A few things that may be of interest:
- Survey articles prepared for the 2018 ICM proceedings are starting to appear on the arXiv, and Peter Scholze (who will be getting a Fields Medal in Rio) has put his on his web-site. His title is p-adic Geometry, and it gives an overview of the ground-breaking work he has been doing over the last few years. The last section tells us that
Currently, the author is trying to understand to what extent it might be true that the “universal” cohomology theory is given by a shtuka relative to Spec Z. It seems that this is a very fruitful philosophy.
For some background about that section, I’d recommend his talk at the 2015 Clay Math conference.
- The New Yorker has a very detailed and interesting profile of Jim Simons and what he is up to with the Flatiron Institute he is now funding here in New York. This new Institute is costing him \$80 million a year, characterized as “a lark” for someone with his assets. David Spergel is running the Center for Computational Astrophysics there, and doing a lot of hiring. When I wrote here about his characterization of multiverse research, his final comment about being able to speak freely because he had tenure left me wondering “wait, what about grants, jobs, etc.?”. From the New Yorker article, I realized that while having tenure may give you some ability to speak freely, having a guy with \$18.5 billion willing to write large checks for you gives you a lot more…
- I’ve just finished teaching a course this semester which concentrated on the formalism for describing geometry in terms of connections and curvature. From the point of view of physicists, this formalism should be of interest because it applies equally well to gauge theory and general relativity. I’d been starting to think again about what light this formalism might shed on how to think about these two subjects together, when last night I noticed a wonderful new article on the arXiv, Gravity and Unification: A review by Krasnov and Percacci.
This article is an extremely lucid and comprehensive survey of the sort of thing I was thinking about, which can be re-expressed as the question of trying to find, at the classical level, a formalism uniting the vector potentials/field strengths of the SM and the different possible fields used to describe geometry in GR. Some of this has a very long history, going back to the things Einstein was trying in his later years. There have been many different ideas that people have tried since then, and the survey article does a great job of both explaining these ideas, as well as indicating why they haven’t worked out.
A couple of the general ideas that have always fascinated me make an appearance in the article. One of these is that of what mathematicians call a “Cartan connection”, the idea that you should think of a geometry as locally looking like a quotient space G/H of two Lie groups. A version of this is known to physicists as the MacDowell-Mansouri formulation, which gets a detailed treatment in the article. Another is the idea of using the fact that the complexified orthogonal group in 4 dimensions breaks up into two pieces, sometimes thought of at the Lie algebra level as self-dual vs. anti-self-dual pieces under the Hodge star operation. A version of this idea is known as the Plebanski formulation, and this decomposition is behind the story of Ashtekar variables. These variable have played a crucial role in modern treatments of GR by Hamiltonian methods, as well as the quantization program of loop quantum gravity.
The focus of the article is on Lagrangian and classical field theory methods for studying these ideas. There’s relatively little about the Hamiltonian story, and also relatively little about the geometry of spinors, two topics that I suspect might provide additional needed insights. For anyone interested in thinking about non-string theory-based ideas about unification of the SM and gravity, there’s a wealth of ideas, references and history here to think about. Perhaps future progress on unification will come from some new breakthrough in this field that shows how to get around the problems identified clearly in this article.
- For surveys of recent work on quantization of gravity and discussion by experts, a good place to look is videos of talks at a recent conference held at the IHES. Videos available here include my fellow Princeton student Costas Bachas surveying the approach growing out of string theory in Holographic Dualities and Quantum Gravity, Carlo Rovelli the opposition in Current Quantum Gravity Theories, Experimental Evidence, Philosophical Implications, and an even-handed overview from Steven Carlip with Why We Need Quantum Gravity and Why We Don’t Have It. Also well-worth watching, both for the talk and the discussion, is Alain Connes on Why Four Dimensions and the Standard Model Coupled to Gravity.
Finally, for fans of Lenny Susskind’s introductory level books on theoretical physics, Andre Cabannes writes to tell me that the most recent volume (which I wrote about here) is being translated by him into French, to appear next year. He also has notes from the lectures on General Relativity, Cosmology, and Statistical Mechanics, for which no book form has yet appeared.
Update: For a detailed account of the event at NYU mentioned here, see this from Jerry Alper.
Bold statement there 😉
I agree with your assessment and sentiment, but some might think you have inside knowledge.
Note that Scholze cite Fargues’ ICM paper, La courbe, which doesn’t seem to be on the arXiv.
Scholze is the easy one, what about the others? Williamson probably, who else?
>>>Scholze is the easy one, what about the others? Williamson probably, who else?
Ciprian Manolescu?
Hugo Duminil-Copin?
See here : https://poll.pollcode.com/44839318
Please, unless you actually know who is getting a Fields medal and want to tell us, let’s leave that speculation game for another time.
Derek Wise wrote a great thesis on Cartan geometry and various formulations of gravity, including the MacDowell-Mansouri formulation – but also the more familiar Palatini formulation, where the SO(3,1) connection and the cotetrad field, which is locally an R^4-valued 1-form, fit together to form a Poincare group connection. What’s nice is that Derek explains it all very geometrically. His paper MacDowell-Mansouri gravity and Cartan geometry is probably the easiest place to start. One doesn’t need to understand Cartan geometry ahead of time: he explains it nicely using a picture of a hamster rolling in a hamster ball.
