36 research outputs found
Spectral triples and manifolds with boundary
We investigate manifolds with boundary in noncommutative geometry. Spectral
triples associated to a symmetric differential operator and a local boundary
condition are constructed. For a classical Dirac operator with a chiral
boundary condition, we show that there is no tadpole.Comment: 18 pages To appear in J. Funct. Ana
Quantum Gravity: Unification of Principles and Interactions, and Promises of Spectral Geometry
Quantum gravity was born as that branch of modern theoretical physics that tries to unify its guiding principles, i.e., quantum mechanics and general relativity. Nowadays it is providing new insight into the unification of all fundamental interactions, while giving rise to new developments in modern mathematics. It is however unclear whether it will ever become a falsifiable physical theory, since it deals with Planck-scale physics. Reviewing a wide range of spectral geometry from index theory to spectral triples, we hope to dismiss the general opinion that the mere mathematical complexity of the unification programme will obstruct that programme
Optimal eigenvalues estimate for the Dirac operator on domains with boundary
We give a lower bound for the eigenvalues of the Dirac operator on a compact
domain of a Riemannian spin manifold under the \MIT bag boundary condition.
The limiting case is characterized by the existence of an imaginary Killing
spinor.Comment: 10 page
A remark on the space of metrics having non-trivial harmonic spinors
Let M be a closed spin manifold of dimension congruent to 3 modulo 4. We give a simple proof of the fact that the space of metrics on M with invertible Dirac operator is either empty or it has infinitely many path components
On a spin conformal invariant on manifolds with boundary
On a n-dimensional connected compact manifold with non-empty boundary
equipped with a Riemannian metric, a spin structure and a chirality operator,
we study some properties of a spin conformal invariant defined from the first
eigenvalue of the Dirac operator under the chiral bag boundary condition. More
precisely, we show that we can derive a spinorial analogue of Aubin's
inequality.Comment: 26 page
Gravitational and axial anomalies for generalized Euclidean Taub-NUT metrics
The gravitational anomalies are investigated for generalized Euclidean
Taub-NUT metrics which admit hidden symmetries analogous to the Runge-Lenz
vector of the Kepler-type problem. In order to evaluate the axial anomalies,
the index of the Dirac operator for these metrics with the APS boundary
condition is computed. The role of the Killing-Yano tensors is discussed for
these two types of quantum anomalies.Comment: 23 page
The Dirac system on the Anti-de Sitter Universe
We investigate the global solutions of the Dirac equation on the
Anti-de-Sitter Universe. Since this space is not globally hyperbolic, the
Cauchy problem is not, {\it a priori}, well-posed. Nevertheless we can prove
that there exists unitary dynamics, but its uniqueness crucially depends on the
ratio beween the mass of the field and the cosmological constant
: it appears a critical value, , which plays a role
similar to the Breitenlohner-Freedman bound for the scalar fields. When
there exists a unique unitary dynamics. In opposite, for
the light fermions satisfying , we construct several asymptotic
conditions at infinity, such that the problem becomes well-posed. In all the
cases, the spectrum of the hamiltonian is discrete. We also prove a result of
equipartition of the energy.Comment: 33 page
A K-theoretical Invariant and Bifurcation for Homoclinics of Hamiltonian Systems
We revisit a K-theoretical invariant that was invented by the first author some years ago for studying multiparameter bifurcation of branches of critical points of functionals. Our main aim is to apply this invariant to investigate bifurcation of homoclinic solutions of families of Hamiltonian systems which are parametrised by tori
The Maslov index in weak symplectic functional analysis
We recall the Chernoff-Marsden definition of weak symplectic structure and
give a rigorous treatment of the functional analysis and geometry of weak
symplectic Banach spaces. We define the Maslov index of a continuous path of
Fredholm pairs of Lagrangian subspaces in continuously varying Banach spaces.
We derive basic properties of this Maslov index and emphasize the new features
appearing.Comment: 34 pages, 13 figures, 45 references, to appear in Ann Glob Anal Geom.
The final publication will be available at http://www.springerlink.com. arXiv
admin note: substantial text overlap with arXiv:math/040613