Return to the Fold

By cjohnson | January 27, 2006 3:05 am

[Warning! This is an unusually technical post.]

So it is an interesting fact that in 1992, after my phd advisor Tim Morris wrote his last paper with me, he never wrote a string theory paper again….until last week. (No, I don’t think it was because of the horror of working with me…. so stop thinking that right now!)

I always considered it a great loss to the field, over the years, as he is quite a remarkable fellow (hey – he seems to have been lauded at a national piano competition again!), and several things happened in the field after he left that I always thought that he would enjoy, and moreover bring an interesting and valuable approach to. On the other hand, he was not lost to physics or high energy physics as a whole, and started a series of really creative and singular work in another area, as I’ll mention below. (Ironically, the scope of some of what we did in that last paper we wrote, with Simon Dalley and Anders Watterstam, has only been fully appreciated in some recent work of mine (with Durham Phd. student James Carlisle and USC undergraduate student Jeff Pennington), described in an earlier post. We knew in 1991-2 that we had something that would now be called “open-closed duality”, but it is part of an even nicer story, as partially uncovered by Klebanov, Maldacena and Seiberg….. read that post for more, and I should do the next part of that story one day.)

So whenever I ran into Tim over the years (sadly, only a very few times since I left) I would rekindle that hope that maybe he’d return to the fold again. I actually had already decided (in my mind) that the best place for him to re-enter the field again might be via what he went off to do. You see, he’s been developing a remarkable scheme for computing in gauge theory, based on the Wilsonian description of RG flow. The scheme allows you to compute using methods completely different from the usual (Feynman diagrammatic). One of the problems that Tim solved was how to make this scheme work for gauge theories. How to do the “Exact Renormalisation Group” (ERG) as it is called scheme in a manifestly gauge invariant way? The whole concept of a cutoff in a gauge theory is plagued by difficulties of definition, at least in the usual approaches we teach our graduate students. Well, it turned out that it is possible, but it required the embedding of the SU(N) gauge group into a supergroup, SU(N|N). (See here and here.) Hmmmm? you might ask… but it works. Well, I kept thinking that surely, this had to make contact with this other way we know from string theory of doing manifestly (non-Feynmann diagrammtic) computations in SU(N) gauge theory, the AdS/CFT correspondence. Further, we know that radial direction in the AdS part of the spacetime is some sort of gauge invariant definition of a cutoff. Clearly, there is “some money to be made” (ho ho ho) in connecting these two approaches. They cannot be unconnected, at least in my mind.

Well, the good news is that last week, Tim returned to the fold, and this is precisely the way he chose to take back home to the world of string theory with a new paper (with Evans and Rosten) showing the connection explicitly. (Welcome back Tim!) I believe that my former fellow Southampton graudate student, my sometime collaborator, and faculty colleague of Tim’s, Nick Evans, had a lot to do with his return (well done Nick!) since in the past he and I have sat and speculated about the content of the above paragraph’s expected connections, and whether we could interest Tim in it or not. I never really understood enough of the ERG approach to know how to proceed, but I did promise myself to look out for a natural appearance of SU(N|N) gauge groups in the context of strings. That was to be my hook….

Well, I missed it. Just been too busy. The paper that had SU(N|N) came out and I missed it, and frankly I feel a bit of a plonker. A few weeks ago, Okuda and Takayanagi wrote a paper introducing the idea of “Ghost D-Branes” (see here) which shows you very naturally how to make SU(N|N) gauge groups. As soon as you see that, and what the role of the ghost D-branes is, it is clear how to AdS/CFT-ize it and make quite natural contact with what the ERG scheme achieves….. This is what Nick Evans, Tim Morris and Tim’s student Oliver Rosten (who gave us here at USC an excellent seminar on the ERG scheme last year, by the way) did in their paper, which is cute, straightforward and all of six pages. Have a read of it. There’s a lot more to be done there, I’m sure.

-cvj

CATEGORIZED UNDER: Academia, Science
  • Moshe

    Hey Clifford, I am puzzled about all the non-unitary stuff floating around in this (scalars with wrong sign kinetic terms, bosonic spinors etc.), do you see how can one avoid the obvious problems?

