Monday, November 01, 2004

Gia's landscape and other new papers

I've decided to remove the article about women in physics - as well as the discussion below the article - because I am afraid that the opinions about this controversial topic that were advocated in that text could be misunderstood in such a way that they would damage Harvard University. At the same moment, I re-enabled anonymous comments under my articles. Thanks for your understanding, and apologies to those who have found the article insulting.

Back to physics:

It may be fun to try to say a couple of words about each hep-th paper that appeared tonight:
  • - Neutrix calculus - This paper seems quite bizarre to me. The authors propose "Hadamard's regularization" of the divergences in quantum field theory, which is based on some specific subtractions of divergent pieces, leading a particular finite value. Well, zeta-function regularization of the sum of integers in string theory may look similar, but in the latter case we have methods to show that such a procedure leads to nice results that satisfy the axioms of CFT, and so forth. In their case, it just seems to me that by removing the infinity, they are converting a correct (divergent) result into a random incorrect (convergent) result that will preserve neither unitarity (of the S-matrix elements, for example), nor locality (of the Green's functions) or gauge symmetries (of everything). The authors don't ask the question whether their theory makes any sense at all. Instead they propose strange statements that non-renormalizable theories are now OK, and so forth. Also, it's not quite clear whether they want to remove the divergence of a particular n-loop amplitude, or the divergence of the asymptotic sum of all terms. In the latter case, the perturbatively non-renormalizable theories are still wrong.
OK, my guess is that we should spend more time with Gia's landscape paper.
  • - Gia Dvali's anthropic solution of the hierarchy problem - There are also other interesting papers on the web, but this one, I think, will attract the attention of many people. The paper is based on Gia's previous work with Vilenkin, but this is my first encounter with these papers.

Gia Dvali joined Nima Arkani-Hamed and Savas Dimopoulos, friends (applied string theorists) who are his co-fathers of the large extra dimensions, and proposed his own version of an anthropically inspired solution of the hierarchy problem.

Recall that Nima and Savas have proposed Split supersymmetry

that argues that it's OK if the Higgs is unnatural and tuned, but with this single sacrifice, one may argue that SUSY should be broken at a high scale and gluinos (dark matter) should be the only light superpartners, which also preserves the success of gauge coupling unification.

Gia Dvali's approach has similar goals and methods, but different "details". Gia's scenario is based on these rather simple ideas:

1. The Higgs mass is effectively a dynamical quantity, which arises from an interaction with a 4-form field strength F4 (the term is Higgs^2.F4^2) induced by some heavy domain walls (2-branes)

2. This mass is dynamical, but discrete. To solve the hierarchy problem, one needs to find a reason why the small masses are preferred. Gia follows Michael Douglas's anthropic counting where the number of vacua measures how natural something is - but in a one-parameter family of vacua only,which may be more meaningful.

Gia's explanation why the small mass is preferred is a kind of "beauty is attractive" argument. It seems that he just postulates that the density of vacua diverges around the desired point because of a power law - without showing why the exponent has the correct sign etc.

(So far, this does not look quite satisfactory to me. You can always say that some quantity XY is small because there is some underlying approximate symmetry that changes XY by a factor, which effectively makes log(XY) a more natural quantity, and the parameter XY is then expected to be much smaller. Is it an explanation? To be more quantitative, I don't know how he gets 10^{-16} as opposed to 10^{-infinity}, for example.)

He says that the required symmetry to make the small mass an "attractor" is a brane conjugation symmetry; he argues that a second Higgs doublet is needed; and he surprisingly relates the Higgs physics with the QCD scale; also, he claims that the quarks' Yukawa couplings are then severely constrained. All these statements may be very interesting.

I have not had time to study all these points in detail yet, but I hope that someone will write her or his understanding below this article. Of course, something like a realistic constraint on the Yukawa couplings is necessary to make the model predictive - otherwise, the amount of input equals the amount of output.

  • - Lorentz violating terms - I don't really understand the motivation of such papers too well. As far as I know, there is no reason to believe that the Lorentz invariance (of local phenomena where curvature can be neglected) is broken by the laws of physics. When we sacrifice Lorentz invariance (which incidentally allows us to violate the CPT theorem, too), the spectrum of possible theories is huge, and I don't understand what's the organizing principle that they only consider this theory and not a much more general theory. Well, yes, maybe if one still requires a certain amount of supersymmetry, like these physicists, the situation may be more constraining. If Lorentz symmetry is broken in string theory, it is always, in some sense, a spontaneous breaking - by matter or condensates of actual fields. Moreover, the authors say that their motivation comes from astrophysics, but they study some Chern-Simons terms in 6 and 10 dimensions.

