45 research outputs found
Exotica or the failure of the strong cosmic censorship in four dimensions
In this letter a generic counterexample to the strong cosmic censor
conjecture is exhibited. More precisely---taking into account that the
conjecture lacks any precise formulation yet---first we make sense of what one
would mean by a "generic counterexample" by introducing the mathematically
unambigous and logically stronger concept of a "robust counterexample". Then
making use of Penrose' nonlinear graviton construction (i.e., twistor theory)
and a Wick rotation trick we construct a smooth Ricci-flat but not flat
Lorentzian metric on the largest member of the Gompf--Taubes uncountable radial
family of large exotic 's. We observe that this solution of the
Lorentzian vacuum Einstein's equations with vanishing cosmological constant
provides us with a sort of counterexample which is weaker than a "robust
counterexample" but still reasonable to consider as a "generic counterexample".
It is interesting that this kind of counterexample exists only in four
dimensions.Comment: LaTeX, 11 pages, 1 figure, the final published versio
Note on a reformulation of the strong cosmic censor conjceture based on computability
In this letter we provide a reformulation of the strong cosmic censor
conjecture taking into account recent results on Malament--Hogarth space-times.
We claim that the strong version of the cosmic censor conjecture can be
formulated by postulating that a physically relevant space-time is either
globally hyperbolic or possesses the Malament--Hogarth property. But it is
known that a Malament--Hogarth space-time in principle is capable for
performing non-Turing computations such as checking consistency of ZFC set
theory. In this way we get an intimate conjectured link between the cosmic
censorship scenario and computability theory.Comment: LaTeX, 9 pages, 1 eps-figure; minor typos corrected and journal
reference adde
S-duality in Abelian gauge theory revisited
Definition of the partition function of U(1) gauge theory is extended to a
class of four-manifolds containing all compact spaces and certain
asymptotically locally flat (ALF) ones including the multi-Taub--NUT spaces.
The partition function is calculated via zeta-function regularization with
special attention to its modular properties. In the compact case, compared with
the purely topological result of Witten, we find a non-trivial curvature
correction to the modular weights of the partition function. But S-duality can
be restored by adding gravitational counter terms to the Lagrangian in the
usual way. In the ALF case however we encounter non-trivial difficulties
stemming from original non-compact ALF phenomena. Fortunately our careful
definition of the partition function makes it possible to circumnavigate them
and conclude that the partition function has the same modular properties as in
the compact case.Comment: LaTeX; 22 pages, no figure
The topology of asymptotically locally flat gravitational instantons
In this letter we demonstrate that the intersection form of the
Hausel--Hunsicker--Mazzeo compactification of a four dimensional ALF
gravitational instanton is definite and diagonalizable over the integers if one
of the Kahler forms of the hyper-Kahler gravitational instanton metric is
exact. This leads to the topological classification of these spaces.
The proof exploits the relationship between L^2 cohomology and U(1)
anti-instantons over gravitational instantons recognized by Hitchin. We then
interprete these as reducible points in a singular SU(2) anti-instanton moduli
space over the compactification leading to the identification of its
intersection form.
This observation on the intersection form might be a useful tool in the full
geometric classification of various asymptotically locally flat gravitational
instantons.Comment: 9 pages, LaTeX, no figures; Some typos corrected, slightly differs
from the published versio
Gravity as a four dimensional algebraic quantum field theory
Based on a family of indefinite unitary representations of the diffeomorphism group of an oriented smooth 4-manifold, a manifestly covariant 4 dimensional and non-perturbative algebraic
quantum field theory formulation of gravity is exhibited. More precisely among the bounded linear operators acting on these representation spaces we identify algebraic curvature tensors hence a net of local quantum observables can be constructed from C*-algebras generated by local curvature
tensors and vector fields. This algebraic quantum field theory is extracted from structures provided by an oriented smooth 4-manifold only hence possesses a diffeomorphism symmetry. In this way classical general relativity exactly in 4 dimensions naturally embeds into a quantum framework.
Several Hilbert space representations of the theory are found. First a “tautological representation” of the limiting global C*-algebra is constructed allowing to associate to any oriented smooth
4-manifold a von Neumann algebra in a canonical fashion. Secondly, influenced by the Dougan–Mason approach to gravitational quasilocal energy-momentum, we construct certain representations what we call “positive mass representations” with unbroken diffeomorphism symmetry. Thirdly, we
also obtain “classical representaions” with spontaneously broken diffeomorphism symmetry corresponding to the classical limit of the theory which turns out to be general relativity.
Finally we observe that the whole family of “positive mass representations” comprise a 2 dimensional conformal field theory in the sense of G. Segal
Gravitational interpretation of the Hitchin equations
By referring to theorems of Donaldson and Hitchin, we exhibit a rigorous
AdS/CFT-type correspondence between classical 2+1 dimensional vacuum general
relativity theory on S x R and SO(3) Hitchin theory (regarded as a classical
conformal field theory) on the spacelike past boundary S, a compact, oriented
Riemann surface of genus greater than one. Within this framework we can
interpret the 2+1 dimensional vacuum Einstein equation as a decoupled ``dual''
version of the 2 dimensional SO(3) Hitchin equations.
