3,031 research outputs found
On polynomially integrable Birkhoff billiards on surfaces of constant curvature
We present a solution of the algebraic version of Birkhoff Conjecture on
integrable billiards. Namely we show that every polynomially integrable real
bounded convex planar billiard with smooth boundary is an ellipse. We extend
this result to billiards with piecewise-smooth and not necessarily convex
boundary on arbitrary two-dimensional surface of constant curvature: plane,
sphere, Lobachevsky (hyperbolic) plane; each of them being modeled as a plane
or a (pseudo-) sphere in equipped with appropriate quadratic
form. Namely, we show that a billiard is polynomially integrable, if and only
if its boundary is a union of confocal conical arcs and appropriate geodesic
segments. We also present a complexification of these results. These are joint
results of Mikhail Bialy, Andrey Mironov and the author. The proof is split
into two parts. The first part is given by Bialy and Mironov in their two joint
papers. They considered the tautological projection of the boundary to
and studied its orthogonal-polar dual curve, which is piecewise
algebraic, by S.V.Bolotin's theorem. By their arguments and another Bolotin's
theorem, it suffices to show that each non-linear complex irreducible component
of the dual curve is a conic. They have proved that all its singularities and
inflection points (if any) lie in the projectivized zero locus of the
corresponding quadratic form on . The present paper provides the
second part of the proof: we show that each above irreducible component is a
conic and finish the solution of the Algebraic Birkhoff Conjecture in constant
curvature.Comment: To appear in the Journal of the European Mathematical Society (JEMS),
69 pages, 2 figures. A shorter proof of Theorem 4.24. Minor precisions and
misprint correction
Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry
A new method to obtain trigonometry for the real spaces of constant curvature
and metric of any (even degenerate) signature is presented. The method
encapsulates trigonometry for all these spaces into a single basic
trigonometric group equation. This brings to its logical end the idea of an
absolute trigonometry, and provides equations which hold true for the nine
two-dimensional spaces of constant curvature and any signature. This family of
spaces includes both relativistic and non-relativistic homogeneous spacetimes;
therefore a complete discussion of trigonometry in the six de Sitter,
minkowskian, Newton--Hooke and galilean spacetimes follow as particular
instances of the general approach. Any equation previously known for the three
classical riemannian spaces also has a version for the remaining six
spacetimes; in most cases these equations are new. Distinctive traits of the
method are universality and self-duality: every equation is meaningful for the
nine spaces at once, and displays explicitly invariance under a duality
transformation relating the nine spaces. The derivation of the single basic
trigonometric equation at group level, its translation to a set of equations
(cosine, sine and dual cosine laws) and the natural apparition of angular and
lateral excesses, area and coarea are explicitly discussed in detail. The
exposition also aims to introduce the main ideas of this direct group
theoretical way to trigonometry, and may well provide a path to systematically
study trigonometry for any homogeneous symmetric space.Comment: 51 pages, LaTe
Invariants and divergences in half-maximal supergravity theories
The invariants in half-maximal supergravity theories in D=4,5 are discussed
in detail up to dimension eight (e.g. R^4). In D=4, owing to the anomaly in the
rigid SL(2,R) duality symmetry, the restrictions on divergences need careful
treatment. In pure N=4 supergravity, this anomalous symmetry still implies
duality invariance of candidate counterterms at three loops. Provided one makes
the additional assumption that there exists a full 16-supercharge off-shell
formulation of the theory, counterterms at L>1 loops would also have to be
writable as full-superspace integrals. At the three-loop order such a
duality-invariant full-superspace integral candidate counterterm exists, but
its duality invariance is marginal in the sense that the full-superspace
counter-Lagrangian is not itself duality-invariant. We show that such marginal
invariants are not allowable as counterterms in a 16-supercharge off-shell
formalism. It is not possible to draw the same conclusion when vector
multiplets are present because of the appearance of F^4 terms in the SL(2,R)
anomaly. In D=5 there is no one-loop anomaly in the shift invariance of the
dilaton, and we argue that this implies finiteness at two loops, again subject
to the assumption that 16 supercharges can be preserved off-shell.Comment: 81 page
Classification of near-horizon geometries of extremal black holes
Any spacetime containing a degenerate Killing horizon, such as an extremal
black hole, possesses a well-defined notion of a near-horizon geometry. We
review such near-horizon geometry solutions in a variety of dimensions and
theories in a unified manner. We discuss various general results including
horizon topology and near-horizon symmetry enhancement. We also discuss the
status of the classification of near-horizon geometries in theories ranging
from vacuum gravity to Einstein-Maxwell theory and supergravity theories.
Finally, we discuss applications to the classification of extremal black holes
and various related topics. Several new results are presented and open problems
are highlighted throughout.Comment: 70 pages; invited review article for Living Reviews in Relativity; v2
some improvements and references adde
From Higher Spins to Strings: A Primer
A contribution to the collection of reviews "Introduction to Higher Spin
Theory" edited by S. Fredenhagen, this introductory article is a pedagogical
account of higher-spin fields and their connections with String Theory. We
start with the motivations for and a brief historical overview of the subject.
We discuss the Wigner classifications of unitary irreducible
Poincar\'e-modules, write down covariant field equations for totally symmetric
massive and massless representations in flat space, and consider their
Lagrangian formulation. After an elementary exposition of the AdS unitary
representations, we review the key no-go and yes-go results concerning
higher-spin interactions, e.g., the Velo-Zwanziger acausality and its
string-theoretic resolution among others. The unfolded formalism, which
underlies Vasiliev's equations, is then introduced to reformulate the
flat-space Bargmann-Wigner equations and the AdS massive-scalar Klein-Gordon
equation, and to state the "central on-mass-shell theorem". These techniques
are used for deriving the unfolded form of the boundary-to-bulk propagator in
, which in turn discloses the asymptotic symmetries of (supersymmetric)
higher-spin theories. The implications for string-higher-spin dualities
revealed by this analysis are then elaborated.Comment: 106 pages, 2 figures. Contribution to the collection of reviews
"Introduction to Higher Spin Theory" edited by S. Fredenhagen. V2: Typos
corrected, acknowledgements and references adde
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