From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Background free again Date: 12 Sep 2000 06:59:47 GMT Newsgroups: sci.physics.research Summary: [missing] In article <8pkakk$3im$1@crib.corepower.com>, Nathan Urban wrote: >In article <8pjbm8$13ls$1@mortar.ucr.edu>, baez@galaxy.ucr.edu (John Baez) >wrote: > Now, I've heard about "F-theory" in 12 dimensions, so I can't help >> but wonder if this is somehow related. Is it? Does it have a limit >> which is a BF-like theory in 12 dimensions, probably with extra constraints? >All I know about F-theory (other than it's 12-dimensional) is that it's >supposed to have 2 timelike dimensions -- presumably that means it has >SO(2,10) in there somewhere. Yeah, that's about all I know about it too - except that "M" stands for "Mother" and "F" stands for "Father". Izhtak Bars keeps writing these papers about "2-time theories", which talk about F-theory - but I've never read anything besides the abstracts so far. I don't even know if there's a classical theory with a Lagrangian associated to F-theory, in analogy to how 11-dimensional supergravity is associated to M-theory. But if there is, I can guess what that Lagrangian should be like, by using that grandiose diagram I wrote down last time: M-theory ^ | imposing extra | constraints BOTT PERIODICITY Smolin's Chern-Simons theory in 3d ---------------------> Chern-Simons-like TQFT in 11d ^ ^ | taking boundary | taking boundary | | BOTT PERIODICITY BF theory in 4d ---------------------------> BF-like theory in 12d | imposing extra | imposing extra | constraints | constraints V BOTT PERIODICITY V 4d quantum gravity ---------------------------> F-theory >Is that incompatible with your conjecture? My conjecture is so vague that this stuff about 2 time dimensions doesn't even faze it! It might, however, offer useful clues. >(Well, you have two conjectures I suppose... 12D BF-like-theory as a limit >of F-theory, and F-theory from 4D QG via Bott periodicity. There are actually a bunch of conjectures rolled up into that diagram: all arrows for which I have no firm evidence (the horizontal ones, the vertical one where we get a Lagrangian for the classical limit of F-theory by imposing extra constraints on a BF-like theory in 12d), and also the commutativity of the squares (which is a way of saying that a bunch of analogies relate the 3d/4d stuff to the 11d/12d stuff). >Probably the >two timelike dimensions don't present much of a problem for BF theory, but >they might for a potential relationship with Lorentzian quantum gravity.) Who knows? The funny thing is that in the 3d/4d stuff it all works regardless of the signature. Why should the signature matter in the 11d/12d stuff? Perhaps because Majorana-Weyl spinors only exist when the number of space dimensions minus the number of time dimensions is a multiple of 8, and Majorana-Weyl spinors are really important in some of these supersymmetric theories. >> It's also a fact >> that the Lagrangian for 10d superstring theory can be elegantly expressed >> by treating the dimensions transverse to the string worldsheet as a copy >> of the octonions: >> >> 10 = 2 + 8 >That holds for all of the superstring theories? Yeah, I think so. They're all very similar, after all! You just get a few minor choices like: do you want the strings open or closed? do you want little waves moving in both directions around your string or just one direction? and so on. Of course the heterotic string is a schizoid hybrid where the left-moving wiggles are described by 10d superstring theory and the right-moving wiggles are described by 26d bosonic string theory, but the 10d part is still very much the same as all the rest. >Where can I read more about this Bott periodicity stuff? I know you've >mentioned it in TWF, I can't remember if you gave intro references. Well, the easy part of Bott periodicity, which every particle physicist should know, is how Clifford algebras and spinors work in various dimensions - and how the pattern repeats with period 8. I described this quite nicely (I thought) in "week82" and "week93", and I gave a wad of references. This is the stuff you should focus on. As for the K-theoretic aspects of Bott periodicity, you shouldn't mess with those too much until you're more in tune with the Tao of Mathematics. For example, once you're comfy with characteristic classes, or classifying spaces, or the whole idea of decategorification. (Note I said "or", not "and" - any one of these will provide a nice point of entry.) As for the relation between