53 research outputs found
The Calderon Projection: New Definition and Applications
We consider an arbitrary linear elliptic first--order differential operator A
with smooth coefficients acting between sections of complex vector bundles E,F
over a compact smooth manifold M with smooth boundary N. We describe the
analytic and topological properties of A in a collar neighborhood U of N and
analyze various ways of writing A|U in product form. We discuss the sectorial
projections of the corresponding tangential operator, construct various
invertible doubles of A by suitable local boundary conditions, obtain Poisson
type operators with different mapping properties, and provide a canonical
construction of the Calderon projection. We apply our construction to
generalize the Cobordism Theorem and to determine sufficient conditions for
continuous variation of the Calderon projection and of well--posed selfadjoint
Fredholm extensions under continuous variation of the data.Comment: 60 pages, 4 figures; revised version; index and list of notation
added; accepted for publication in J. Geom. Phys; v3 contains a few minor
correction
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
Instantons on cylindrical manifolds and stable bundles
Let Sigma be a smooth complex curve, and let S be the product ruled surface
Sigma \times CP^1. We prove a correspondence conjectured by Donaldson between
finite energy U(2)-instantons over the cylinder Sigma \times S^1 \times R, and
rank 2 holomorphic bundles over S whose restrictions to the divisors at
infinity are stable.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol5/paper24.abs.htm
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
On the rho invariant for manifolds with boundary
This article is a follow up of the previous article of the authors on the
analytic surgery of eta- and rho-invariants. We investigate in detail the
(Atiyah-Patodi-Singer)-rho-invariant for manifolds with boundary. First we
generalize the cut-and-paste formula to arbitrary boundary conditions. A priori
the rho-invariant is an invariant of the Riemannian structure and a
representation of the fundamental group. We show, however, that the dependence
on the metric is only very mild: it is independent of the metric in the
interior and the dependence on the metric on the boundary is only up to its
pseudo--isotopy class. Furthermore, we show that this cannot be improved: we
give explicit examples and a theoretical argument that different metrics on the
boundary in general give rise to different rho-invariants. Theoretically, this
follows from an interpretation of the exponentiated rho-invariant as a
covariantly constant section of a determinant bundle over a certain moduli
space of flat connections and Riemannian metrics on the boundary. Finally we
extend to manifolds with boundary the results of Farber-Levine-Weinberger
concerning the homotopy invariance of the rho-invariant and spectral flow of
the odd signature operator.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-22.abs.htm
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
Gauge Theoretic Invariants of, Dehn Surgeries on Knots
New methods for computing a variety of gauge theoretic invariants for
homology 3-spheres are developed. These invariants include the Chern-Simons
invariants, the spectral flow of the odd signature operator, and the rho
invariants of irreducible SU(2) representations. These quantities are
calculated for flat SU(2) connections on homology 3-spheres obtained by 1/k
Dehn surgery on (2,q) torus knots. The methods are then applied to compute the
SU(3) gauge theoretic Casson invariant (introduced in [H U Boden and C M
Herald, The SU(3) Casson invariant for integral homology 3--spheres, J. Diff.
Geom. 50 (1998) 147-206]) for Dehn surgeries on (2,q) torus knots for q=3,5,7
and 9.Comment: Version 3: minor corrections from version 2. Published by Geometry
and Topology at http://www.maths.warwick.ac.uk/gt/GTVol5/paper6.abs.htm
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
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