    Alternatively, I could go read the original papers, looks interesting.

  • Elliot

    Layman’s request,

    Clifford. Is there a way to give a basic definition of Ads/CFT correspondence for neophytes like me. Anything that doesn’t jump directly into higher level mathematics would be greatly appreciated.

    Thanks,

    Elliot

  • http://eskesthai.blogspot.com/2006/01/spacetime-101.html Plato

    I apologize for the sins of a layman here. Could one ever have been lead to a clear understanding of the world you inhibit Clifford?

    Maybe this is a trend (your compadre) is dispelling in regard’s to Peter Woit’s hard question about the “restraints” that were needed and applied to reality?


    Because string theories are by nature relativistic, adding energy to a string is equivalent to adding mass, by Einstein’s relation E = mc2. Therefore, the separation between D-branes controls the minimum mass open strings may have.

    Would Dvali’s comparative findings of hitting metal plates be possible in sight of these examples, or early universe phase transitions to today. Is that too vague?

  • http://eskesthai.blogspot.com/2006/01/spacetime-101.html Plato

    I see layman minds think alike? :)

    The “horizon” becomes a interesting place on that space inside the blackhole.


    Conformal field theory is most often studied in two dimensions where there is a large group of local conformal transformations coming from holomorphic functions.

  • http://blogs.discovermagazine.com/cosmicvariance/clifford/ Clifford

    Moshe,

    I think that the Ghost Branes paper probably talks about this. I learned of it only last night and it is long, so cannot tell you its contents. Also, I imagine there is a complementary discussion in Tim’s SU(N|N) formalism….. or…..maybe one of the authors of the paper might see this and tell us the answer…..

    Elliot,

    Ok…maybe I’ll do something more extensive later. Right now, the basic result is that the physics of string theory in the spacetime which is the product of five dimensional anti-de sitter and a five sphere is equivalent to a four dimensional SU(N) gauge theory. Specifically there is a relation between the coupling strength of the gauge theory, the rank, N, of its gauge group on the one hand, and the parameters of the string theory and the ten dimensional spacetime. Where we have a reliable spacetime description, the gauge theory is strongly coupled and N is large.

    The equivalence between these two apparently completely different types of physics (they don’t even match spacetimes!!) is the dictionary that comes with the AdS/CFT correspondence, and the whole thing generalises to several more interesting spacetimes and gauget theories….

    This is all very good, since it teaches us new things about gauge theory at strong coupling (and hence properties of the strong nuclear interactions at low energy) by studying the physics of strings in interesting ten dimensional spacetimes.

    Some of us like this stuff a lot, since whether or not you care about whether strings have anything to do with quests for “theories of everything” (whatever that means), there is a pragmatic approach in mind: We get to learn about stuff that we know for sure that Nature uses and cares about:- strongly coupled gauge theory.

    Cheers,

    -cvj

  • Moshe

    Thanks Clifford, meanwhile the words Pauli-Villars came to my mind and put it at ease, at least for now.

  • http://blogs.discovermagazine.com/cosmicvariance/clifford/ Clifford

    Yes…. Pauli-Villars is exactly what Tim’s approach works into the SU(N|N) stuff, if I recall, and so that’s probably the connection.

    Cheers,

    -cvj

  • Chris W.

    In comment 5, gauget theories? I assume that’s just a typo.

  • Elliot

    Clifford,

    O. K. I get the picture that this is not a easy thing to simply and clearly explain (without the math) but it seems that what this is telling me is that there is a linkage or mapping between two theoretical superstructures 1-strongly coupled gauge theory and 2-superstrings and that they “pass” from 4 to 5 dimensions and that by “fiddling” with string parameters it affects (theoretically of course!) the coupling strengths in the gauge theory or visa versa that the gauge theory may provide information about the string parameters. Is that pretty close or should I just wait for another day on this?

    Elliot

  • http://eskesthai.blogspot.com/2006/01/spacetime-101.html Uber

    Why Why WHy, and, if, if,….

