Wednesday, November 17, 2004

Orbifold tachyons from SUGRA and other papers

Last time when I commented all the articles, we were impressed how interesting and serious they were. Tonight it's slightly easier to describe all the hep-th papers on the web, but some of them still look interesting:

This paper by Matthew Headrick and Joris Raeymaekers is obviously interesting. Consider the nonsupersymmetric orbifold of type II string theory - Adams-Polchinski-Silverstein (APS) type of orbifold - on C/Z_n for large n. You know that there are many tachyons in the twisted sectors. For large n, it makes sense to T-dualize around the angular direction of C/Z_n. You get some SUGRA solution. Well, many people have definitely looked at the orbifold in this way, using T-duality. But Matt and Joris finally consider the obviously interesting limit in which n is sent to infinity, but you keep n times alpha' fixed. It's some kind of zero slope limit, but the "lightest" tachyons (=closest to being massless) whose squared masses are comparable to (-1/alpha' n) survive this limit because these squared masses are exactly inverse to the quantity that is kept fixed.

Note that the tachyons come from twisted sectors. By interpreting the angle as a circle, they're winding strings. It means that in the T-dual picture, they are momentum modes of a field in the dual string theory which is effectively supergravity. Finally, the present authors calculate some interactions of the momentum modes in supergravity - which exist off-shell - and they show that they agree with the couplings of the different tachyons calculated from the CFT - which only exist on-shell. That's very interesting. The main thing I worry about is that the result is perhaps not too unexpected because instead of the orbifold, one might work directly with the "limiting CFT" on the thin cone and its T-dual.

This author constructs some new solutions of the modified Ricci-flatness equations, something that is necessary for a CFT to be well-defined. You know that Ricci flatness is the right equation of motion only if you have no fluxes and if the dilaton is constant. If the dilaton is not constant, the Ricci tensor is nonzero - Einstein's equations get a source. He or she does not quite want to talk about a non-constant dilaton. Instead, he or she focuses on another generalization of the CFT and Ricci flatness - namely a CFT with an extra complex field that has an "anomalous dimension". My understanding is that it's just a matter of notation whether you say that a field has an "anomalous dimension", or whether you redefine it by a function of the dilaton, and you allow the dilaton to vary. If my understanding is correct, Nitta has effectively found new solutions of the combined Einstein's equations with some dilaton-gradient source. These solutions have either a U(N) isometry, or an O(N) isometry. It's because the coordinates of his or her manifolds are explicitly written using U(N) or O(N) covariant coordinates, and the metric only depends on the quadratic invariants. Some of these solutions can be interpreted as generalizations or deformations of the Ricci-flat metric of the conifold with an extra linear dilaton - at least that's my impression.

We often say that the maximally supersymmetric Yang-Mills theory in four dimensions is "finite" because of powerful supersymmetric cancellations. Well, what is exactly is finite? Clearly, there are operators with anomalous dimensions that must be regulated and that are cutoff-dependent, and so forth. So it's not everything that is finite. However, the effective action is something that should be finite - it sort of computes the correlators of the elementary fields in which the divergences are supposed to cancel between the bosons and fermions. In this paper, these two guys try to prove the finiteness of the N=4 effective action in the N=1 superspace language. The effective action of N=4 can be written in this form, due to the Slavnov-Taylor identity, as long as we allow the superfields to be "dressed". They must go through some un-controversial steps to show that the R-symmetry anomaly cancels due to the N=2 supersymmetry, and they re-check that the one-loop beta-function vanishes. Nevertheless, the final result is that once you express the effective action of the N=4 gauge theory in terms of the dressed fields, all the terms become independent of the UV cutoff, and the effective theory is therefore finite.

OK, there was a Lorentz violating paper last time, too, and many comments may be repeated. I still don't understand the motivation behind these models. The space of possible non-relativistic non-stringy quantum field theories is huge. I don't really feel what constrains it. For relativistic theories, we may label all fields by their dimension - which is their dimension with respect to space as well as time. However, for non-relativistic theories, we must introduce separate spatial and temporal dimensional analyses, and I don't think that we can really distinguish "renormalizable" theories from "non-renormalizable" theories. Of course, this can be done if we write down a non-relativistic theory as a deformation of a relativistic theory, and this is what this groups does, too. Nevertheless the number of new terms, once you allow the Lorentz symmetry to be broken, is again huge. Moreover, I think that the lessons of 1905 are serious lessons, and breaking the Lorentz invariance explicitly should also be accompanied by breaking of the rotational invariance, and I see no reasons to do so. Let's stop criticism for a while.