OK, let's now go from CPT violation to M2-branes.

  • - Supermembranes with central charge - The authors like to talk about symplectic manifolds and calibrations, but I think that there may be interesting physics in the paper, too - although the number of equations is very small. If I understand well, they want to wrap an M2-brane n times on a genus g Riemann surface. If you want to connect all these n sheets into a single membrane, you must put n special branching points on the Riemann surface. And the authors argue that these special points give the membrane a sort of central charge - it's not quite clear to me yet what this central charge of a membrane is. There is a T-dual perspective with D2-D0 bundles. I remain skeptical about the paper, especially because the authors claim strange things, e.g. that the spectrum of a membrane becomes discrete, and so forth. The interactions of membranes can never be set to zero, and a membrane is always able to emit a smaller membrane if it has enough energy, which should make the spectrum continuous in all cases.

Now a generalization of the ADM mass for branes.

  • - Y-ADM mass and positivity theorems - In a system with infinite branes, we may want to generalize the ADM mass to a quantity that measures the mass density, as opposed to the total (divergent) mass. This is probably behind the letters Y-ADM. The ADM mass usually assumes a Killing vector - d_a v_b + d_b v_a = 0. In the Y-ADM business, they generalize the vector v_a - which is a one-form - into a p-form, but otherwise keep the equation to be "ab-symmetrization of the derivative of the p-form vanishes". They discuss how the positive energy theorem generalizes in this case, which may be interesting.

Sergio Ferrara et al. write about gauging an abelian algebra in supergravity.

  • - Special quaternionic manifolds - They start with the fact that the moduli space of type II compactifications on Calabi-Yau have the structure of a special quaternionic, times a special Kähler manifold. Because it is always natural for special quaternionic manifolds to make a certain algebra a gauge symmetry, they study the geometrical conditions when it's possible to do so. I suppose that the actual stringy compactifications with fluxes have the property that this algebra is gauged, and this algebra has the interpretation of shifts of the RR-fields, or something like that.

Something about mirror symmetry for supermanifolds.

  • - Toric CY supermanifolds - When they apply the mirror symmetry manipulations to certain supermanifolds (Sethi; A. Schwartz; Aganagič and Vafa) - fermionic extensions of complex weighted projective spaces - they find a relation between the super Calabi-Yau constraint on the A-model side, and the homogeneity condition on the B-side, plus some quantitative relations.

We want more topological string theory!

  • - Self-dual YM theory - The author proposes new thick, fattened manifolds, those that suffer from obesity, as the target space for the B-model in order to describe the self-dual part of the N=4 Yang-Mills theory in d=4 in a twistor-like language. This guy obviously knows many things about the subject, more than me, and moreover I don't quite understand what the rules of the game are. Is it just about finding some quantities that resemble the bosonic part of the self-dual part of the YM theory? He is doing Penrose-Ward transforms - is it something that can always be done, or is there some non-trivial restriction?

Now a K-theory for D-branes paper.

  • - S-duality for K-theory - The authors (Igor Kříž sounds like a Czech name, and I should know him) study K-theory for type IIB in a B-field background. K-theory is not invariant under S-duality of type IIB. Well, I think it never will because K-theory is just a language to describe the specific objects that can be classically obtained from spacetime filling D-branes - i.e. objects whose tension scales like 1/g. The authors show that indeed, one can't make K-theory S-dual, even if he generalizes twisted K-theory into generalized biased genetically modified twisted K-theory. Well, the full generalization of K-theory invariant under all dualities knows, in a sense, about the whole "theory of everything", and I always found K-theory as a limited description that is very relevant for a few great special examples mostly due to Sen, but otherwise K-theory was a part of the abstract nonsense that tries to propagate into physics.

Now a PhD thesis.

  • - Black hole production - That's a massive, 300-page text about black hole pair production in 3, 4, and higher dimensions, with solutions, different types of the instantons signalling the instability, causal structure of the solutions, including the addition of dilaton couplings, electric charges, and angular momentum. A rather impressive work, although I guess that it is more or less a review of existing literature on the subject.

Black hole production is cool. What about another paper on black hole production?