    Pauli-Villars and gauge theory….oh my poor head. This “stuff” is hard to grok.:)I like it too though, as the “fantasy” of mine progresses towards reality.

    Now “ghosts”…. their going to have a field day :)

  • http://thomas.loc.gov X

    So strings are just a fancy way of regulating quantum field theories?

  • http://blogs.discovermagazine.com/cosmicvariance/clifford/ Clifford

    Elliot: On the “passing” from higher to lower dimensions (a holographic relationship) I have already written on the blog. See here (when I explained my bad physics joke):

    http://blogs.discovermagazine.com/cosmicvariance/2005/09/08/bad-physics-joke-explained-part-i/

    On the rest of what you said: It is not that the string parameters affect the parameters of the gauge theory… it is that they are given in terms of them…. they are rewritings of each other….if you know one, you know the other. So in this way, you have a formalism (the string theory) where you can compute a number. There is then a dictionary that tells you what that number as computed for you in terms of the other formalism (the gauge theory). This is particularly good when it turns out that the string theory is in a regime where you can compute with relative ease, and the dictionary is telling you that it is giving you answers about the gauge theory where it is hard to compute. So it is a new window (using strings) on regimes of the gauge theory where you’ve previously not been able to proceed.

    X: yes, but much much much more than that.

    Cheers,

    -cvj

  • Takuya

    Hi Clifford,

    Non-unitarity due to the wrong spin-statistics relations and the wrong signs in the kinetic terms is avoided when ghost branes are coincident with usual branes and are canceled. That is, when doing a scattering experiment, if you prepare the in-state so that only usual particles are present, the ghost particles are never created and the probability is conserved. In this case, the system with ghost branes are equivalent to a system without ghost branes. If you believe ghost branes are there, they are there. If not, they are not there.

    When ghost branes and usual branes do not completely cancel out, unitarity is lost. This is also true in the SU(N|N) regularization of SU(N) theories, where the Higgs vev (corresponding to moving ghost branes off usual branes) is turned on to break SU(N|N) to SU(N)x SU(N). The Higgs vev is the cut-off scale. The non-unitarity does not affect the physical SU(N) part if you take the cut-off to infinity. But to do RG flow, you need to keep the cut-off finite. It is not completely clear to me how your Ph.D advisor and his collaborators deal with the pathologies, but I think taking N large while fixing the ‘t Hooft parameter helps protect the physical SU(N) from non-unitarity.

    Best,

    Takuya

  • Moshe

    Takuya, that was an interesting paper…I am also confused, as above, by the configuration where the supergroup is broken to SU(N), assuming one can make sense out of “giving VEV” to non-unitary scalars. In some sense this is what you do in Pauli-Villars regularization but there is much more structure here (like various interactions between the normal sector and the ghost sector). It is plausible all the answers are in the literature…

    Actually, I am also confused by the equivalence between SU(m,n) and SU(m-n) that you mentioned, is the statement that “mixed” correlations, with some ghosts and some physical states, all vanish? is that fact obvious or does it need some work to show?

  • Takuya

    Moshe,

    Thanks for your comments. The extra structure is what makes the regularization nicer than the usual PV regularization. It is gauge invariant, for example. Dr. Morris started by working with PV regularization, and I imagine, he found the supergroup structure by asking for good properties like gauge invariance and various cancellations in Feynman diagrams. Ghost D-branes turned out to provide a natural explanation of these nice properties.

    The statement of equivalence can be made at various levels. In our paper, we discussed the correlation functions of gauge invariant operators and showed that the results are the same for SU(N|M) and SU(N-M). The statement I made above (same as the statement you guessed), that scattering ordinary particles doesn’t produce ghost particles, can also be easily understood by considering double-line Feynman diagrams. The conservation of indices ensures that if the incoming particles have bosonic indices, the outgoing ones must also have bosonic indices. The summed indices produce supertraces of one, which is N-M. This proves the equivalence. -cheers, Takuya

  • http://eskesthai.blogspot.com/2006/01/cosmic-rays-collisions-and-strangelets.html PLato
  • Moshe

    Thanks Takuya.

  • http://blogs.discovermagazine.com/cosmicvariance/clifford/ Clifford

    Takuya:- Thank you very much!

