exceptional generalized geometry



Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous *

          \array{ && id &\dashv& id \ && \vee && \vee \ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \ && \bot && \bot \ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }



          Lie theory, ∞-Lie theory

          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)×O(d)O(d,d)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.


          Higher supersymmetry

          Compactification of 11-dimensional supergravity on a manifold of dimension 7 preserves N=1N = 1 supersymmetry precisely if its generalized tangent bundle has G-structure for the inclusion

          SU(7)E 7(7) SU(7) \hookrightarrow E_{7(7)}

          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 XX preserves N=2N = 2 supersymmetry precisely if the generalized tangent bundle TXT *XT X \otimes T^* X in the NS-NS sector admits G-structure for the inclusion

          SU(3)×SU(3)O(6,6). SU(3) \times SU(3) \hookrightarrow O(6,6) \,.

          This is reviewed in (GLSW, section 2).


          • David Baraglia, Leibniz algebroids, twistings and exceptional generalized geometry, Journal of Geometry and Physics 62 (2012), pp. 903-934 (arXiv:1101.0856)

          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

          E6,E7, E8-geometry is discussed in

          (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

          • Machiko Hatsuda, Kiyoshi Kamimura, M5 algebra and SO(5,5)SO(5,5) duality (arXiv:1305.2258)

          Relation to Borcherds superalgebras is surveyed and discussed in

          • Jakob Palmkvist, Exceptional geometry and Borcherds superalgebras (arXiv:1507.08828)

          black branes in the exotic spacetime are discussed in

          The string- and membrane sigma-models on exceptional spacetime (the “exceptional sigma models”) are discussed in

          Last revised on April 24, 2018 at 11:57:59. See the history of this page for a list of all contributions to it.