  • - Multiple black holes from trans-Planckian collisions - Well, we can feel suspicious about the calculational machinery - the black holes are treated as elementary particles (which may or may not be a fine approach to reproduce the results from quantum gravity) - nevertheless the result, that is meant to kill a previous result, sounds pretty plausible. The production of many black holes is suppressed which means, I hope, that the production of a single black hole is preferred. I am not sure whether we should trust the power laws (the dependence on the energy). Also, it seems as an unnecessary restriction that the author focused on d=4 black holes only. Could not he make the analysis for all dimensions simultaneously?

The last paper must be about twistors...

  • - Niels Bohr, Lance Dixon et al. - The authors investigate the arguments of Cachazo, Svrček, and Witten, for the holomorphic anomaly affecting the unitary cuts of one-loop amplitudes. The present authors reduce the supersymmetry from N=4 to N=1, which also reduces the direct connections to string theory and topological string theory (and frankly, also my interest in this calculation). Their results are positive - the results match what you expect from collinearity in the twistor space, even though I don't exactly know what the rules of the game are for the twistors describing the N=1 theories. Most likely, it does not matter.


  1. re: neutrix calculus
    I had a quick look at the paper... and I agree it is a bit bizarre. They assume Ward identity, it would have made more sense if they showed it to hold in their formalism. But they also find the
    standard result for the running coupling, which I didn't expect. Surely an incorrect renormalization scheme would not lead to the correct running?

  2. I appreciate the further information on Gia. The historical infomration is very important here, that leads to further developements by these individuals postulating the extra dimensions.

    Knowing you are a busy man and short of patience, I could not help but point out the historical struggle that might have deviated sometime in our histories, to find that its product might have been found in how John Baez views the issues of quantum gravity as a LQGist, or how pythagroean virtues are now demonstrated in the harmonics of that sound identification.

    So I thought about this, and developed this Blog today, in the hopes this history as it has develope through the theoretcial, and the physics of, might have found some contributions by indivduals that would thwart misconceptions about what dimensions is revealing according to those extra dimensions.

    Is this unreasonable?

  3. Hi Lubos,

    I see you found another outlet for your thoughts... Regarding your comments on the last paper (Dixon et. al.): do you have a specific reason to suspect the string description is more plausible for the N=4 theory rather than any (say conformal) N=1 theory?



  4. This comment has been removed by a blog administrator.

  5. Hi Moshe! Welcome here!

    Yes, if one stays in the context of Witten, the N=4 theory is special because its amplitudes follow from the B-model on CP(3|4) which is a Calabi-Yau supermanifold. It's always the case for CP(N|N+1). The symmetry of that is SL(4|4,C), which is the complexification of the full superconformal group SU(2,2|4).

    In this sense, the number of supersymmetries must be equal to the dimension of the fundamental representation of SU(2,2) which is the conformal group.

    I don't know the right spaces for the topological B-models that would give you N=1 theories, although Witten speculated about some possible candidates.

    (Incidentally, I had to erase an obscene post that an obnoxious user keeps on posting under many different articles. The only thing that I can appreciate is that the person learned how to use Copy-And-Paste.)

    All the best

  6. Lubos,

    Did you ever find out what was up with the T. Padmanabhan paper
    on theories of spin-2 particles not necessarily rolling up to GR?

    Also want to say that I do very much appreciate your postings on physics.

  7. Hi Arun! Thanks for your warm words.

    If I simplify, there were two major problems with Paddy's paper:

    1. He did not understand that when you start with the spin 2 field with a gauge invariance, the only nontrivial way to get interactions is to add nonlinear terms *both* to the actions, but also to the gauge invariance. He never modified the gauge invariance, and was working with the linearized gauge invariance only (that has the partial derivative instead of the covariant derivative). Of course, if you do so, then you won't be able to write down any interactions whatsoever, and the free quadratic action will be the only gauge-invariant action.

    Of course, the correct solution is to add the nonlinear terms to the gauge-invariance, too, which converts the linearized gauge invariance to diff invariance with delta g_{ab} = nabla_a v_b + nabla_b v_a, and then the usual diffeomorphism invariant actions - starting with the Einstein-Hilbert term - are the only solutions.

    2. Paddy did not carefully check the gauge invariance of his actions. The actions that he constructed were clearly not gauge-invariant under his gauge invariance.

    These two errors meant that Paddy could not have found the correct actions (of general relativity), and the actions that he found were wrong (not gauge-invariant).

    I posted a longer - but perhaps less illuminating - posting both to sci.physics.research and sci.physics.strings one day after Paddy published his paper. So try to find it out if you want to see some more details.


  8. Lubos,

    Re: Paddy's paper - one of the things which remains puzzling to me is how we get the non-analytic term in the expansion parameter.