    -cvj

  • http://eskesthai.blogspot.com/2006/01/quark-gluon-plasma-ii-strangelets.html PLato

    Can model assumptions be effective way in which to see how experimental processes are lining up with current theoretical data? I,m not sure, but if we had identified flat spacetime in this model usage, within context relativistic considerations, then this would be quite a profound thing would it not?


    The collisions are strange: PHENIX can identify particles that contain strange quarks, which are interesting since strange quarks are not present in the original nuclei so they all must be produced. It is expected that a Quark-Gluon Plasma will produce a large amount of strange quarks. In particular, PHENIX has measured lambda particles. There are more lambda particles seen than expected.

    Maybe I have reverted back to fantasy? :)

  • Elliot

    Clifford,

    Thanks. The reference to the TASI lectures looks promising.

    Elliot

  • AndrewP

    One must be very cautious in extrapolating current data to infinitely accelerating acceleration of the universe until the big rip is achieved. A real world example about increasing acceleration makes that clear. Imagine that you are on the far side of an asteroid that is falling toward the earth. You observe that you are accelerating away from distant astronomical bodies and toward others without apparent explanation. But since you can’t see the earth (you are on the far side), you assume that you are being moved by some mysterious dark energy. The acceleration is accelerating. Eventually it reaches 9.8 m/s2. Then smack – you hit the ground and become a giant mushroom cloud.

  • boreds

    Hi

    I actually don’t fully understand why SU(N|N) is supposed to be dual to flat space in the ERG paper. I’d expect that, as N–>M, the SU(N|M) theory will be dual to increasingly highly-curved AdS_5xS^5. (Even if in the case N=M the near horizon limit isn’t really sensible).

  • http://blogs.discovermagazine.com/cosmicvariance/clifford/ Clifford

    Hi,

    I don’t understand why it would be highly curved AdS in that limit. The sources of curvature seem to cancel each other out exactly (at least above a certain distance scale), and so flat space is pretty much the only option, I would say.

    Cheers,

    -cvj

  • Moshe

    I think the statement is much more simple, SU(n|m) is equivalent in some sense to SU(n-m), and is reaized as the theory on appropriate numbers of D3 and ghost D3 in flat space as usual (no decoupling limits). So when n=m one is left without any D3-branes, only with flat space. I don’t think one can take any meanigful near horizon limit with this story, maybe I am wrong.

  • boreds

    I agree with what you have both said, particularly that the nh limit isn’t meaningful in the case n=m.

    But isn’t SU(n-m) when (n-m) is small dual to AdS_5xS^5 with small AdS length? If I am not missing something obvious then the same should go for SU(n|m) when n-m is small.

    I agree the whole geometry is flat when there are equal numbers of D and ghost-branes, but don’t understand why the gauge theory is `dual’ to this flat geometry, as stated in the paper—probably I am missing something.

  • boreds

    another thing that puzzles me is that the flat space isometries won’t match up with the conformal group etc anymore

  • http://atdotde.blogspot.com Robert

    I think I have another problem with the ghost branes. So I sent an email to the authors saying

    I am surprised that by multiplying a boundary state |D> by (-1) you get a different state.
    As you claim, a D-brane is turned into a ghost brane with opposite charges and tension.
    I am a bit surprised by this since I would have thought that strictly speaking a state is not
    given by a vector in a Hilbert space but by a ray in such a space. That is, only by a vector
    up to normalisation by a non-zero complex number.

    This would also follow from boundary CFT where the boundary state is defined by the condition
    that certain operators (for example T, the left handed energy momentum tensor and its
    right handed counterpart T-bar) agree on the boundary:

    T|boundary>= T-bar|boundary>

    But such equation again would characterise |boundary> at best up to a scalar multiple. So
    I am quite surprised that a factor of (-1) has such drastic consequences. Maybe you
    could comment on this?

    Once we start transforming states by multiplication by (-1), nothing stops us from
    considering exp(i phi)|D> for some phase exp(i phi). By a similar argument as yours,
    I would conclude that this rotated state has a complex mass and charge since the
    phase also multiplies amplitudes like

    Is that correct?