More precisely, we prove that if over S with a fixed conformal class a real
solution of the SO(3) Hitchin equations with induced flat SO(2,1) connection is
given, then there exists a certain cohomology class of non-isometric, singular,
flat Lorentzian metrics on S x R whose Levi--Civita connections are precisely
the lifts of this induced flat connection and the conformal class induced by
this cohomology class on S agrees with the fixed one.
Conversely, given a singular, flat Lorentzian metric on S x R the restriction
of its Levi--Civita connection gives rise to a real solution of the SO(3)
Hitchin equations on S with respect to the conformal class induced by the
corresponding cohomology class of the Lorentzian metric.Comment: LaTeX, 14 pages, no figures; compared with the previous version,
Proposition 2.4 and its proof presented in a more clear for
Spin(7)-manifolds and symmetric Yang--Mills instantons
In this Letter we establish a relationship between symmetric SU(2)
Yang--Mills instantons and metrics with Spin(7)-holonomy. Our method is based
on a slight extension of that of Bryant and Salamon developed to construct
explicit manifolds with special holonomies in 1989.
More precisely, we prove that making use of symmetric SU(2) Yang--Mills
instantons on Riemannian spin-manifolds, we can construct metrics on the chiral
spinor bundle whose holonomies are within Spin(7). Moreover if the resulting
space is connected, simply connected and complete, the holonomy coincides with
Spin(7).
The basic example is the metric constructed on the chiral spinor bundle of
the round four-sphere by using a generic SU(2)-instanton of unit action; hence
it is a five-parameter deformation of the Bryant--Salamon example, also found
by Gibbons, Page and Pope.Comment: 10 pages, no figures, LaTeX. More references have been added; but
this version differs from the published on
Szimmetria és Csoporthatások az Algebrai Topológiában = Symmetry and Group Actions in Algebraic Topology
Pályázatunk három, lazán összefüggő problémakört érint. Több kiemelkedő eredményt értünk el a Thom polinomok, és általánosabban, az ekvivariáns obstrukciók elméletében. A Thom sorok bevezetése és a Morin szingularitások Thom polinomjainak kiszámítása a terület legfontosabb eredményei az utóbbi években. A geometriai oldalon, komoly előrehaladást értünk el a hiperkahler modulusterek geometriájának leírásában, és sikerült bebizonyítanunk a Batyrev-Materov tükör reziduum sejtést tórikus orbifoldokra. Végezetül, projektünk algebrai eredményei között megemlítjük új 2-karakterisztikai jelenségek felfedezését az ortogonális csoport reprezentációelméletében, és a Zamolodcsikov periodicitási sejtés bizonyítását Y-rendszerekre. Ezenkívül, új algebrai egyenlőtlenségeket találtunk szemidefinit mátrixokra, és ezeket felhasználva megjavítottuk a legjobb ismert alsó becslést valós lineáris funkcionálok szorzatára. | The project deals with three loosely interconnected areas of mathematics. We obtained a number of outstanding results in the theory of Thom polynomials, and more generally, in equivariant obstruction theory. In particular, the introduction of Thom series, and the calculation of the Thom polynomials of Morin singularities are the most important advances in the subject in the last few years. On the more geometric side, we made serious progress in the description of the geometry of hyperkahler moduli spaces, and proved the Batyrev-Materov mirror residue conjecture for toric orbifolds. Finally, the more algebraic results of our project include discovering new characteristic-2 phenomena in the representation theory of the orthogonal group, and proving the Zamolodchikov periodicity conjecture for Y-systems. We also found new algebraic inequalities for semidefinite matrices, and using these, improved the best known lower bound on products of real linear functionals
A rigidity theorem for nonvacuum initial data
In this note we prove a theorem on non-vacuum initial data for general
relativity. The result presents a ``rigidity phenomenon'' for the extrinsic
curvature, caused by the non-positive scalar curvature.
More precisely, we state that in the case of asymptotically flat non-vacuum
initial data if the metric has everywhere non-positive scalar curvature then
the extrinsic curvature cannot be compactly supported.Comment: This is an extended and published version: LaTex, 10 pages, no
figure
Geometric construction of new Yang-Mills instantons over Taub-NUT space
In this paper we exhibit a one-parameter family of new Taub-NUT instantons
parameterized by a half-line. The endpoint of the half-line will be the
reducible Yang-Mills instanton corresponding to the Eguchi-Hanson-Gibbons L^2
harmonic 2-form, while at an inner point we recover the Pope-Yuille instanton
constructed as a projection of the Levi-Civita connection onto the positive
su(2) subalgebra of the Lie algebra so(4).
Our method imitates the Jackiw-Nohl-Rebbi construction originally designed
for flat R^4. That is we find a one-parameter family of harmonic functions on
the Taub-NUT space with a point singularity, rescale the metric and project the
obtained Levi-Civita connection onto the other negative su(2) subalgebra of
so(4). Our solutions will possess the full U(2) symmetry, and thus provide more
solutions to the recently proposed U(2) symmetric ansatz of Kim and Yoon.Comment: 13 pages, LaTex, no figure