From a nice post bu U Screiber:
Urs Schreiber
Shared publiclyYesterday 7:23 PM
Here is a mad title: M-Theory via HoTT and now here is why it might not be so mad after all.
11-dimensional supergravity is exceptional -- in the technical sense of exceptionality in mathematics. For instance its Lagrangian is controled by an exceptional 7-cocycle on a super-Lie 3- algebra which is itself the extension classified by an exceptional 4-cocycle on a super-Minkowski Lie algebra [0]. These numbers 4 and 7 correspond to the numbers 2 and 5 in the statement that in supergravity there is a 2-brane and a 5-brane.
Here is one more way in which 11d SuGra is rather remarkable: a solution to the SuGra equations of motion is precisely and already a higher super-Cartan geometry (locally modeled on extended 11d super-Minkowski spacetime) which is first-order integrable relative to the intrinsic torsion of super-Minkowski space.
This remarkable statement follows from a computation by Howe in 1997 [1]. Remarkable as it is, it doesn't seem to have been followed up much [2]. (But please let me know if I am missing some relevant references here.)
What's remarkable about this statement is that theories of gravity in lower dimension/with less supersymmetry have as solutions also certain Cartan geometries, but subject to further differential equations. Trying to rewrite these equations, or the Lagrangians that they follow from, in a suggestive way is an industry all in itself. For 11d SuGra the statement is simply: once one has a Cartan geometry of the given flavor, then there are no further equations, no further constraints.
This is curious, because the concept "infinitesimally integrable Cartan geometry relative to a torsion model space" is a concept that is "elementary" in the technical sense: this may be stated (and naturally so) in the internal language of elementary infinity-toposes, (and hence, with enough patience , in homotopy type theory). Or so I claim in [3].
I spoke about this last week at the MathPhys meeting in Srni [4]. And I will be speaking about it again this Thursday in Göttingen [5].
[0] http://ncatlab.org/schreiber/show/The+brane+bouquet
[1] http://ncatlab.org/nlab/show/torsion+constraints+in+supergravity#Howe97
[2] http://www.physicsoverflow.org/26352/11d-sugra-from-the-torsion-constraint
[3] http://ncatlab.org/schreiber/show/Obstruction+theory+for+parameterized+higher+WZW+terms
[4] http://www.math.muni.cz/~srni/
[5] http://www.uni-math.gwdg.de/grad/Schreiber_29.01.15.html
11-dimensional supergravity is exceptional -- in the technical sense of exceptionality in mathematics. For instance its Lagrangian is controled by an exceptional 7-cocycle on a super-Lie 3- algebra which is itself the extension classified by an exceptional 4-cocycle on a super-Minkowski Lie algebra [0]. These numbers 4 and 7 correspond to the numbers 2 and 5 in the statement that in supergravity there is a 2-brane and a 5-brane.
Here is one more way in which 11d SuGra is rather remarkable: a solution to the SuGra equations of motion is precisely and already a higher super-Cartan geometry (locally modeled on extended 11d super-Minkowski spacetime) which is first-order integrable relative to the intrinsic torsion of super-Minkowski space.
This remarkable statement follows from a computation by Howe in 1997 [1]. Remarkable as it is, it doesn't seem to have been followed up much [2]. (But please let me know if I am missing some relevant references here.)
What's remarkable about this statement is that theories of gravity in lower dimension/with less supersymmetry have as solutions also certain Cartan geometries, but subject to further differential equations. Trying to rewrite these equations, or the Lagrangians that they follow from, in a suggestive way is an industry all in itself. For 11d SuGra the statement is simply: once one has a Cartan geometry of the given flavor, then there are no further equations, no further constraints.
This is curious, because the concept "infinitesimally integrable Cartan geometry relative to a torsion model space" is a concept that is "elementary" in the technical sense: this may be stated (and naturally so) in the internal language of elementary infinity-toposes, (and hence, with enough patience , in homotopy type theory). Or so I claim in [3].
I spoke about this last week at the MathPhys meeting in Srni [4]. And I will be speaking about it again this Thursday in Göttingen [5].
[0] http://ncatlab.org/schreiber/show/The+brane+bouquet
[1] http://ncatlab.org/nlab/show/torsion+constraints+in+supergravity#Howe97
[2] http://www.physicsoverflow.org/26352/11d-sugra-from-the-torsion-constraint
[3] http://ncatlab.org/schreiber/show/Obstruction+theory+for+parameterized+higher+WZW+terms
[4] http://www.math.muni.cz/~srni/
[5] http://www.uni-math.gwdg.de/grad/Schreiber_29.01.15.html
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