted) holonomy
contained in SU(n) and classify all such manifolds of dimension four. In this
way we provide necessary conditions for the existence of such structures on
hermitian manifolds. Then we apply our results to solutions of the string
equations and show that such solutions admit various cohomological restrictions
like for example that under certain natural assumptions the plurigenera vanish.
We also find that under some assumptions the string equations are equivalent to
the condition that a certain vector is parallel with respect to the Bismut
connection.Comment: 25 pages, Late
New Solutions of the Yang-Baxter Equation Based on Root of 1 Representations of the Para-Bose Superalgebra U[osp(1/2)]
New solutions of the quantum Yang-Baxter equation, depending in general on
three arbitrary parameters, are written down. They are based on the root of
unity representations of the quantum orthosymplectic superalgebra \\U, which
were found recently. Representations of the braid group are defined
within any tensorial power of root of 1 \\U modules.Comment: 11 pages, PlainTe
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