What theory do they consider? Take the usual Maxwell action in four dimensions, and imagine that you decide to deform it in some way, by adding another action. What is the other action you know for a U(1) gauge field? Well, the Chern-Simons action. There is a problem: the usual Chern-Simons term only exists in three dimensions. However, that's not a problem for those who don't care about the Lorentz invariance. Just multiply the Chern-Simons 3-form with a general vector, i.e. a 1-form, to get a 4-form, and you can integrate the product over the spacetime. The vector picks a priviliged direction which is OK with you. Because the Chern-Simons action has an epsilon in it, you will break not only the Lorentz symmetry but also the CPT symmetry - which can happen once the Lorentz symmetry is gone. To make the things even more confusing, add an external current J that couples to the gauge field via the J.A term.

That was too natural so far. Let's make something more fancy. Reduce this four-dimensional Lorentz invariant theory to three spacetime dimensions (dimensional reduction). Moreover, don't reduce it along the priviliged vector discussed previously, but along a more general vector. In this case, the three-dimensional Lorentz symmetry will still be violated. If you write down some terms, you will discover mass terms of various types. Just do it and derive the equations of motion and solve them and draw several graphs. And don't forget to be excited that the results pick a priviliged reference frame (even though you know that it was your starting point). It's probably a good feature to violate the Lorentz symmetry.

Finally, let me admit that I am totally lost. I have no idea why they're doing what they're doing, whether it should be a physically realistic model or a mathematically interesting one: I just don't see the meaning of it all. This type of activity is what most of us would be doing today if we had no string theory. Combining random terms that apparently follow no deeper or organizational principles - terms extracted from an infinite chaotic ocean of arbitrary terms and their combinations - terms that are much more ugly and unjustified than the theories that are known to work. Sorry for being so skeptical; I might simply be dumb.

These colleagues first repeat a lot of the commercials about "Causal Dynamical Triangulations" that they've already written in many previous papers. The starting points are very obvious and sort of naive: try to define the path integral of quantum gravity in a discretized form. (It's like spin foams in loop quantum gravity, but you don't necessarily require that the details will agree.) OK, so how can you discretize a geometry? You triangulate it into simplices, and you imagine that every simplex has a region of flat Minkowski spacetime in it.

(That's not like loop quantum gravity - the latter assumes that there is no geometry "inside" the spin foam simplices - the geometry is concentrated at the singular points and edges of the spin foam.)

Then you write down the Einstein-Hilbert action many times and you emphasize that it is discretized. There are many other differences from loop quantum gravity: while the minimal positive distance in loop quantum gravity is sort of Planckian, in the present case they want to send the size of the simplices to zero and the regulator should be unphysical. Of course that if you do it, you formally get quantized general relativity with all of its problems: as soon as the resolution becomes strongly subPlanckian, the fluctuation of the metric tensor becomes large. The path integral will be dominated by heavily fluctuating configurations where the topology changes a lot and where the causal relations are totally obscured - and the results of these path integrals will be non-renormalizably divergent - at least if you expand them perturbatively. But this is simply what a correct, authentic quantization of pure gravity gives you.

These authors are doing something different in one essential aspect. They don't want to sum over all configurations, all metrics - the objects that you encounter in the foamy GR path integral above. They don't do it because they sort of know that pure GR at subPlanckian distances is rubbish. Instead, they truncate the path integral to contain "nice and smooth" configurations only. The allowed configurations they include must be not only nice, but they must have the trivial causal diagram as well as a fixed topology - namely S3 x R in their main example. Well, if you restrict your path integral to configurations that look nice, it's not surprising that your final pictures will look nice and similar to flat space, too. But it by no means implies that you have found a physical theory.

Any path integral that more or less works simply must be dominated by configurations that are non-differentiable almost everywhere, by the very nature of functional integration and by the uncertainty principle. One can often show that the path integral localizes, but that's just a result of theorems and calculations. One cannot define the path integral to include smooth and causal histories only. Such a definition simply violates the uncertainty principle as well as locality, if you make some global constraints on the way how your 3-geometry can look like. Consequently, it also violates general covariance, and you won't decouple the unphysical polarizations. If you also make global constraints about the allowed shapes as functions of time that cannot be derived from local constraints, you will also violate unitarity.

As far as I know, none of these "discrete gravity" people ever asked the question whether these theories are physical, unitary, and so forth. In fact, the subset of the "discrete gravity" people called the "loop quantum gravity" people declares quite openly that they don't care about unitarity at all. Unitarity is actually one of their enemies and they claim that it follows from time-translation symmetry, which is of course a misunderstanding of total basics of physics: unitarity is about hermiticity of the Hamiltonian, if one exists, while time-translation symmetry is about its time-independence. Unitarity is one of the concepts that must be destroyed by the revolution in physics that they've been planning for quite some time. ;-) They're just not getting that unitarity is about the probabilities being non-negative numbers that sum up to one, and that this rule must work in any context in quantum physics. I am afraid that the rest of the "discrete gravity" people does not bother to check these elementary physics questions either. That may be a good point to stop criticism because everyone knows it anyway that I don't believe that this line of research will lead to any usable new physics because it simply neglects some totally essential features of quantum physics.