K-theory and octonions, that's a fact that hardly anybody knows about, and I intend to exploit it to the hilt before everyone catches on. You can read about it in "week105" - but don't tell anyone I said so. >Did Smolin and Ying exhibit a direct relationship (Bott-periodic or >otherwise) between 3D CS theory and their CS-like 11D TQFT, or was that >part of your conjecture? Do you think there's a subtle relationship >there or should one expect such a relationship to exist? They exhibited a relationship that had nothing in particular to do with Bott periodicity: just that the Lagrangians look damn similar. For 3d abelian CS theory it's A ^ dA where A is a 1-form. For their theory it's a bunch of terms like A ^ dA ^ dA where A is now a 3-form. 1 + 2 = 3, 3 + 4 + 4 = 11, get it? Actually *this* much is not new (which is why I've actually had this conjecture for a while). You'll see Distler talking about the same sort of term in his post on the usual approach to 11d supergravity. What's new is that Ying and Smolin's theory start with a TQFT and then get 11d supergravity by imposing extra constraints. The relation to Bott periodicity, if it exists, is bound to be more subtle. But I bet a doughnut that it exists. ============================================================================== From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Background free again Date: 12 Sep 2000 18:46:04 GMT Newsgroups: sci.physics.research In article <8pcckk$4kf$1@geraldo.cc.utexas.edu>, Jacques Distler wrote: >John Baez wailed: >>I don't like formulas with dozens of terms bristling with indices [...] >Oh c'mon! 11-D supergravity (classically, at least) is the simplest of >all the supergravity theories. The lower dimensional theories are much >more complicated. Yes, I know - that's why I'm trying to understand the 11d case. But anyway, you're doing the right thing: maybe you'll convince me that 11d supergravity is very simple. Last night I woke up at 3:30 am and decided I had to learn about it, so I grabbed the only text at hand which contained the Lagrangian. It was the 2nd volume of Green, Schwarz and Witten. Like you, they tried to convince me it was simple. They start out by saying: "The eleven-dimensional supergravity theory only involves 3 different fields: the vielbein e^A_M, a Majorana gravitino psi_M, and a 3-form potential A_{MNP}." You say: >11-D supergravity has just two bosonic fields: a metric, g, and a 3-form >"gauge field" C. So far so good - I know how to switch between a metric and a vielbein, your 3-form C is their A, and you are being even kinder than they are, by postponing the fermionic field (the gravitino). >The bosonic part of the action contains the usual Einstein-Hilbert term, >a |dC|^2 kinetic term for the 3-form gauge field, and a "Chern-Simons" >term, > > \integral C \wedge dC \wedge dC > >The only odd thing about this Chern-Simons term is that it has the >"wrong" normalization. Normally, the coefficient of such a Chern-Simons >term should be quantized so that it is invariant under "large" gauge >transformations. This one is actually NOT gauge invariant (its >coefficient is 1/6, when it should be an integer). To see that >everything works out, and indeed that this funny coefficient was >actually NECESSARY to cancel certain anomalies, is a slightly involved >tale. Okay, you are being nice by warning me that the coefficient will look scary. Green Schwarz and Witten say "we settle here for simply quoting the formula" (which already gets me nervous), and then KABOOM: L = -(1/2k^2) e R - (1/2) e psibar_M Gamma^{MNP} D_N((1/2)omega + omegahat) psi_P - (1/48) e F^2_{MNPQ} - (sqrt(2)k/384) e (psibar_M Gamma^{MNPQRS} psi_S + 12 psibar^N Gamma^{PQ} psi^R) (F + Fhat)_{NPQR} - (sqrt(2)k/3456) epsilon^{M1 ... M11} F_{M5 ... M8} A_{M9 ... M11} I had to write this down just to explain to people why I was complaining! 384? 