synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Euler-Lagrange equation, de Donder-Weyl formalism?,
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
Given a Lorentzian manifold, then its causal structure is the information at each tangent space of the subspace of lightlike tangent vectors – the “light cones”. This information distinguishes timelike from the spacelike vectors. Hence if one thinks of the Lorentzian manifold as modelling a spacetime in physics, then this information encodes, at each point, the directions along which causal influences may propagate in this spacetime.
This causal stucture is closely related to the underlying conformal structure. One may also define a concept of a manifold with causal structure or causal manifold without reference to a concept of Lorentzian pseudo-Riemannian structure (Bannier 88, Rainer 99, Khavkine 12).
In the formalization of quantum field theory in terms of locally covariant AQFT a QFT over a background field of classical gravity is axiomatized as a causally local net of observables on a category of Lorentzian manifolds. Making more of the degrees of freedom of gravity become quantized themselves would mean to replace the latter with a category of manifolds with less structure than Lorentzian manifold structure. Since the causal structure is necessary to express “local net” at all, one idea is to consider local nets on a category of causal manifolds.
This idea is mentioned as motivation for developing concepts of causal manifolds for instance in Rainer 99, and in a more sophisticated version in Khavkine 12.
Ulrich Bannier, On generally covariant quantum field theory and generalized causal and dynamical structures, Communications in Mathematical Physics, 118(1):163–170, March 1988
Martin Rainer, Cones and causal structures on topological and differentiable manifolds (arXiv:gr-qc/9905106)
Renee Hoekzema, On the Topology of Lorentzian manifolds, Essay as a part of the 2010-2011 lecture on “Quantum Fields in curved spacetimes” by W.G. Unruh 2011 (pdf)
Igor Khavkine, Characteristics, Conal Geometry and Causality in Locally Covariant Field Theory (arXiv:1211.1914)
Last revised on January 20, 2021 at 07:19:43. See the history of this page for a list of all contributions to it.