The geometric role of symmetry breaking in gravity

I suffered a blogging derailment at the end of 2011. Now I’m eager to get back on track, and I’ve got a bunch of stuff I want to write about.

Right at the end of the year, I wrote a short conference proceedings article:

The geometric role of symmetry breaking in gravity

The point of the paper is that the mathematics physicists are most familiar with because of "spontaneous symmetry breaking" plays a somewhat different role in gravitational physics, as a key ingredient of Cartan geometry.

Anyway, this paper is at a level of detail that could work just as well as a blog article, and it fits with what I’ve been talking about on this blog, so I’m putting a version here.

The success of spontaneous symmetry breaking in condensed matter and particle physics is famous. It explains second order phase transitions, superconductivity, the origin of mass via the Higgs mechanism, why there must be at least three generations of quarks, and so on. These applications are by now standard material for modern textbooks.

Much less famous is this: broken symmetry links the geometry of gauge fields to the geometry of spacetime. This, in my view, is the main role of symmetry breaking in gravity.

An early clue came in 1977, when MacDowell and Mansouri wrote down an action for general relativity using a connection for the (anti-) de Sitter group, but invariant only under the Lorentz group. Though their work was surely inspired by spontaneous symmetry breaking, it was Stelle and West who first made their action fully gauge invariant, breaking the symmetry dynamically using a field y locally valued in (anti-) de Sitter space.

Whether one breaks the symmetry dynamically or `by hand,’ the broken symmetry of the MacDowell–Mansouri connection plays the geometric role of relating spacetime geometry to the geometry of de Sitter space. This is best understood using Cartan geometry, a generalization of Riemannian geometry originating in the work of Élie Cartan, in which the geometry of tangent spaces is generalized—in this case, they become copies of de Sitter space. But to explain how this works, and how symmetry breaking is involved, it helps to back up further.

In geometry, inklings of spontaneous symmetry breaking date from at least 1872, in the work of Felix Klein. Ironically, to study a homogeneous space Y, with symmetry group G, one first breaks its perfect symmetry, artificially giving special significance to some point y\in Y. This gives an isomorphism Y \cong G/G_y as G-spaces, where G_y is the stabilizer of y, allowing algebraic study of the geometry. While Y itself has G symmetry, this description of it is only invariant under the subgroup G_y. Different algebraic descriptions of Y are however related in a G-equivariant way, since G_{gy} = gG_y g^{-1}.

This is strikingly similar to spontaneous symmetry breaking in physics. There, one really has a family of minimum-energy states, related in a G-equivariant way under the original gauge group G. Singling out any particular state |0\rangle as `the’ vacuum breaks symmetry to G_{|0\rangle}.

Cartan took Klein’s ideas a dramatic step further, getting an algebraic description of the geometry of a nonhomogeneous manifold M, by relating it `infinitesimally’ to one of Klein’s geometries Y. Just as Klein geometry uses broken symmetry to get an isomorphism Y\to G/G_y, in Cartan geometry, the broken symmetry in a G connection induces an isomorphism e\colon T_xM \to \mathfrak{g}/\mathfrak{g}_y for each tangent space. This is just the coframe field, also called the soldering form since identifying T_xM with \mathfrak{g}/\mathfrak{g}_y\cong T_yY effectively solders a copy of Y to M, at each point x. These copies of Y are then related via holonomy of the Cartan connection, which can be viewed as describing `rolling Y along M without slipping’ (see my paper on MM gravity, and also Appendix B of Sharpe’s book).

Physics history unfortunately skips over Cartan geometry. The Levi-Civita connection is adequate for the standard metric formulation of general relativity, and more general kinds of connections played no vital role in physics until some time later. When these eventually were introduced in Yang–Mill theory, they served a purpose far removed from spacetime geometry. Yang–Mills gauge fields are really just the principal connections of Ehresmann, who, building on Cartan’s ideas, liberated connections from their bondage to classical geometry. Ehresmann’s definition, which lacks the crucial `broken symmetry’ in Cartan’s original version, has just the flexibility needed for gauge fields in particle physics, which are concerned only with the geometry of an abstract `internal space’—a bundle over spacetime, rather than spacetime itself. On the other hand, Cartan’s original version is better when it comes to studying gravity.

