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)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(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)
In the context of T-duality and in particular of differential T-duality one considers (as discussed in detail there) fiber products of two torus-fiber bundles together with a circle 2-bundle on this, (with connection).
In some disguise, this has been called $B_n$-geometry (Baraglia). The T-duality interpretation is made explicit in Bouwknegt
Here “$B_n$” refers to the special orthogonal group of the form $SO(n+1,n)$, which appears as the structure group of a generalized tangent bundle tensored with a line bundle (the Poincare line bundle of the T-duality correspondence).
We give the interpretation of $B_n$-geometry in higher differential geometry.
For $c_{conn},c'_{conn} \colon X \to \mathbf{B}U(1)$ modulating two circle principal bundles with conection, a differential T-duality structure is a choice of trivialization of their cup product class. From this we get the pasting diagram of homotopy pullbacks of smooth $\infty$-stacks
Here $\tau$ is the morphism that modulates the circle 2-bundle on the fiber product of the two circle bundles.
(…)
The term $B_n$-geometry was introduced in
A review is in
The relation to T-duality is made clear around slide 80 of
A discussion of the higher Lie theoretic aspects is in
Last revised on March 17, 2018 at 08:44:42. See the history of this page for a list of all contributions to it.