    I got a reply which I am not sure I am supposed to quote. It said something along the lines of the boundary state being a classical object from the target space perspective and thus a prefactor would matter. For example by multiplying by N I would get a state of N coincident branes.

    I must say this does not really convince me. Could anybody else please say something about this?

  • http://golem.ph.utexas.edu/~distler/blog/ Jacques Distler

    The open-string partition function can be written, in the closed-string channel, as
    Z=. Adding Chan-Paton factors, formally multiplies Z by a factor of N N’.

    So, indeed, the normalization of the boundary state matters. That’s part of the data that goes into normalizing the path integral in the presence of boundaries.

  • http://golem.ph.utexas.edu/~distler/blog/ Jacques Distler

    Hmmm. That was amusing. Let’s try that equation again, shall we?

    <B|q^{H}|B’>

  • http://golem.ph.utexas.edu/~distler/blog/ Jacques Distler

    Whoa! Cool! I can mess up, not only my own comment, but succeeding ones as well. WordPress is da Bomb!

  • http://atdotde.blogspot.com Robert

    Shouldn’t I normalize this somehow with ?

  • http://atdotde.blogspot.com Robert

    ? was langle B|B rangle

  • http://atdotde.blogspot.com Robert

    I could ask the same question in ordinary QFT: I can ‘probe’ the charge of a particle by computing the amplitude for a T-channel exchange of a photon. That should be proportional to e^2. However, I do not get ghost electrons with positive charge and negative mass by multiplying external electron lines by (-1).

  • http://golem.ph.utexas.edu/~distler/blog/ Jacques Distler

    Shouldn’t I normalize this somehow with <B|B’>

    No! Because <B|B’>=∞ !

    The Boundary State is not a normalizable state in the Hilbert space.

  • http://atdotde.blogspot.com Robert

    So, if I multiply by a phase do I really get branes with complex tension?

  • http://golem.ph.utexas.edu/~distler/blog/ Jacques Distler

    Multiply by whatever factor you want, and you can get whatever nonsense you desire.

    The normalization of the boundary state is fixed by demanding that the open string partition function is properly normalized in the presence of boundaries.

    Perhaps it might be helpful to note that one can, technically, put either bosonic or fermionic Chan-Paton factors on the boundary; you get the same answer either way. Usually, this is thought of as an either/or proposition. The ghost D-brane proposal is to allow both bosonic and fermionic Chan-Paton factors in the same theory.

  • Oliver

    Hi,

    I can shed some light on the issue of unitarity at finite cutoff. As effectively said already, if the cutoff is sent to infinity, all fields in the spontaneously broken SU(N|N) theory become infintely massive, with the exception of the physical SU(N) field (and an unphysical copy which is decoupled). One of the crucial ingredients of our ERG approach – which is built in from the start – is that the partition function is invariant under the flow. Since we know that we are dealing with just SU(N) YM at the top end of the flow, we know that we must be dealing with the same theory everywhere along the flow. So, when computing physical quantities, the unphysical fields serve only to regularize the physical theory, and do not spoil unitarity. This works at finite N.

    Oliver.

  • http://www.hep.phys.soton.ac.uk/~evans Nick Evans

    Hi Guys. Thanks for your interest in our paper on ghost branes and SU(N|N) regularization.

    I’ll let Olly try to convince you about regulators – as you’ve deduced it’s basically the same story as Pauli Villars though.

    Flat space as a dual I can try to be helpful for you though. Stay in the pure AdS/CFT Correspondence – move the D3 onto a 5-sphere. The geometry is AdS outside the sphere and flat space inside. What does that mean? It means the theory is totally higgsed at the scale corresponding to the radius of the sphere and below that scale is a mass gap. The gravity description of this “nothing” is flat space. Our host, Clifford, played precisely this game in his enhancon papers removing repulsons and replacing with flat space…

    cheers Nick

  • http://eskesthai.blogspot.com/2006/01/landscape-of-neighborhood.html PLato

    Layman scratching

    amazing…that viscosity measures(D brane analysis?) could have been hidden in all this talk?

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