Concretely, a Cartan geometry may be thought of as a connection on a principal bundle (with Ehresmann’s now standard definition) together with a section y of the associated Y bundle. As an example, let us write a version of the MacDowell–Mansouri action, using de Sitter space Y\cong G/H = \mathrm{SO}(4,1)/\mathrm{SO}(3,1) as the corresponding Klein geometry:

I[A,y]  = \int \mathrm{tr}(F_y \wedge {\star}_y   F_y)

The Cartan connection (A,y) consists of an \mathrm{SO}(4,1) connection A and a locally de Sitter-valued field y. F is the curvature of A, calculated by the usual formula, and F_y is its \mathfrak{g}_y-valued part, where \mathfrak{g}_y \cong \mathfrak{so}(3,1) \cong \Lambda^2\mathbb{R}^{3,1} has Hodge star operator \star_y.

I have described additional examples of Cartan-geometric formulations of various gravity theories elsewhere, and there are many more. But besides the diversity of specific examples, there are deep reasons that gravity, or any related "gauge theory of geometry," should be framed in the language of Cartan geometry. This is the subject of geometric "equivalence theorems."

In fact, if one believes semi-Riemannian metrics are fundamental in classical gravity, one is forced to accept Cartan connections as equally fundamental. The reason for this is Cartan’s method of equivalence, a process for proving that specified kinds of "raw geometric data" are equivalent to corresponding types of Cartan geometry. In the case of Riemannian geometry, solving the "equivalence problem" leads to the following theorem:

Theorem: A Riemannian metric determines a unique torsion-free Cartan geometry modeled on Euclidean space; conversely, a torsion-free Cartan geometry modeled on Euclidean space determines a Riemannian metric up to overall scale (on each connected component).

Proof: See Sharpe’s book.

Physically, the "overall scale" in the converse just represents a choice of length unit. One can also show that deformed versions (or "mutations") of Euclidean geometry, namely hyperbolic and spherical geometry, lead to Cartan geometries that carry the same information. The Lorentzian analogs of these results are the real reason de Sitter and anti de Sitter geometries work in MacDowell–Mansouri gravity.

Riemannian geometry is but one application of the equivalence method. There are analogous theorems, for example, in conformal geometry or Weyl geometry, relating various types of conformal structures to Cartan geometries that take the model Y to be an appropriate kind of homogeneous conformal model. Sharpe’s book contains some such theorems, and some significant work has been done on applications of conformal Cartan geometry—which often goes by the name "tractor calculus"—in physics. (See, e.g. this paper and references therein.)

For now, I just want to describe one more application of Cartan geometric thinking in gravitational theory. Besides spacetime geometry, one can also use Cartan’s ideas to describe the geometry of space.

Wheeler’s term "geometrodynamics" originally referred to the of evolution of spatial geometries in the metric sense. This has sometimes been contrasted with "connection dynamics" (see e.g. here or here). In light of the above equivalence theorem, however, there seems little point in establishing any technical distinction between geometrodynamics and connection dynamics, at least if we mean connections in the Cartan-geometric sense. The metric and connection pictures have their own advantages, but the equivalence theorem suggests we should be able to translate exactly between the two.

In recent work with Steffen Gielen, we take an explicitly Cartan-geometric approach to evolving spatial geometries. In this case, the symmetry breaking field y lives in 3d hyperbolic space \mathrm{SO}(3,1)/\mathrm{SO}(3), and can be interpretated as a field of observers, since the spacetime coframe field converts it into a unit timelike vector field. This can be dualized via the metric to a unit covector field, which we might call a field of co-observers. Just as observers determine a local time direction, co-observers determine local space directions, by taking their kernel. Our strategy in the Hamiltonian formulation is to fix a field of co-observers—the infinitesimal analog of picking a spacetime folitation—but let the field of observers be determined dynamically, as part of determining the metric.

The result is a model in which the observer field plays a two part symmetry breaking role: first splitting spacetime fields into spatial and temporal parts, but then also acting as the symmetry breaking field in Cartan geometry of space. This gives a Cartan-geometric Hamiltonian framework in which the spatial fields fit neatly and transparently into their spacetime counterparts and transform in an equivariant way under local Lorentz symmetry.

Thanks to the equivalence theorem, this may be viewed as a concrete link between connection dynamics and geometrodynamics in the original sense.

