48 research outputs found
Explicit formulas for GJMS-operators and -curvatures
We describe GJMS-operators as linear combinations of compositions of natural
second-order differential operators. These are defined in terms of
Poincar\'e-Einstein metrics and renormalized volume coefficients. As special
cases, we find explicit formulas for conformally covariant third and fourth
powers of the Laplacian. Moreover, we prove related formulas for all Branson's
-curvatures. The results settle and refine conjectural statements in earlier
works. The proofs rest on the theory of residue families.Comment: 84 pages, revised argument in proof of Theorem 3.1, corrected typo
Q-Curvature, Spectral Invariants, and Representation Theory
We give an introductory account of functional determinants of elliptic operators on manifolds and Polyakov-type formulas for their infinitesimal and finite conformal variations. We relate this to extremal problems and to the Q-curvature on even-dimensional conformal manifolds. The exposition is self-contained, in the sense of giving references sufficient to allow the reader to work through all details
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
Translation to Bundle Operators
We give explicit formulas for conformally invariant operators with leading term an m-th power of Laplacian on the product of spheres with the natural pseudo-Riemannian product metric for all m
Positive mass theorem for the Paneitz-Branson operator
We prove that under suitable assumptions, the constant term in the Green
function of the Paneitz-Branson operator on a compact Riemannian manifold
is positive unless is conformally diffeomophic to the standard
sphere. The proof is inspired by the positive mass theorem on spin manifolds by
Ammann-Humbert.Comment: 7 page
Boundary dynamics and multiple reflection expansion for Robin boundary conditions
In the presence of a boundary interaction, Neumann boundary conditions should
be modified to contain a function S of the boundary fields: (\nabla_N +S)\phi
=0. Information on quantum boundary dynamics is then encoded in the
-dependent part of the effective action. In the present paper we extend the
multiple reflection expansion method to the Robin boundary conditions mentioned
above, and calculate the heat kernel and the effective action (i) for constant
S, (ii) to the order S^2 with an arbitrary number of tangential derivatives.
Some applications to symmetry breaking effects, tachyon condensation and brane
world are briefly discussed.Comment: latex, 22 pages, no figure
Worldline approach to quantum field theories on flat manifolds with boundaries
We study a worldline approach to quantum field theories on flat manifolds
with boundaries. We consider the concrete case of a scalar field propagating on
R_+ x R^{D-1} which leads us to study the associated heat kernel through a one
dimensional (worldline) path integral. To calculate the latter we map it onto
an auxiliary path integral on the full R^D using an image charge. The main
technical difficulty lies in the fact that a smooth potential on R_+ x R^{D-1}
extends to a potential which generically fails to be smooth on R^D. This
implies that standard perturbative methods fail and must be improved. We
propose a method to deal with this situation. As a result we recover the known
heat kernel coefficients on a flat manifold with geodesic boundary, and compute
two additional ones, A_3 and A_{7/2}. The calculation becomes sensibly harder
as the perturbative order increases, and we are able to identify the complete
A_{7/2} with the help of a suitable toy model. Our findings show that the
worldline approach is viable on manifolds with boundaries. Certainly, it would
be desirable to improve our method of implementing the worldline approach to
further simplify the perturbative calculations that arise in the presence of
non-smooth potentials.Comment: 19 pages, 6 figures. Minor rephrasing of a few sentences, references
added. Version accepted by JHE
Spectral action for torsion with and without boundaries
We derive a commutative spectral triple and study the spectral action for a
rather general geometric setting which includes the (skew-symmetric) torsion
and the chiral bag conditions on the boundary. The spectral action splits into
bulk and boundary parts. In the bulk, we clarify certain issues of the previous
calculations, show that many terms in fact cancel out, and demonstrate that
this cancellation is a result of the chiral symmetry of spectral action. On the
boundary, we calculate several leading terms in the expansion of spectral
action in four dimensions for vanishing chiral parameter of the
boundary conditions, and show that is a critical point of the action
in any dimension and at all orders of the expansion.Comment: 16 pages, references adde
Zeta function determinant of the Laplace operator on the -dimensional ball
We present a direct approach for the calculation of functional determinants
of the Laplace operator on balls. Dirichlet and Robin boundary conditions are
considered. Using this approach, formulas for any value of the dimension, ,
of the ball, can be obtained quite easily. Explicit results are presented here
for dimensions and .Comment: 22 pages, one figure appended as uuencoded postscript fil
Twistor geometry of a pair of second order ODEs
We discuss the twistor correspondence between path geometries in three
dimensions with vanishing Wilczynski invariants and anti-self-dual conformal
structures of signature . We show how to reconstruct a system of ODEs
with vanishing invariants for a given conformal structure, highlighting the
Ricci-flat case in particular. Using this framework, we give a new derivation
of the Wilczynski invariants for a system of ODEs whose solution space is
endowed with a conformal structure. We explain how to reconstruct the conformal
structure directly from the integral curves, and present new examples of
systems of ODEs with point symmetry algebra of dimension four and greater which
give rise to anti--self--dual structures with conformal symmetry algebra of the
same dimension. Some of these examples are analogues of plane wave
space--times in General Relativity. Finally we discuss a variational principle
for twistor curves arising from the Finsler structures with scalar flag
curvature.Comment: Final version to appear in the Communications in Mathematical
Physics. The procedure of recovering a system of torsion-fee ODEs from the
heavenly equation has been clarified. The proof of Prop 7.1 has been
expanded. Dedicated to Mike Eastwood on the occasion of his 60th birthda