5,425 research outputs found
Automorphic forms: a physicist's survey
Motivated by issues in string theory and M-theory, we provide a pedestrian
introduction to automorphic forms and theta series, emphasizing examples rather
than generality.Comment: 22 pages, to appear in the Proceedings of Les Houches Winter School
``Frontiers in Number Theory, Physics and Geometry'', March 9-21, 2003; v2:
minor changes and clarifications, section 3.5 on pure spinors has been
rewritte
Conformal Orthosymplectic Quantum Mechanics
We present the most general curvature obstruction to the deformed parabolic
orthosymplectic symmetry subalgebra of the supersymmetric quantum mechanical
models recently developed to describe Lichnerowicz wave operators acting on
arbitrary tensors and spinors. For geometries possessing a
hypersurface-orthogonal homothetic conformal Killing vector we show that the
parabolic subalgebra is enhanced to a (curvature-obstructed) orthosymplectic
algebra. The new symmetries correspond to time-dependent conformal symmetries
of the underlying particle model. We also comment on generalizations germane to
three dimensions and new Chern--Simons-like particle models.Comment: 27 pages LaTe
Supersymmetric Quantum Mechanics and Super-Lichnerowicz Algebras
We present supersymmetric, curved space, quantum mechanical models based on
deformations of a parabolic subalgebra of osp(2p+2|Q). The dynamics are
governed by a spinning particle action whose internal coordinates are Lorentz
vectors labeled by the fundamental representation of osp(2p|Q). The states of
the theory are tensors or spinor-tensors on the curved background while
conserved charges correspond to the various differential geometry operators
acting on these. The Hamiltonian generalizes Lichnerowicz's wave/Laplace
operator. It is central, and the models are supersymmetric whenever the
background is a symmetric space, although there is an osp(2p|Q) superalgebra
for any curved background. The lowest purely bosonic example (2p,Q)=(2,0)
corresponds to a deformed Jacobi group and describes Lichnerowicz's original
algebra of constant curvature, differential geometric operators acting on
symmetric tensors. The case (2p,Q)=(0,1) is simply the {\cal N}=1 superparticle
whose supercharge amounts to the Dirac operator acting on spinors. The
(2p,Q)=(0,2) model is the {\cal N}=2 supersymmetric quantum mechanics
corresponding to differential forms. (This latter pair of models are
supersymmetric on any Riemannian background.) When Q is odd, the models apply
to spinor-tensors. The (2p,Q)=(2,1) model is distinguished by admitting a
central Lichnerowicz-Dirac operator when the background is constant curvature.
The new supersymmetric models are novel in that the Hamiltonian is not just a
square of super charges, but rather a sum of commutators of supercharges and
commutators of bosonic charges. These models and superalgebras are a very
useful tool for any study involving high rank tensors and spinors on manifolds.Comment: 39 pages, LaTeX, fixed typos, added refs, final version to appear in
CM
Conformal Invariance of Partially Massless Higher Spins
We show that there exist conformally invariant theories for all spins in d=4
de Sitter space, namely the partially massless models with higher derivative
gauge invariance under a scalar gauge parameter. This extends the catalog from
the two known gauge models -- Maxwell and partially massless spin 2 -- to all
spins.Comment: 10 pages Late
Renormalized Volume
We develop a universal distributional calculus for regulated volumes of
metrics that are singular along hypersurfaces. When the hypersurface is a
conformal infinity we give simple integrated distribution expressions for the
divergences and anomaly of the regulated volume functional valid for any choice
of regulator. For closed hypersurfaces or conformally compact geometries,
methods from a previously developed boundary calculus for conformally compact
manifolds can be applied to give explicit holographic formulae for the
divergences and anomaly expressed as hypersurface integrals over local
quantities (the method also extends to non-closed hypersurfaces). The resulting
anomaly does not depend on any particular choice of regulator, while the
regulator dependence of the divergences is precisely captured by these
formulae. Conformal hypersurface invariants can be studied by demanding that
the singular metric obey, smoothly and formally to a suitable order, a Yamabe
type problem with boundary data along the conformal infinity. We prove that the
volume anomaly for these singular Yamabe solutions is a conformally invariant
integral of a local Q-curvature that generalizes the Branson Q-curvature by
including data of the embedding. In each dimension this canonically defines a
higher dimensional generalization of the Willmore energy/rigid string action.
Recently Graham proved that the first variation of the volume anomaly recovers
the density obstructing smooth solutions to this singular Yamabe problem; we
give a new proof of this result employing our boundary calculus. Physical
applications of our results include studies of quantum corrections to
entanglement entropies.Comment: 31 pages, LaTeX, 5 figures, anomaly formula generalized to any bulk
geometry, improved discussion of hypersurfaces with boundar
Conformal hypersurface geometry via a boundary Loewner-Nirenberg-Yamabe problem
We develop a new approach to the conformal geometry of embedded hypersurfaces
by treating them as conformal infinities of conformally compact manifolds. This
involves the Loewner--Nirenberg-type problem of finding on the interior a
metric that is both conformally compact and of constant scalar curvature. Our
first result is an asymptotic solution to all orders. This involves log terms.
We show that the coefficient of the first of these is a new hypersurface
conformal invariant which generalises to higher dimensions the important
Willmore invariant of embedded surfaces. We call this the obstruction density.
For even dimensional hypersurfaces it is a fundamental curvature invariant. We
make the latter notion precise and show that the obstruction density and the
trace-free second fundamental form are, in a suitable sense, the only such
invariants. We also show that this obstruction to smoothness is a scalar
density analog of the Fefferman-Graham obstruction tensor for Poincare-Einstein
metrics; in part this is achieved by exploiting Bernstein-Gel'fand-Gel'fand
machinery. The solution to the constant scalar curvature problem provides a
smooth hypersurface defining density determined canonically by the embedding up
to the order of the obstruction. We give two key applications: the construction
of conformal hypersurface invariants and the construction of conformal
differential operators. In particular we present an infinite family of
conformal powers of the Laplacian determined canonically by the conformal
embedding. In general these depend non-trivially on the embedding and, in
contrast to Graham-Jennes-Mason-Sparling operators intrinsic to even
dimensional hypersurfaces, exist to all orders. These extrinsic conformal
Laplacian powers determine an explicit holographic formula for the obstruction
density.Comment: 37 pages, LaTeX, abridged version, functionals and explicit
invariants from previous version treated in greater detail in another postin
Metric projective geometry, BGG detour complexes and partially massless gauge theories
A projective geometry is an equivalence class of torsion free connections
sharing the same unparametrised geodesics; this is a basic structure for
understanding physical systems. Metric projective geometry is concerned with
the interaction of projective and pseudo-Riemannian geometry. We show that the
BGG machinery of projective geometry combines with structures known as
Yang-Mills detour complexes to produce a general tool for generating invariant
pseudo-Riemannian gauge theories. This produces (detour) complexes of
differential operators corresponding to gauge invariances and dynamics. We
show, as an application, that curved versions of these sequences give geometric
characterizations of the obstructions to propagation of higher spins in
Einstein spaces. Further, we show that projective BGG detour complexes generate
both gauge invariances and gauge invariant constraint systems for partially
massless models: the input for this machinery is a projectively invariant gauge
operator corresponding to the first operator of a certain BGG sequence. We also
connect this technology to the log-radial reduction method and extend the
latter to Einstein backgrounds.Comment: 30 pages, LaTe
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