It is conceivable that gravity descends from a more fundamental theory with larger gauge group, and so fits into the tradition of symmetry breaking in gauge theories. Such ideas are clearly worth pursuing (see, e.g. papers by Percacci or Randono). At the same time, we should not ignore the lesson of Cartan geometry: broken symmetry is the means to establishing exact correspondence between geometric structures living on tangent spaces on one hand and connections on the other.

Thanks to John Baez, Julian Barbour, James Dolan, Andy Randono and Steffen Gielen for helpful discussions.

(By the way, someone might still remember that in the last post on this blog, I promised to explain some things. I still plan to do that later… )

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21 Responses to “The geometric role of symmetry breaking in gravity”

  1. Derek Wise Says:

    Over on Google+, where I had announced this blog article, Garrett Lisi and I got into a discussion about using Cartan geometry in unified theories, where you really only want to think of *part* of the connection as a Cartan connection. He agreed to let me move the conversation over here to the blog. So, here’s the beginning of what we had said:

    Garrett Lisi – Hey Derek,
    The main problem I have with Cartan geometry for model building is that one is stuck with a base manifold of the same dimension as some homogenous space. For example, if you’re working with G/H = SO(7,1) / SO(3,1)xSO(4), your base manifold is 16D. I’m wondering though… is there some way to choose a smaller distinguished subspace of G to model on? What if, for example, you first chose a SO(4,1) in SO(7,1), then modeled on a 4D SO(4,1)/SO(3,1) subspace? Looked at in a physics way, this might be equivalent to choosing a vev for a Higgs in the 4 rep of the SO(4). Any thoughts on that?

    Derek Wise – Hi Garrett!
    Suppose we’ve got groups G” < G' < G. I guess you want to do gauge theory with a G connection, but get Cartan geometry modeled on G'/G''. First, if we've got any chance of this working, I guess we need to be able to reduce the principal G bundle P not just down to a principal G' bundle P', but all the way down to a principal G'' bundle G''. But this is just a global issue, and I can always do it if the topology is simple enough…

    If the geometry G/G' happens to be reductive (for example, if it is a symmetric space) then pulling the connection A back to P', it decomposes into a G' connection A' and another piece. Pulling this A' back further to P'', we can think of it as a Cartan connection modeled on G'/G''.

    In your example, this should work fine. SO(7,1)/SO(4,1) isn't a symmetric space but it is reductive.

    Garrett Lisi – OK, cool, I’m feeling better about this Cartan geometry stuff for unified physics model building then, now that I can see how to get it down to a 4D base. (And yes, I am after a G connection over a 4D base.) That 16D corresponds to a 4 vector rep under the SO(3,1) times a 4 vector rep under the SO(4). The “choosing” of a SO(4,1) in SO(7,1) then corresponds to choosing some constant background value for the SO(4) vector, 4, consistent with the Higgs mechanism.

    So, next question: I see a Cartan geometry sometimes described as “allowing part of a group manifold to go wobbly.” More specifically, for a Cartan geometry built from H < G, one has a base, M, modeled on G/H, with a non-zero curvature of the G connection over M. I suppose it's the entire space, HxM, that then looks like G with the G/H part allowed to wobble. Does this sound right? And, if so, what does this mean for the G'' < G' < G construction, with M modeled on G'/G''? I suppose G'/G'' is a subspace of G, but if we're to think of G as built of M modeled on G'/G'' and other stuff, what's the other stuff? I suppose G'', and its centralizer in G, which we can call G''* (unless there's a better name) and… the other "half" of G/(G''xG''*) in some sense? Or does this no longer make sense?

    I suppose what I'm wondering now is if a base M modeled on G'/G'', with a G connection over it, can be thought of as G with its G'/G'' subspace allowed to go wobbly. My guess would be no, but I'm not clear on it yet. (Plus, it's late here!)

    Garrett Lisi – Ideally, I want to start with G and have a subspace M modeled on G’/G’ ‘ go wobbly, producing a bundle with fibers G’ ‘, and G’ ‘ *, and… the fiber left over that I’m not sure how to describe. It’s something like… (G/(G’ ‘xG’ ‘ *))/(G’/G’ ‘) but that’s not right. Maybe we should work with the normalizer rather than centralizer of G’ ‘?