3456?? That last term is your C ^ dC ^ dC, and yes, that coefficient sqrt(2)k/3456 sure looks funny. >The fermionic lagrangian is also quite simple. There's just one fermion >field -- the gravitino, and aside from its usual kinetic term, it has a >coupling to G=dC. > >That's it: just 3 fields (2 bosons and one fermion) and a very simple >Lagrangian. Well, I guess my problem is that I don't consider the Lagrangian I just wrote down "very simple". Are there ways to make it look simpler? (Using a metric instead of a vielbein might be a slight improvement, and using more differential form notation and fewer indices too, but is that all one can do?) Suppose I just want to understand it at the classical level for a while, so I'm not worrying about anomaly cancellation. How much do those coefficients matter? >Now, I suppose you will want to know what sort of geometry is associated >to the 3-form gauge field (as principal fiber bundles are associated to >1-form gauge fields). > >I wish I could tell you, but if I did, . . . I'd have to kill you :-). Actually, believe it or not, this is the one thing I feel I *do* have a chance of understanding. I know that if you categorify the concept of bundle (or sheaf, really) you get the concept of "gerbe", and if you categorify the concept of connection you get the concept of "connective structure"... and a connective structure on a U(1) gerbe is a 2-form. The exterior derivative of the connective structure is then a 3-form called its "curving", analogous to the curvature. And though people haven't thought about the higher cases as much, this stuff should keep on going. For example, Breen has a nice book on "2-gerbes", which are doubly categorified analogues of bundles (really sheaves). Although he doesn't talk about the analog of connections here, it's gotta be that a U(1) 2-gerbe has a 3-form as the right sort of "connection". And so on for higher n-gerbes.... In fact, once you learn about it, this abstract nonsense fits in perfectly with the fact that M-theory has 2-branes in it. There's a nice relationship between n-branes and connective structures on n-gerbes, just like between point particles and connections on bundles. So this all seems very nice to me; the problem is that I don't understand the geometry of the 11d supergravity Lagrangian very well. In other words: I can handle the 3-form; it's all the rest that I don't have any geometric insight into! Any help with this would be appreciated. But maybe it's already getting a little less scary. ============================================================================== From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Background free again Date: 12 Sep 2000 21:45:11 GMT Newsgroups: sci.physics.research In article , Aaron Bergman wrote: >John Baez wrote: >> All sorts of things about 3d >> Chern-Simons theory become clearer if you use its relationship to 4d >> BF theory. >BF? Yeah, this is the simplest TQFT of all. There's a version that works in any dimension: the Lagrangian is tr(B ^ F) where F is the curvature of a connection on the principal G-bundle P over a spacetime of dimension n, and B is an Ad(P)-valued (n-2)-form. Or in lowbrow lingo: F is locally like a 2-form taking values in the Lie algebra of G, while B is locally an (n-2)-form taking values in the Lie algebra of G. [Moderator's note: EXTREMELY lowbrow, Ensign. --Mr. Spock] You can add an extra "cosmological constant term" in dimensions 3: tr(B ^ F + lambda B ^ B ^ B) and 4: tr(B ^ F + lambda B ^ B) The 4d theory with cosmological constant term is closely related to 3d Chern-Simons theory as follows. The equations of motion for 4d BF theory say among other things that B is proportional to F, so "on shell" the Lagrangian is proportional to tr(F ^ F) which for a 4-manifold without boundary is just the 2nd Chern class of the bundle P (up to a constant factor). This allows one to switch back and forth between BF theory and the theory with Lagrangian tr(F ^ F). In either case, the equations of motion for A work out to be vacuous: i.e., the 2nd Chern class doesn't depend on the connection A. But now suppose we work on a 4-manifold with boundary. Then the integral of tr(F ^ F) *does* depend on A; if we vary A, we get a boundary term. This boundary term is proportional to the variation of the Chern-Simons action tr(A ^ dA + (2/3)A ^ A ^ A) on the boundary 3-manifold. So BF theory in 4d has Chern-Simons theory in 3d as a kind of "boundary value theory". This has all sorts of cool consequences - the coolest being that since 4d general relativity is a BF theory with extra constraints, 4d quantum gravity induces a Chern-Simons theory on the event horizon of a black hole, which you can use to compute the black hole's entropy. (I should mutter something pompous about the holographic hypothesis at this point for maximum effect, but I'm not up to it.) Similarly one would expect that an 11d TQFT with a Lagrangian like tr(C ^ dC ^ dC), where C is a 3-form, would be the "boundary value theory" of an 12d theory with Lagrangian like tr(dC ^ dC ^ dC) which would be some funky analog of the 3rd Chern class for a connective structure on a 2-gerbe! So I want to