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)
What are now called Hitchin functionals are functions/functionals on spaces of differential n-forms on manifolds of dimension 6 or 7, for $n = 3$ and/or $n =4$. They are such that their critical points characterize holomorphic 3-forms of complex manifolds in dimension 6, as associative 3-forms of G2-manifolds in dimension 7.
Let $X$ be a closed oriented smooth manifold of dimension 6.
For $\Omega \in \wedge^3 \Gamma(T^* X)$ a differential 3-form on $X$, write
${\vert \lambda \Omega \vert} \in (\wedge^6 \Gamma(T^* X))^{\otimes 2}$
$\sqrt{\vert \lambda \Omega \vert} \in \wedge^6 \Gamma(T^* X)$.
The Hitchin function on 3-forms is the function
which sends 3-form to the integration of differential forms of this 6-form over $X$
Let $X$ be 7-dimensional.
Let $\Omega \in \wedge^3 \Gamma(T^* X)$ be a stable differential form?. This determines a Riemannian metric $\g_\Omega$ on $X$. Write $\star_\Omega$ for the corresponding Hodge star operator. The Hitchin functional now is the function that takes stable forms to
A differential 3-form $\Omega$ such that $\lambda(\Omega) \lt 0$ is a critical point of the above functional precisely if there is the structure of a complex manifold on $X$ such that $\Omega$ is the real part of a non-vanishing holomorphic 3-form.
This is (Hitchin, theorem 13).
(…)
The original articles are
Reviews with a perspective on the role of the functionals in physics/gravity/string theory/topological M-theory include
Last revised on January 9, 2013 at 05:44:54. See the history of this page for a list of all contributions to it.