    Derek Wise – You’re right: for Cartan geometry modeled on G/G’ there’s ultimately a principal G’ bundle P’, whose total space locally looks like H x M, and you can think of P (locally!) as a deformed version of G, which is still nice and homogenous in the “G’-directions”, but potentially lumpy in the “G/G’ directions”. This is a very useful way to think of Cartan connections in terms of the total space – my favorite way of thinking about Cartan connections, actually!

    I said before that when the geometry is reductive, you can split the G connection, pulled back to a G’ bundle, into a G’ connection and another piece. And now, I guess you’re asking, more or less, what that other piece is. It’s a section of an associated bundle with fiber Lie(G)/Lie(G’), where Lie(H) means the Lie algebra of the Lie group H. It’s the coframe field, as you probably know, in this case.

    In our case with the nested subgroups G”<G'<G, similar stuff works. In particular, you'll ultimately have a 1-form on a principal G'' bundle, and this will split into a bunch of pieces, depending on how Lie(G) splits into irreducible representations of G''. In the example you were mentioning, where
    G = SO(7,1)
    G' = SO(4,1)
    G'' = SO(3,1),
    your connection is originally a Lie(G)-valued 1-form, but having reduced to G'' using a symmetry-breaking field, Lie(G) will split into a direct sum of:
    Lie(G'') — here's where your "spin connection" for gravity lives
    Lie(G')/Lie(G'') — where the coframe lives
    Lie(G)/Lie(G') — where everything else has to go
    The last piece won't be irreducible, of course, though I'm too lazy to work out the irreps. I'm sure you know all this anyway!

    Note that all of the splitting is occurring at the level of Lie algebras, not groups, though. So, a reductive geometry G/G' is one in which you can think of Lie(G)/Lie(G') as a subrepresentation of the G' representation Lie(G), even though it is most naturally a quotient representation. But, there's generally no very good way to think of G/G' as a subspace of G. I don't quite see how you're trying to use centralizers and/or normalizers so far.

    Garrett Lisi – It is more clear (for me as well) to think in terms of parts of the Lie algebras and representation spaces, since I have more experience with those. But I would like to better understand this stuff in terms of Lie group manifolds and subspaces. For example, you say there’s no very good way to think of G/G’ as a subspace of G, but can’t we think of G, as a Lie group manifold, as G = G’ x (G/G’) ? I suppose that requires knowing the specifics of how G/G’ is embedded in G, which from the entire space picture means choosing a specific section of G?

    The other thread is this matter of how to describe fibers over G’/G’ ‘ in such a way that the resulting entire space looks locally like G.

    Derek Wise – In general, G won’t have the topology of G’ x (G/G’). Also, G – > G/G’ need not have any sections at all. :-) It’s a principal G’ bundle, so it has a section if and only if it is trivial.

    The principal G’ bundle will look “infinitesimally” like G. In other words, its tangent spaces will look like Lie(G), with the specific isomorphism from a tangent space to Lie(G) given by the connection itself. This is what you need for Cartan geometry.

  2. Garrett Says:

    (Continuing the G+ discussion here. Hmm, wonder if there’s latex? \psi )

    So, right, G won’t in general be G’ x (G/G’). But it can be, for some cases. What about the de Sitter spacetime case — isn’t M=SO(4,1)/SO(3,1) a subspace of SO(4,1)? Hmm, even if G can’t always be G’ x (G/G’), can’t G/G’ always be embedded in G?

    • Derek Wise Says:

      Yep, this is a \LaTeX-friendly site, as long as you know the trick, which you apparently do.

      G/G' can’t always be embedded in G. The simplest counterexample I can think of is the case of the circle, as a homogeneous space of G=\mathbb{R}, with stabilizer G'=\mathbb{Z}. There’s obviously no embedding of the circle into the line.

      Jim Dolan and I have talked quite a bit about cases where there is a “natural” section of the bundle G \to G/G'. A section of such a bundle is a (smooth or continuous) way to pick, for each point x in the base, a specific symmetry that takes my favorite point (the one stabilized by G') to the point x. I could spell out what “natural” means here, but essentially it means that the symmetry is some obvious canonical thing, like “translation along a geodesic” or something like that. We’ve always called spaces like this “strongly homogeneous”—I’m not sure if there’s some more standard name for them in the literature. Alas, de Sitter space is not strongly homogeneous.

