724 research outputs found
Complexes of Discrete Distributional Differential Forms and their Homology Theory
Complexes of discrete distributional differential forms are introduced into
finite element exterior calculus. Thus we generalize a notion of Braess and
Sch\"oberl, originally studied for a posteriori error estimation. We construct
isomorphisms between the simplicial homology groups of the triangulation, the
discrete harmonic forms of the finite element complex, and the harmonic forms
of the distributional finite element complexes. As an application, we prove
that the complexes of finite element exterior calculus have cohomology groups
isomorphic to the de Rham cohomology, including the case of partial boundary
conditions. Poincar\'e-Friedrichs-type inequalities will be studied in a
subsequent contribution.Comment: revised preprint, 26 page
The Calabi complex and Killing sheaf cohomology
It has recently been noticed that the degeneracies of the Poisson bracket of
linearized gravity on constant curvature Lorentzian manifold can be described
in terms of the cohomologies of a certain complex of differential operators.
This complex was first introduced by Calabi and its cohomology is known to be
isomorphic to that of the (locally constant) sheaf of Killing vectors. We
review the structure of the Calabi complex in a novel way, with explicit
calculations based on representation theory of GL(n), and also some tools for
studying its cohomology in terms of of locally constant sheaves. We also
conjecture how these tools would adapt to linearized gravity on other
backgrounds and to other gauge theories. The presentation includes explicit
formulas for the differential operators in the Calabi complex, arguments for
its local exactness, discussion of generalized Poincar\'e duality, methods of
computing the cohomology of locally constant sheaves, and example calculations
of Killing sheaf cohomologies of some black hole and cosmological Lorentzian
manifolds.Comment: tikz-cd diagrams, 69 page
Localization over complex-analytic groupoids and conformal renormalization
We present a higher index theorem for a certain class of etale
one-dimensional complex-analytic groupoids. The novelty is the use of the local
anomaly formula established in a previous paper, which represents the bivariant
Chern character of a quasihomomorphism as the chiral anomaly associated to a
renormalized non-commutative chiral field theory. In the present situation the
geometry is non-metric and the corresponding field theory can be renormalized
in a purely conformal way, by exploiting the complex-analytic structure of the
groupoid only. The index formula is automatically localized at the automorphism
subset of the groupoid and involves a cap-product with the sum of two different
cyclic cocycles over the groupoid algebra. The first cocycle is a trace
involving a generalization of the Lefschetz numbers to higher-order fixed
points. The second cocycle is a non-commutative Todd class, constructed from
the modular automorphism group of the algebra.Comment: 38 pages. v2: some inconsistencies with the use of pseudogroups have
been fixe
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