At any rate, they show that these strange rules of the game admit some big-bang big-crunch cosmological solution described by some collective coordinates (a nice picture animates in front of your eyes), and they construct or propose a wave function of the Universe that depends on the observable representing the "3-volume of the Universe".


  1. Lubos,

    I came across your review of the Ambjorn/Loll paper.
    Since you criticize essentially all non-superstring approaches
    to quantum gravity, I would like to point out that most
    of your remarks do not apply to the original Regge approach.
    I wrote a reply on my webpage

    Wolfgang Beirl

  2. Dear Wolfgang,

    it's not quite clear to me why you think that my criticism does not apply to the Regge calculus. Regge calculus is a good tool for (classical) numerical relativity, but that's it. Claiming that it is a viable approach to quantum gravity is a misconception.

    I've now read your paper hep-lat/0309002 - three pages is not too difficult to read. Are you serious that you consider it an evidence that one can define quantum gravity by a discretization process? The only reason why things seem to get finite numbers in your approach is that you truncate the integrals (3) over "q" - the simplices are allowed to oscillate, but just a little bit, right? ;-) This cutoff is completely unphysical, non-unitary, and has very little to do with gravity. It's much worse than just making a cutoff in Feynman diagrams. What are you exactly trying to do in this paper and why?

    Your phase transition may be an interesting transition for this particular model, but it has certainly no relevance for real quantum gravity. It's just an artifact of your formalism and the way how you try to regularize some integral that may have been remotely inspired by general relativity. The arguments on your web page whose URL you mentioned are sort of cute - one sentence says that one paper (unavailable) shows that the machinery is unitary, and another paper shows that it is non-singular. But these two papers talk about two different models. The non-singular ones (yours) is non-unitary, and the unitary one is not well-defined at all because it is singular.

    If you want to get a theory of quantum gravity, it must be simultaneously non-singular *and* unitary.

    Concerning the "hopes" near the conjectured 2nd order phase transition point, I am not sure what's your goal. As far as I know, the goal in quantum gravity is not to find a critical behavior of some discrete system. Quantum gravity is not a conformal fixed point (except in the context of AdS/CFT, which is a totally different story because holography is involved). Quantum gravity is a theory that locally reduces to special relativity, but that also predicts a gravitational force - everything in a framework that satisfies the postulates of quantum mechanics. It's not clear whether we talk about the same thing.


  3. And by the way, the most relevant reference that you should have mentioned to advocate Quantum Regge Calculus was Martin Rocek and Ruth Williams (1981) and Giorgio Immirzi (1997). However I guess that you know what's the problem with those.

  4. Lubos,

    thank you for taking the time to read our paper.

    Just one comment to your comments:
    The parameter q in our integral does truncate the
    edge lengths but it is (relatively) easy to see that
    we are still dealing with full Regge caluclus (we can
    capture all possible geometries). The lattice does NOT
    fluctuate "just a little bit". The neat trick is to
    understand that q is an unphysical number which can be
    absorbed in the (bare) coupling constants. But it
    allows to control the statistics/numeric.


  5. Dear Wolfgang,

    thanks for your interesting message. Yes, the value of the cutoff is the only new parameter that you add, and therefore you can rescale it away if it is finite - but the full theory has an infinite value of Q which cannot be rescaled to any finite number.


  6. Lubos,

    the values of our coupling parameters at the critical
    point may (or may not) imply q = infty; there is no 'first
    principle' which determines that q has to be infinite.
    IF we find a 2nd order critical point we can discuss the
    physical properties of the continuum limit. But I would
    postpone this discussion until we do find such a point ...
    (So far there are only some hints.)

    In order to get back to your original post, the discussion
    cannot really be about string vs. quantum-gravity theories
    (after all 2D gravity + N matterfields is equivalent to string
    theory in N dimensions and can be simulated via dynamical
    triangulation). The division is obviously about the "super"
    in superstring theories. Your believe seems to be that the
    supersymmetric "balance" between bosonic and fermionic degrees
    of freedom is necessary for a consistent theory.

    My point (which is shared by many others) is that the lack
    of experimental evidence for supersymmetry suggests that we
    examine (and rule out) other alternatives as well.
    This does not prevent us from analyzing supersymmetric
    variants of dynamical triangulation and other lattice models
    and people have started to work on those.

    Thank you again for your time and interest,
    Wolfgang Beirl

    PS: The discovery of cosmic strings would be very exciting,
    unfortunately they would not provide evidence for super-
    symmetry (at least not immediately).

  7. Dear Wolfgang,

    good luck with your calculations, especially the 2nd order phase transition region if there is one.

    Right, SUSY is not proved by a cosmic string - but some sort of grand-unified physics at the right scale is. SUSY is what the LHC may be good for, if for anything. ;-)


  8. Lubos,

    thank you and good luck to your project(s) as well,