      I’m not immediately sure whether there’s some “unnatural” embedding of SO(4,1)/SO(3,1) into SO(4,1), but I also don’t know what I would do with such an embedding if I had one, so I haven’t bothered trying to figure out if it works topologically or, harder, isometrically, with the Killing metric on SO(4,1).

  3. Garrett Says:

    Ah, cool. The mathematics here is even more interesting than I thought. On the physics side, I do know one thing we could do with an embedding of SO(4,1)/SO(3,1) into SO(4,1). We could then say that our physical spacetime is a subspace of SO(4,1), and of some larger group containing SO(4,1). The next step would be to build models to describe this in more detail, including the dynamics of how the larger group can go wobbly in this way, and the resulting structure.

    • Derek Wise Says:

      Personally, I think I’m happier with spacetime being a quotient of a group, rather than a subspace, and this is what generalizes to the principal bundle setting. Spacetime might not be naturally viewed as a subspace of a principal bundle over spacetime (i.e. the principal bundle may not have a section) but it’s always a quotient of the principal bundle by a group action.

      In a sense, this is what Cartan geometry is all about. The homogeneous geometry isn’t a natural subspace of the group, but a natural quotient space. Likewise, in Cartan geometry, we mimic this “infinitesimally” by making the tangent spaces look like quotient vector spaces of the corresponding Lie algebras.

      Mathematically, of course, it’s always interesting when it turns out you can think of something that’s most naturally a quotient-thing as being a sub-thing instead. For example, in Cartan geometry, a reductive geometry is one where I can think of the quotient \mathfrak{g}/\mathfrak{g'} as a subrepresentation of the G' representation \mathfrak{g}. But this doesn’t necessarily mean G/G' fits inside G in a natural way, even though it happens infinitesimally.

      I guess in the unified theory context you’re talking about, you actually do both—first take a subgroup of the full gauge group, and then take a quotient, to get down to spacetime.

  4. Garrett Says:

    Yes, for the SO(3,1) \subset SO(4,1) \subset SO(7,1) case it’s harder to justify a 4D base. In general, for physics model building, one uses simplicity as a guide (despite some popular models to the contrary). If we start with a Lie group, we get all sorts of cool geometric stuff: manifold, Killing vector fields and their Lie algebra, the Maurer-Cartan form, frame, metric, etc. Starting with that and allowing a subspace to go wavy seems natural to me, with the pullback of the Maurer-Cartan form now interpreted as the Cartan connection over the subspace and allowed to vary a little, producing a corresponding frame, metric, and connection over what was the subspace. Choosing a subspace which inherits structure this way seems more natural than choosing subgroups and constructing a symmetric space to then use as the model for a base which then is used as part of a related fiber bundle structure with connection. It’s true that the latter approach will work in cases the former won’t, but as a physicist I only really care about one case — even if I don’t know which one yet!

    But, whatever our aesthetic differences of opinion, the math here is interesting common ground. I suppose I should next figure out if and how to embed de Sitter in SO(4,1).

    • Derek Wise Says:

      As a physicist, I only really care about one case—even if I don’t know which one yet!

      That’s a nice quote. :-) I don’t know if it’s necessarily true, but it does nicely encapsulate a common outlook that has kept physics—or more precisely “fundamental physics”—moving forward. (Even though different people have different senses of “forward.”

      I don’t think it’s too hard to justify the 4d base for the case where we start out with SO(7,1). It’s just that I think part of the process naturally has to do with sub-things and part has to do with quotient-things. To me, both ideas are simple and natural. What seems unnatural is trying to force something that naturally wants to be a quotient-thing to be a sub-thing instead.

      On the other hand, I’m not saying I don’t like your idea here. It would be interesting to try viewing spacetime as sitting inside the de Sitter group, and getting different geometries by changing the embedding. This would mean viewing most of the interesting aspects of spacetime geometry as “extrinsic” geometry of this de Sitter “brane” in SO(4,1). I’m not sure how much of the geometry of general relativity could potentially be captured in this way.

      • Garrett Says:

        Well, a problem with the attitude I expressed in that quote is that people tend to form strong opinions about which case is the important one, which doesn’t help progress much if it’s wrong. So I try to be open.