see a paper where someone talks about the latter sort of theory. ============================================================================== From: Aaron Bergman Subject: Re: Background free again Date: 12 Sep 2000 21:45:38 GMT Newsgroups: sci.physics.research In article <200009090654.e896sxA25438@math-cl-n03.ucr.edu>, baez@galaxy.ucr.edu (John Baez) wrote: > Actually, believe it or not, this is the one thing I feel I *do* > have a chance of understanding. I know that if you categorify the > concept of bundle (or sheaf, really) you get the concept of "gerbe", > and if you categorify the concept of connection you get the concept > of "connective structure"... and a connective structure on a U(1) > gerbe is a 2-form. The exterior derivative of the connective structure > is then a 3-form called its "curving", analogous to the curvature. This works fine for when you have one brane, but the $64,000 question is what happens when you stack up a whole bunch of these branes. For D-branes, you get a U(N) gauge theory. What happens when you start stacking things that have gerbes on them? Still, even the 3 form of M-theory isn't that simple. It doesn't have integer quantization conditions. For G = dC, we have G / 2\pi - \lambda / 2 \in H^4(M,Z) where \lambda is half the first Pontryagin class. Interestingly, according to Witten, E_8 bundles are completely determined by (1/16\pi^2) tr F /\ F which is a four form. E_8 shows up all over the place with M-theory. I'm not sure it's quite understood just what the deep relation is, however. Aaron -- Aaron Bergman ============================================================================== From: baez@galaxy.ucr.edu Subject: Re: Geometric Quantization Date: 12 Sep 2000 21:48:24 GMT Newsgroups: sci.physics.research In article <8pl31m$3kg$1@nnrp1.deja.com>, wrote: >John Baez wrote: >By the way, here's one cute example of how fermions are related >to quaternions - in addition to the fact that they rhyme. [Explains how the time reversal operator T for nonrelativistic particle of half-integer spin must be a conjugate-linear operator with T^2 = -1, thus giving an action of the quaternions I,J,K where I = i, J = T, and K = IJ.] >The same is true for any QFT and the CPT operator, I guess? You're saying that for any quantum field theory the CPT operator is a conjugate-linear operator with square equal to -1? I don't see why. But just to show you how ignorant I am, I had thought CPT was linear until you brought this up. I figured: P is linear, T is conjugate- linear, and of course C is conjugate-linear since it's called "charge conjugation". But reading over some books I see that, at least for spinor fields, C is linear! That would make CPT conjugate- linear, just like you say. I still don't see why it's *always* conjugate-linear, or why its square is always -1. I should have taken more courses on particle physics when I was a youth, dammit. Ignorance sucks. >This is funny, because it allows us to think of the whole Standard Model, >say, as a quaternionic quantum system. That would be cool if true! >Moreover - it's funny, but CPT seems to be the only symmetry realized >by a conjugate-linear operator - so there are no ambiguities in the >definition! That's cool. If its square is -1 we get a quaternionic structure on our Hilbert space; if its square is 1 we get a real structure. Either way it's interesting, but the quaternionic case seems cooler. >> Presto! Fermions are quaternionic, bosons real. >Oops, a correction - the fermionic superselction sector (I.e. the one >with half-integer spin, and an odd number of fermions) is a >quaternionic system, and the bosonic one a real system. Hey, wait a minute, get your story straight here! :-) Are you now trying to tell me that (CPT)^2 is (-1)^F where F is the fermion number? >Wait a second - why did you refer to a non-relativistic particle, then? I occasionally dally in the lowly depths of nonrelativistic physics; it's a form of slumming. :-) Seriously, I'm trying to teach Toby how to tell which unitary self-dual irreps are real and which ones are quaternionic. The simplest interesting example is the group SU(2), so I did that. This is a fun example for lots of reasons: everyone knows the irreps from their study of nonrelativistic quantum mechanics, they're *all* self-dual, and SU(2) is just the unit quaternions. If you want, we could figure out how it goes for the Poincare group. Are fermions quaternionic here too? >It seems to be true for the relativistic one too. Hmm, now I'm really interested.