31 research outputs found
Conformal Dirichlet-Neumann Maps and Poincaré-Einstein Manifolds
A conformal description of Poincaré-Einstein manifolds is developed: these structures are seen to be a special case of a natural weakening of the Einstein condition termed an almost Einstein structure. This is used for two purposes: to shed light on the relationship between the scattering construction of Graham-Zworski and the higher order conformal Dirichlet-Neumann maps of Branson and the author; to sketch a new construction of non-local (Dirichlet-to-Neumann type) conformal operators between tensor bundles
Metric connections in projective differential geometry
We search for Riemannian metrics whose Levi-Civita connection belongs to a
given projective class. Following Sinjukov and Mikes, we show that such metrics
correspond precisely to suitably positive solutions of a certain projectively
invariant finite-type linear system of partial differential equations.
Prolonging this system, we may reformulate these equations as defining
covariant constant sections of a certain vector bundle with connection. This
vector bundle and its connection are derived from the Cartan connection of the
underlying projective structure.Comment: 10 page
The Research of Thomas P. Branson
The Midwest Geometry Conference 2007 was devoted to the substantial mathematical legacy of Thomas P. Branson who passed away unexpectedly the previous year. This contribution to the Proceedings briefly introduces this legacy. We also take the opportunity of recording his bibliography. Thomas Branson was on the Editorial Board of SIGMA and we are pleased that SIGMA is able to publish the Proceedings
Conformal geometry of embedded manifolds with boundary from universal holographic formulæ
Indexación ScopusFor an embedded conformal hypersurface with boundary, we construct critical order local invariants and their canonically associated differential operators. These are obtained holographically in a construction that uses a singular Yamabe problem and a corresponding minimal hypersurface with boundary. They include an extrinsic Q-curvature for the boundary of the embedded conformal manifold and, for its interior, the Q-curvature and accompanying boundary transgression curvatures. This gives universal formulæ for extrinsic analogs of Branson Q-curvatures that simultaneously generalize the Willmore energy density, including the boundary transgression terms required for conformal invariance. It also gives extrinsic conformal Laplacian power type operators associated with all these curvatures. The construction also gives formulæ for the divergent terms and anomalies in the volume and hyper-area asymptotics determined by minimal hypersurfaces having boundary at the conformal infinity. A main feature is the development of a universal, distribution-based, boundary calculus for the treatment of these and related problems. © 2021https://www-sciencedirect-com.recursosbiblioteca.unab.cl/science/article/pii/S0001870821001389?via%3Dihu
The twistor spinors of generic 2- and 3-distributions
Generic distributions on 5- and 6-manifolds give rise to conformal structures
that were discovered by P. Nurowski resp. R. Bryant. We describe both as
Fefferman-type constructions and show that for orientable distributions one
obtains conformal spin structures. The resulting conformal spin geometries are
then characterized by their conformal holonomy and equivalently by the
existence of a twistor spinor which satisfies a genericity condition. Moreover,
we show that given such a twistor spinor we can decompose a conformal Killing
field of the structure. We obtain explicit formulas relating conformal Killing
fields, almost Einstein structures and twistor spinors.Comment: 26 page
A holonomy characterisation of Fefferman spaces
We prove that Fefferman spaces, associated to non--degenerate CR structures
of hypersurface type, are characterised, up to local conformal isometry, by the
existence of a parallel orthogonal complex structure on the standard tractor
bundle. This condition can be equivalently expressed in terms of conformal
holonomy. Extracting from this picture the essential consequences at the level
of tensor bundles yields an improved, conformally invariant analogue of
Sparling's characterisation of Fefferman spaces.Comment: AMSLaTeX, 15 page
Smooth metric measure spaces, quasi-Einstein metrics, and tractors
We introduce the tractor formalism from conformal geometry to the study of
smooth metric measure spaces. In particular, this gives rise to a
correspondence between quasi-Einstein metrics and parallel sections of certain
tractor bundles. We use this formulation to give a sharp upper bound on the
dimension of the vector space of quasi-Einstein metrics, providing a different
perspective on some recent results of He, Petersen and Wylie.Comment: 33 pages; final versio
Current Exchanges for Reducible Higher Spin Multiplets and Gauge Fixing
We compute the current exchanges between triplets of higher spin fields which
describe reducible representations of the Poincare group. Through this
computation we can extract the propagator of the reducible higher spin fields
which compose the triplet. We show how to decompose the triplet fields into
irreducible HS fields which obey Fronsdal equations, and how to compute the
current-current interaction for the cubic couplings which appear in
ArXiv:0708.1399 [hep-th] using the decomposition into irreducible modes. We
compare this result with the same computation using a gauge fixed (Feynman)
version of the triplet Lagrangian which allows us to write very simple HS
propagators for the triplet fields.Comment: 26 pages, 1 table; v3 some clarifications and references added, typos
corrected. Published versio
Ambient metrics for -dimensional -waves
We provide an explicit formula for the Fefferman-Graham-ambient metric of an
-dimensional conformal -wave in those cases where it exists. In even
dimensions we calculate the obstruction explicitly. Furthermore, we describe
all 4-dimensional -waves that are Bach-flat, and give a large class of
Bach-flat examples which are conformally Cotton-flat, but not conformally
Einstein. Finally, as an application, we use the obtained ambient metric to
show that even-dimensional -waves have vanishing critical -curvature.Comment: 17 pages, in v2 footnote and references added and typos corrected, in
v3 remark in the Introduction about Brinkmann's results corrected and
footnote adde
Holographic Description of Gravitational Anomalies
The holographic duality can be extended to include quantum theories with
broken coordinate invariance leading to the appearance of the gravitational
anomalies. On the gravity side one adds the gravitational Chern-Simons term to
the bulk action which gauge invariance is only up to the boundary terms. We
analyze in detail how the gravitational anomalies originate from the modified
Einstein equations in the bulk. As a side observation we find that the
gravitational Chern-Simons functional has interesting conformal properties. It
is invariant under conformal transformations. Moreover, its metric variation
produces conformal tensor which is a generalization of the Cotton tensor to
dimension . We calculate the modification of the holographic
stress-energy tensor that is due to the Chern-Simons term and use the bulk
Einstein equations to find its divergence and thus reproduce the gravitational
anomaly. Explicit calculation of the anomaly is carried out in dimensions
and . The result of the holographic calculation is compared with that of
the descent method and agreement is found. The gravitational Chern-Simons term
originates by Kaluza-Klein mechanism from a one-loop modification of M-theory
action. This modification is discussed in the context of the gravitational
anomaly in six-dimensional theory. The agreement with earlier
conjectured anomaly is found.Comment: 24 pages, Latex; presentation re-structured, new references adde