        It does look like de Sitter can be embedded in SO(4,1), starting from an Iwasawa decomposition, as nicely described here:
        It looks to be pretty interesting… Half of de Sitter is embedded in the identity component of SO(4,1)? Hmm, wonder which half. Is it a half that can be covered by a metric like
        dt^2 - e^{2t/ \alpha} dx^2
        ? Because I’m unjustifiably fond of that one. (Sometimes I wonder if I’m little more than a collection of biases.) (No need to answer, I’m just thinking out loud and will check.)

        OK, so if we have an embedding, one does wonder about other embeddings. Now… you expressed the idea that one might get interesting geometry from different embeddings. But, let me ask something: The curvature 2-form of the Maurer-Cartan form on the Lie group G vanishes. And, as a Lie(G) valued form, the pullback of the Maurer-Cartan form to an embedded manifold is going to resemble the Cartan connection, but with vanishing curvature (this is the part I’m not sure about, is it true?) because it equals the pullback of the curvature over G. Ah, but you mentioned extrinsic curvature, so I guess that’s different and what you’re suggesting.

        But I’m interested in the Cartan connection and its curvature, so it’s no fun if it vanishes. But I do think a de Sitter spacetime embedded in SO(4,1), and hence in SO(7,1) and larger groups, may be a great place to start as a model, as one could then naturally start with all the nice Lie group structure and geometry, pull it back to the embedded subspace, and then describe this structure going wobbly via “extended” Cartan geometry. This, as you informed me, won’t work for every Cartan geometry for arbitrary groups, but since de Sitter embeds in this case, it may work for the one that matters! What do you think?

        • Derek Wise Says:

          Well, a problem with the attitude I expressed in that quote is that people tend to form strong opinions about which case is the important one, which doesn’t help progress much if it’s wrong. So I try to be open.

          Right. I agree.

          Now… you expressed the idea that one might get interesting geometry from different embeddings.

          I was actually only trying to interpret what you had in mind, and apparently jumping to the wrong conclusion.

          You’re right that if you think of the Maurer-Cartan form (a Lie(G)-valued 1-form on G) as a connection on the trivial G-bundle over G, then it is “flat”. But, I wasn’t thinking of just pulling back the Maurer-Cartan form and thinking of that as a Cartan connection on your embedded copy of G/H. Rather, I was thinking of the Maurer Cartan form as a coframe field, giving the metric on G by pulling back the Killing form. Then, given an embedding G/H –> G, if such exists, we can pull back again to get the metric on G/H. We could imagine deforming the embedding, thus deforming the metric. In the cases we’re thinking about, there’s essentially a unique torsion-free Cartan geometry determined by this metric. This would be given by some (locally) Lie(G)-valued 1-form, but I don’t know off hand how it relates to the pullback of the Maurer-Cartan form itself along the embedding.

          This, as you informed me, won’t work for every Cartan geometry for arbitrary groups, but since de Sitter embeds in this case, it may work for the one that matters! What do you think?

          I personally tend to think about geometric ideas that work more generally than this (except that particular examples of kinds of geometry of course can have nice special features that are good to take advantage of). So far, I’m not convinced this embedding idea helps me do something I couldn’t do better with the usual quotient-like constructions, but it’s fun to think about.

  5. Garrett Says:

    Checked, and yes, that metric does give the correct half of de Sitter space. Nice. Still interested in your thoughts on this stuff. Also… something remarkable that I hadn’t before realized: de Sitter isn’t just a subspace of SO(4,1), it’s a subgroup!

    • Derek Wise Says:

      Hmm. It would be very interesting if de Sitter space could be viewed as a subgroup of SO(4,1). I’m having doubts, though! Can you tell me what this group is, in some explicit way? If it’s true, then after I pick one point in de Sitter space to serve as the identity, there should be a nice geometric interpretation of what the product of two other points is.

