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Atrium > Mathematics, Physics & Philosophy: Category theoretic setting for Klein geometry

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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 24th 2011

    I was wondering what the best categorical setting for Klein geometry might be. You could argue that it’s just the 2-category of (Lie) groups, where monomorphisms are Klein geometries. 2-morphisms are useful for relating conjugate subgroups, i.e., figures of the same kind in a geometry.

  2. You have probably in mind to define Klein geometries as the inclusion H↪GH\hookrightarrow G of a closed (Lie) subgroup. But you could choose the projection G→G/HG\to G/H either which is a HH-principal bundle which suggests considering appropriate stacks. Moreover the homogenous space G/HG/H need not be a group and thus I am not aware of the purposes for which the Hilsum-Skandallis 2-category of Lie groupoids would be a sensible setting for klein geometries. As I understand it, the main point in which a Klein geometry (locally) might differ from an orbifold is the set of conditions imposed on the action.This suggests the techniques and objects (e.g. geometric stacks) used in the study of the latter.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 27th 2011

    Thanks for this. I’m caught up in the festive season still. But back to work soon.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 28th 2011

    There are some comments rouhgly along the lines that Stephan mentions at higher Klein geometry.

    I expect a general statement holds true here: the general theory will certainly want to live in the context of stacks on smooth manifolds. But for certain classes of applications one may want to restrict attention to more specific models.

    What are the central motivating applications of Klein geometry that we want to better understand by embedding them in a good theoretical context?

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 29th 2011

    What are the central motivating applications of Klein geometry that we want to better understand by embedding them in a good theoretical context?

    The original motivation was to understand the relationships between existing geometries: Euclidean, affine, elliptic, hyperbolic, projective; to understand how a more restrictive geometry can be seen as a subgroup of symmetries of the broader geometry which fix a certain figure (line at infinity, etc.). Klein’s own writings are worth reading here. Then there’s Coxeter’s understanding projective duality as an outer automorphism on PGL(V)PGL(V).

    But then there’s much more. Have you seen Kisil’s recent programme Erlangen Programme at Large: An Overview?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 29th 2011

    Have you seen Kisil’s recent programme Erlangen Programme at Large: An Overview?

    I had not. Thanks for the link. I have added it to Klein geometry and to Erlangen program.

    I briefly glanced at the sections related to quantum mechanics. I did not spend enough time with the text, but right now I am not sure what about these sections is “Erlangen program” as opposed to the standard story. But maybe what now is the standard story once was a visionary program?

    What are the central motivating applications of Klein geometry that we want to better understand by embedding them in a good theoretical context?

    The original motivation was to understand the relationships between existing geometries: Euclidean, affine, elliptic, hyperbolic, projective; to understand how a more restrictive geometry can be seen as a subgroup of symmetries of the broader geometry which fix a certain figure (line at infinity, etc.). Klei

    Okay, sure. But do you have some more concrete questions in mind that we could try to usefully apply some higher topos theory to? I’d enjoy to, but I am not sure which examples motivate you.

    There is one class of examples of (higher) Klein geometries that I know the full relevance of: these are the (higher) Klein super-geometries given by the inclusion of the super-Lorentz group into things like the supergravity 3-group or the supergravity 6-group, etc. Their higher Cartan geometry is 11-dimensional supergravity and its descendants.

    This is the beginning of a vast story. But even here, I am not sure exactly what the incantation “Erlangen program” can conjure up that is not already present by standard means.

    But I’d be happy to have us all toy around with these gadgets more here. Lots of fun things to be found there.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 30th 2011

    Ok, so I agree the incantation “Erlangen program” doesn’t assist us much. Why this has cropped up now is that I’ve been invited to a workshop at Oberwolfach on Explicit Versus Tacit Knowledge in Mathematics in a week’s time. I thought I’d take the opportunity to talk about my experience of categorifying the Erlangen program. In that many attending are historians of mathematics, there’s at least a decent chance that they’ll know a lot about Klein’s work. If you look at the abstract for the meeting, you’ll see that it calls in part for us to look at tacit knowledge manifesting today online:

    In contemporary mathematics, blogs and Wikis – the most famous probably being Terence Tao’s – currently provide an extended form of oral culture in which less formal, formerly tacit approaches are written down and opened to a broad mathematical public according to shifting and variable rules.

    So I thought I’d reflect on what happened in the Klein 2-geometry postings.