Thanks for reminding me of that John. That paper is a wonderfully clear explanation of Cartan geometry and how to formulate GR using it.
On the update: Alper’s article is a detailed account of how, uninvited, he tries to get closer to the speakers (before the discussion) and two members of the audience, just to show off and obtain some personal advice. And in addition presenting them rather unfavorably for not showing him the warmth he was expecting. I don’t know if he’ll report on the event itself, but the time spent on this article doesn’t tell anyone anything about it (except maybe how great he believes Nima is, yes, unrelated to the event).
The first paragraphs about today’s students in high profile universities is rather interesting though.
I read Alper’s piece and am pleased, though not surprised, to discover that Hossenfelder is not obviously susceptible to flattery. Speaking from experience, being a thorn in the side of academia is much easier, not to say effective, if you retain a foothold in it, so let us just hope that she finds a way to do that.
Reading Jerry Alper’s piece, I could not help thinking that the reason Peter had so much trouble remembering him was because during their three-hour interview Peter’s mind was too focused on keeping the conversation on topic to remember the interviewer.
tulpoeid,
I think you’re being unfair to Alper. This was an event aimed at the public, so he was certainly invited, and having the audience interact with the speakers was part of the goal there (the audience was not that large). He’s a fan of the speakers, it was a quite appropriate place for him to meet them and have a conversation. While (just like in the piece he wrote about talking to me a while ago) his writing style is very personal and largely about himself and how he experiences things, it is pretty much factually accurate. Yes, I didn’t immediately recognize him (he did prompt me quickly, probably I would have remembered who he was in a few moments, although I’ve gotten very bad at recalling names). Yes, the quote from me about Sabine was accurate. I was there for part of his conversation with her, and what he writes is consistent with what I remember.
Peter,
Thanks! My paper on MacDowell-Mansouri gravity seems to have helped a lot of people understand Cartan geometry and its relationship to gravity, and that makes me really happy!
Since you mentioned the Ashtekar approach in the original post here, I’ll point out that Steffen Gielen and I worked out the precise sense in which the real Ashtekar-Barbero formulation is a theory of evolving spatial Cartan connections, starting with our paper Spontaneously broken Lorentz symmetry for Hamiltonian gravity, and then with some of the geometric ideas explained further in Geometrodynamics and Lorentz symmetry.
This of course is not quite the same thing as the Plebanski-type symmetry breaking you mentioned. I’ve never worked out the precise sense in which the Plebanski formulation is a case of Cartan geometry (is it?!), but I think the above papers (and other work with Steffen Gielen) give the “right” way to understand real Ashtekar variables geometrically.
-Derek
Maybe I exaggerated my feeling, triggered by the negative-in-disguise portrayal of those who didn’t respond according to his expectation. (“Uninvited” was about the repeated attempts at professional cordiality.)
It may be helpful to notice that, in different terminology, Cartan geometry is fairly familiar to many physicsists, who may call it “first-order formulation of gravity” or similar and regularly use it when discussing gravity coupled to fermions.
An excellent but widely underappreciated (forgotten?) textbook all based on this perspective is “Supergravity and Superstrings – A Geometric Perspective“.
The authors don’t use the words “Cartan geometry”, but anyone who knows the subject will immdiately recognize that this is precisely their “geometric perspective” (working locally, of course).
While the aim of this textbook is to go further to super Cartan geometry and then to higher Cartan geometry, the entirely of part one (in volume one) is about standard gravity in terms of Cartan geometry.
So Cartan geometry is basically the same thing as the vielbein formalism? Then even I would be able to understand.
Yes. For some reason where in maths texts the term “Cartan geometry” became established, physics texts stick to Cartan’s original terminology (Cartan 22) and speak of the “Cartan moving frame method”.
Of course mathematicians have developed the theory with more generality and precision, a seminal textbook is Cap-Slovák 09.
Another terminology issue to be aware of is that many physicists who do say “Cartan geometry” are concerned with an independent issue, namely the question whether encoding the field of gravity in a “Cartan connection” = “vielbein + spin connection” instead of in a metric tensor (which is what Cartan geometry itself is about, and which is well known and uncontroversial) suggests ways to rewrite the Einstein-Hilbert Lagrangian in ways (e.g. Palatini-Cartan-Holst form) that are then often subject to much speculation about a deeper nature of gravity.
In view of this it is remarkable that, while the on-shell equivalence of the Einstein-Hilbert action to the Palatini-Cartan-Holst action was long known, the equivalence of the two Lagrangian field theories on the level of phase spaces was established only this year, by Cattaneo-Schiavina 17.
Derek and Urs,
Thanks for the references!
