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
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
A variant of the idea of generalized complex geometry given by passing from generalization of complex geometry to generalization of exceptional geometry. Instead of by reduction of structure groups along inclusions like $O(d)\times O(d) \to O(d,d)$ it is controled by inclusions into split real forms of exceptional Lie groups.
This serves to neatly encode U-duality groups in supergravity as well as higher supersymmetry of supergravity compactifications. See also at exceptional field theory for more on this.
Compactification of 11-dimensional supergravity on a manifold of dimension 7 preserves $N = 1$ supersymmetry precisely if its generalized tangent bundle has G-structure for the inclusion
of the special unitary group in dimension 7 into the split real form of E7. This is shown in (Pacheco-Waldram 08).
One dimension down, compactification of 10-dimensional type II supergravity on a 6-manifold $X$ preserves $N = 2$ supersymmetry precisely if the generalized tangent bundle $T X \otimes T^* X$ in the NS-NS sector admits G-structure for the inclusion
This is reviewed in (GLSW, section 2).
Survey slides include
Reviewes include
Daniel Persson, Arithmetic and Hyperbolic Structures in String Theory (arXiv:1001.3154)
Nassiba Tabti, Kac-Moody algebraic structures in supergravity theories (arXiv:0910.1444)
Original articles include
K. Koepsell, Hermann Nicolai, Henning Samtleben, An exceptional geometry for d=11 supergravity?, Class.Quant.Grav.17:3689-3702,2000 (arXiv:hep-th/0006034)
Chris Hull, Generalised Geometry for M-Theory, JHEP 0707:079 (2007) (arXiv:hep-th/0701203)
Paulo Pires Pacheco, Daniel Waldram, M-theory, exceptional generalised geometry and superpotentials, JHEP 0809:123,2008 (arXiv:0804.1362)
Mariana Graña, Jan Louis, Aaron Sim, Daniel Waldram, $E_{7(7)}$ formulation of $N=2$ backgrounds (arXiv:0904.2333)
G. Aldazabala, E. Andrésb, P. Cámarac, Mariana Graña, U-dual fluxes and generalized geometry, JHEP 1011:083,2010 (arXiv:1007.5509)
Mariana Graña, Francesco Orsi, $N=1$ vacua in Exceptional Generalized Geometry (arXiv:1105.4855)
Mariana Graña, Francesco Orsi, N=2 vacua in Generalized Geometry, (arXiv:1207.3004)
André Coimbra, Charles Strickland-Constable, Daniel Waldram, $E_{d(d)} \times \mathbb{R}^+$ Generalised Geometry, Connections and M theory (arXiv:1112.3989)
E6,E7, E8-geometry is discussed in
Christian Hillmann, Generalized E(7(7)) coset dynamics and D=11 supergravity, JHEP 0903 (2009) 135 (arXiv:0901.1581)
Hadi Godazgar, Mahdi Godazgar, Hermann Nicolai, Generalised geometry from the ground up (arXiv:1307.8295)
Olaf Hohm, Henning Samtleben, Exceptional Form of $D=11$ Supergravity, Phys. Rev. Lett. 111, 231601 (2013) (arXiv:1308.1673)
(see also at 3d supergravity – possible gaugings).
The E10-geometry of 11-dimensional supergravity compactified to the line is discussed in
The E11-geometry of 11-dimensional supergravity compactified to the point is discussed in
The generalized-U-duality+diffeomorphism invariance in 11d is discussed in
For the worldvolume theory of the M5-brane this is discussed in
Relation to Borcherds superalgebras is surveyed and discussed in