    Here are some issues:

    • I think I believed that the process would be quite straightforward. Perhaps we could say that it ought to have been - there’s nothing mysterious about quotienting a 2-group by a sub-2-group, or n-groups in general.
    • But then I think I expected Klein 2-geometry to give us a range of examples, some we might have already known. We never seemed to find 2-geometries that were obvious parallels to the rigid Euclidean or hyperbolic geometry of 1-geometry.
    • John liked to bundle together the Erlangen program with Galois theory, as in this lecture. So why is there not a 2-Galois theory?
    • What kinds of thing should we be looking for if as you say “This is the beginning of a vast story… Lots of fun things to be found there”?
    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeDec 30th 2011
    • (edited Dec 30th 2011)

    I think I believed that the process would be quite straightforward.

    Yes, that’s what I would think, too. As described at higher Klein geometry, the basic idea is very simple. A simple (but interesting) special case of what I’d call the general theory of principal infinity-bundles in a cohesive ∞\infty-topos.

    We never seemed to find 2-geometries that were obvious parallels to the rigid Euclidean or hyperbolic geometry of 1-geometry.

    Those nn-supergeometries that I mentioned, are of this form. (Remember that Poincaré geometry is Euclidean geometry for Lorentzian signature). For n=2n = 2 the relevant example to look at is the super-2-group that John Huerta wrote his thesis about. With the sub-2-group "Lorentz group times shifted linear group" this is a super Klein 2-geometry for the Lorentzian geometry appearing in 10-dimensional supergravity.

    So why is there not a 2-Galois theory?

    There is. In my writeup this is the content of section 2.3.9. Some of this is also on the nnLab here.

    What kinds of thing should we be looking for if as you say "This is the beginning of a vast story… Lots of fun things to be found there"?

    Generally, 11-dimensional supergravity is rich. It is an exceptional structure sitting like a focal point in the space of differential geometric structures. From it originate a number of whole subfields of formal theoretical physics.

    Here "exceptional" is as in “exceptional simple Lie group E 8E_8”. The next higher Kac-Moody “groups” E 9E_9, E 10E_{10} and E 11E_{11} have been argued to to encode all of 11d supergravity together with its UV-completion.

    What D’Auria-Fré have secretly shown here (not in these words, of course) is that 11-d sugra is all about the higher Cartan supergeometry of the supergravity Lie 3-algebra/supergravity Lie 6-algebra, which in turn exists due to the finitely many exceptional cocycles on the super Poincaré Lie algebra.

    But, while many people are excited about 11-dimensional supergravity, the D’auria-Fré insight has not propagated far. Their textbook is out of print, and the community is unaware of what the crucial point of these books is. Even the authors don’t really seem to appreciate what they have found. So there are lots of things to be investigated here.

    For instance this aspect, which makes the connection to higher Galois theory, too (section 4.3.2.2):

    To study the higher Cartan geometry of the supergravity Lie 3-algebra, it helps to study its automorphism super Lie 4-algebra. One finds that its degree-0 part is the “polyvector extension” of the super Poincaré Lie algebra that goes by the name "M-theory algebra". This plays a big role in the investigation of 11-d sugra. But it is only the degree-0 part of a higher super Lie algebra that nobody ever considered. From higher Galois theory/higher bundle theory we know that this is the structure Lie 4-algebra of the super 3-gerbe of 11d sugra instanton moduli. Studying this will be of utmost interest.

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 31st 2011

    Thanks, that’s very helpful.

    Can you turn your higher Galois theory into theorems in number theory? We’ve had Minhyong Kim telling us at the Cafe about how not all of the information from the nonabelian fundamental group has been extracted. I wonder how higher homotopy might feature.

    There’s a useful account in the MathOverflow answer mentioned here. It would be good to see what sense can be made of his thought that there’s something TQFT-ish in the air.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeDec 31st 2011

    Hi David,

    in that MathOverflow post that you point to, Minhyong Kim indicates that the correct definition of the fundamental group in the context of covering theory/Galois theory is the automorphism group of the “fiber functor” that sends coverings to their fibers, the one that in the algebraic context is denoted

    π 1 et \pi_1^{et}

    due to Grothendieck. This definition, or rather the statement that this definition indeed gives the correct fundamental group, is what the disucssion in that section 2.3.9 generalizes to higher Galois theory.

    It would be good to see what sense can be made of his thought that there’s something TQFT-ish in the air.

    I have to say that I am not following the aim of this.

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