Thomas Larsson,
I’d say you have a Cartan geometry when you have a Cartan connection. You can do this generally for a pair H,G of Lie groups, H a subgroup of G, identifying the tangent space of your manifold with Lie G/Lie H. The case of GR is G=Poincare, H=Lorentz. In this case the spin connection and vielbein together are the components of the Cartan connection, with the curvature of this connection having components the usual curvature of the spin connection, and the torsion.
Urs, I disagree — Cartan geometry is not just the same things as the method of moving frames. Frames are an important ingredient in Cartan geometry, and in certain cases (reductive geometries) the Cartan connection can be split into a coframe field and a connection for the stabilizer group. This special case explains the geometry behind first order formulations of gravity. But, Cartan geometry is much more general (and frankly much more “geometric”) than the usual physics understanding of first order gravity. It’s about generalizing the whole of Klein’s Erlangen Program to the differential setting. It’s also about solving the “equivalence problem” relating “raw” geometry (defined in terms of, say, smoothly varying structures on tangent spaces) to Cartan connections with special choice of groups and conditions on curvature. Sharpe’s book has some great examples of this, for example, relating conformal geometry to Cartan connections based on either Weyl or Möbius models.
Derek, careful with suggesting that theoretical physicists don’t understand these phenomena only because they use different words than you do. This attitude will backfire.
If you look at the first part of “Supergravity and Superstrings – A geometric perspective” that i keep recommending, you’ll see that it is exactly the geometric perspective of globalizing Klein’s geometry that drives the development of their concepts.
Of course they don’t say the words “Klein geometry” or “Erlanger program” (we have to distinguish between names and the mathematical reality that they refer to) but they start by considering the homogenous coset spaces that are the Klein geometry hallmark and then find their crucial technique by globalizing these. In their re-invention of Cartan geometry this way, they came up with interesting ways of thinking about these structures: For instance they say “soft group manifold” for the differential form structure which globalizes Maurer-Cartan forms on Kleinian cosets to Cartan geometries. The terminology didn’t catch on, but it reflects precisely the understanding that Cartan geometry is a “softening” of Kleinian geometry
Also, these authors are well aware that this is more general than just the case of Minkowski = Poincaré/Lorentz: In that textbook they discuss also the dS and AdS case, and, crucially, they use all this only as the stepping stone to do super-Cartan geometry, which comes about from considering cosets of the super-Poincaré and super-(anti)de Sitter groups.
This perspective keeps being developed to great generality by physicists: In approaches to make U-duality geometric one considers Cartan geometry over ever larger cosets, culminating in something like Cartan geometry modeled on E11 modulo its maximal compact sub-group (or sub-thing).
So once you allow for the fact that not every physicists who studies Cartan geometry uses the same language as you do, you’ll find that the Cartan geometry physicists have been and are studying is extremely general and powerful. It serves to pay attention to that.
(Not the least, Cartan geometry is pretty much the only way to do supergravity — noticing that the only way to define local model super-spacetimes is as coset of supergroups — and that this is what physicists are doing has been particularly highlighted by John Lott 01.)
While conformal Cartan geometry is not discussed in this particular textbook, they do discuss this in their published articles. (But I won’t point to the arXiv now, not to distract you from opening their fantastic textbook!)
Urs,
I certainly don’t intend to suggest that physicists don’t understand something because they don’t phrase it in the language of mathematicians, or vice-versa. And I agree that there’s a lot of great work in the physics literature related to Cartan geometry, including the D’Auria-Fré formulation, a bunch of work on first order gravity, and the tractor calculus literature, to name a few. But, when you answer the question “So Cartan geometry is basically the same thing as the vielbein formalism?” with “Yes” I have to disagree. That’s just a small piece of Cartan geometry.
Derek
Towards the beginning of the paper of Cattaneo and Schiavina (CS) referenced by Urs’s page on first order formulations of gravity, they write that they hope to reproduce for the Palatini-Holst (PH) action the success they’d had with the Einstein-Hilbert (EH) action. CS proved in 2015 the EH action gave a (Batalin Vilkovisk) BV theory which, on manifolds with boundary, moreover gave a proper (add Fradkin) BV-BFV theory, in the sense of Cattaneo, Mnev, and Reshetikhin (CMR). Given a solution of the corresponding master equation, this would give a quantization of gravity.
In Schiavina’s thesis, he proves the Palatini-Holst action’s minimal BV theory doesn’t allow a CMR-BV-BFV theory – the kernel of the pre-boundary two form isn’t of uniform rank changes as you move around the space of fields. Worse still, when you impose the Half-shell constraint (without which the connection is underconstrained), the pre-boundary two form isn’t presymplectic. So this equivalence of on-shell phase spaces *doesn’t seem* to extend to their derived locii (similarly for Pleibanski- while MacDowell Mansouri doesn’t seem to have this problem).
Their more recent paper suggests either perturbing the action by a boundary to overcome this problem or considering boundary preserving variations. They explore the latter and obtain something again inequivalent to Einstein-Hilbert.
When CS first began publishing their work two years ago, I thought this stuff would become a hot topic in QG. Whenever I’d run into Michele on campus, I didn’t get the sense other physics people cared about it. Good to see this work making its way into the collective consciousness.