  6. Garrett Says:

    Yes! Most of the details are in the paper I linked above. Basically, from the SO(4,1) generators, choose four:
    H, N_i
    that generate de Sitter when exponentiated, following an Iwasawa decomposition. The N_i are nilpotent, [N_i,N_j]=0, and are unit root vectors, [H,N_i]=N_i. Using a nice matrix representation for these, we can exponentiate to get explicit group elements,
    A = e^{-\frac{x^0}{\kappa} H}
    N = e^{\frac{1}{\kappa} x^i N_i}
    which multiply to give the explicit de Sitter group elements, NA(x). As you say, this gives a product of de Sitter points:
    NA(x^0,x^i) NA(x^{'0},x^{'i}) = NA(x^0+x^{'0},e^{-\frac{x^0}{\kappa}}x^{'i}+x^i)
    And, after differentiation, matrix multiplication, and hyperbolic trig identities, we get the Maurer-Cartan form on this subgroup,
    (NA)^-d(NA) = - \frac{1}{\kappa} dx^0 H + \frac{1}{\kappa} e^{\frac{x^0}{\kappa}} dx^i N_i
    recognizable as a de Sitter frame, at which point one can say “Holy crap!”

  7. Garrett Says:

    One can avoid matrix pain by using Clifford algebra instead, such as
    H = - \frac{1}{2} \gamma_{04}
    N_i = \gamma_{0i} + \gamma_{i4}
    giving the explicit group elements
    A = cosh(\frac{x^0}{2\kappa}) - 2 H sinh(\frac{x^0}{2\kappa})
    N = 1 + \frac{1}{\kappa} x^i N_i
    Or, hmm, probably better to use H = - \gamma_{04} to avoid some of those 2’s.

  8. Garrett Says:

    There is a problem though, that I’m not sure how to solve. Since the N_i are nilpotent, the Killing form pulled back to dS is degenerate. As far as I know, one can choose invariant metrics on Lie group manifolds other than the “nice” one, but I’m not familiar with how to justify that.

  9. Garrett Says:

    Thoughts, Derek?

  10. Garrett Says:

    Ah, I figured out what’s going on with the nilpotent N_i. Turns out it’s a feature and not a bug.

    • Derek Wise Says:

      Oh yeah? Does this mean you now have a nice geometric understanding of these nilpotent generators?

      (By the way, sorry for dropping this thread before — just busy with other things about Cartan geometry and I didn’t have much to contribute immediately in the direction you were going.)

      • Garrett Says:

        Yes. I was thinking the Maurer-Cartan form for this subgroup would be the de Sitter spacetime frame, but it’s not — it’s the de Sitter MacDowell-Mansouri connection. :)

        Ah, nice paper. Hmm, I’ll have to write up this de Sitter subgroup stuff.

      • Derek Wise Says:

        So, Garrett, let me see if I understand what you are saying. The Maurer-Cartan form is a 1-form on SO(4,1), with values in the Lie algebra of SO(4,1). If I’m reading correctly into your claim, you mean that there is a subgroup X of SO(4,1) such that if I pull back the Maurer-Cartan form to X, this 1-form is a (flat, obviously) Cartan connection where the model geometry is de Sitter space SO(4,1)/SO(3,1)?

        If I were trying to guess how this “X” works, I might start by breaking the Lie algebra into representations of SO(3,1):

        \mathfrak{so}(4,1) = \mathfrak{so}(3,1) \oplus \mathfrak{p}

        and then exponentiating in the \mathfrak{p} directions. This will locally give a 4d submanifold of SO(4,1) that looks like de Sitter space, but it sure won’t give me a subgroup, since \mathfrak{p} isn’t a subalgebra. And also, it’ll only work locally. So, I guess you must mean something different.

        • Garrett Says:

          Yes, all correct, except it is a different X. The Lie algebra generators we exponentiate are the H and the N_\pi, defined a few posts up.

        • Derek Wise Says:

          OK, right, you said that already. Sorry.

          Somehow along this other “X” I guess you’ve still got to break the tangent space of SO(4,1) into some particular copy of so(3,1) and a complement, so that the Cartan connection consists of an SO(3,1) connection plus a coframe field. What’s the “symmetry breaking” field in this case?

          Anyway, to really understand this, what I would still like is a geometric description of how, after choosing a point to serve as the identity, any point in de Sitter space gives me an isometry of de Sitter space, and a geometric understanding of why these particular isometries form a subgroup of all of the isometries. I suspect that a really good geometric description of this should not involve talking about the Lie algebras.

          But anyway, if you are writing stuff up about this de Sitter subgroup, and if you don’t already have such a geometric description, maybe we can figure it out after I read what you’ve got.

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