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Smolin: LQG-style Mtheory

  1. Mar 17, 2005 #1

    marcus

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    http://arxiv.org/abs/hep-th/0503140
    A quantization of topological M theory
    Lee Smolin
    20 pages

    "A conjecture is made as to how to quantize topological M theory. We study a Hamiltonian decomposition of Hitchin's 7-dimensional action and propose a formulation for it in terms of 13 first class constraints. The theory has 2 degrees of freedom per point, and hence is diffeomorphism invariant, but not strictly speaking topological. The result is argued to be equivalent to Hitchin's formulation. The theory is quantized using loop quantum gravity methods. An orthonormal basis for the diffeomorphism invariant states is given by diffeomorphism classes of networks of two dimensional surfaces in the six dimensional manifold. The hamiltonian constraint is polynomial and can be regulated by methods similar to those used in LQG.
    To connect topological M theory to full M theory, a reduction from 11 dimensional supergravity to Hitchin's 7 dimensional theory is proposed. One important conclusion is that the complex and symplectic structures represent non-commuting degrees of freedom. This may have implications for attempts to construct phenomenologies on Calabi-Yau compactifications."
     
    Last edited: Mar 17, 2005
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  3. Mar 18, 2005 #2

    Chronos

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    When Smolin talks, I listen. On the surface, it looks like a challenge to M-theory, but having read it, that does not appear to be the case. Good link.
     
  4. Mar 18, 2005 #3

    marcus

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    I think you are right. the paper seems more like part of a conversation that he is having with string theorists Cumrun Vafa, Robbert Dijkgraaf, et al
    (the authors of http://arxiv.org/hep-th/0411073 )
    and with differential geometer Nigel Hitchin.
    In the acknowledgements there are thanks to Dijkgraaf, Gukow, Hitchin, Neitzke, Vafa. Smolin's paper was apparently written in haste and still has lots of typos to correct. Some sections, like section 3 maybe, could be expanded. I think the paper could be seen as a response to
    hep-th/0411073.

    If that is the way to look at it, then one would have to go back and see what stands out in the earlier paper
     
    Last edited: Mar 18, 2005
  5. Mar 18, 2005 #4

    marcus

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    just to have it handy, since it seems to go with Smolin's paper or viceversa, here is the Dijkgraaf et al abstract:

    Topological M-theory as Unification of Form Theories of Gravity
    Robbert Dijkgraaf, Sergei Gukov, Andrew Neitzke, Cumrun Vafa
    65 pages, 2 figures


    "We introduce a notion of topological M-theory and argue that it provides a unification of form theories of gravity in various dimensions. Its classical solutions involve G_2 holonomy metrics on 7-manifolds, obtained from a topological action for a 3-form gauge field introduced by Hitchin. We show that by reductions of this 7-dimensional theory one can classically obtain 6-dimensional topological A and B models, the topological sector of loop quantum gravity in 4 dimensions, and Chern-Simons gravity in 3 dimensions. We also find that the 7-dimensional M-theory perspective sheds some light on the fact that the topological string partition function is a wavefunction, as well as on S-duality between the A and B models. The degrees of freedom of the A and B models appear as conjugate variables in the 7-dimensional theory. Finally, from the topological M-theory perspective we find hints of an intriguing holographic link between non-supersymmetric Yang-Mills in 4 dimensions and A model topological strings on twistor space."
     
    Last edited: Mar 18, 2005
  6. Mar 18, 2005 #5

    marcus

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    Last edited: Mar 18, 2005
  7. Mar 19, 2005 #6

    selfAdjoint

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    Just reading a bit in these papers, an important point that Smolin makes is that the DGNV Topological M-theory isn't really topological, since he shows it has local degrees of freedom. Two of them in fact, and they don't commute with each other.

    He further shows that both of these degrees of freedom are needed to specify a paticular Calabi-Yau manifold to compact your string physics on, but if they don't commute, when you come to quantize the theory, you can't specify them both at the same time because of uncertainty. Bummer! It seems amazing to me that nobody noticed this little problem before!
     
    Last edited: Mar 19, 2005
  8. Mar 19, 2005 #7

    marcus

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    can you explain what difference it makes? I noticed him saying that too: that there were a couple of local degrees of freedom. But I couldnt interpret it. Why should that be inconvenient, for what DGNV are up to? sorry if this is a naive question.

    maybe it is merely disconcerting, that DGNV didnt notice something and had to wait for Smolin to point it out? but maybe it is not a serious impediment to what they want to do? (feeling very tentative about trying to interpret, these is unfamiliar terrain for me)
     
    Last edited: Mar 19, 2005
  9. Mar 19, 2005 #8

    Kea

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    Don't know if this is helpful:

    Generalized Calabi-Yau manifolds
    Nigel Hitchin
    http://www.arxiv.org/abs/math.dg/0209099

    It wasn't referenced by Smolin. It ends up talking about gerbes. Note that Hitchin is a very important mathematician in the history of cohomological field theory and the maths of String theory in general. The above paper is expanded upon in the 2003 thesis of Hitchin's student

    Generalized complex geometry
    Marco Gualtieri
    http://arxiv.org/abs/math.DG/0401221

    Regards
    Kea :smile:
     
    Last edited: Mar 19, 2005
  10. Mar 19, 2005 #9

    selfAdjoint

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    It sure isn't naive, I am just on the outer erdges of the math myself. I'm going to research it, including the Hitchins paper Kea linked to (Thanks Kea!) and I'll post something hopefully tomorrow.

    The idea of Calabi-Yau manifolds having two structures, a metric and a symplectic, was new to me; in fact I never knew of any connection between C-Y and symplectic manifolds at all.
     
  11. Mar 20, 2005 #10

    Kea

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    Thanks selfAdjoint. I've been busy with work, mountain accidents (which I have a tendency to get mixed up in) and other adventures.

    DGNV don't reference the latter Hitchin stuff either, which makes me think (perhaps erroneously) that this is why Smolin didn't look at it. It would be good if we could get some comments from an M-theory person, such as Sati, but I'm not sure if any of them read this site.

    Cheers
    Kea :smile:
     
  12. Mar 20, 2005 #11

    marcus

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    Hi Kea, sorry to hear about the mountain accidents. Trust it is not you but someone you know, hope they are all right.

    Just to clarify about the Hitchin papers (not for you or sA so much as in case anyone else is listening). The two recent papers by Hitchin that DGNV cited (besides the earlier "twistor" one from 1981) were

    http://arxiv.org/math_DG/0010054
    http://arxiv.org/math_DG/0107101

    and these are exactly the two that Smolin referenced in his paper, which seems appropriate because his paper is in response to theirs and initiates a Loop treatment of what DGMV presented as topological M-theory.
     
  13. Mar 21, 2005 #12

    selfAdjoint

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    Right, Marcus. The Hitchen "super-Calabi-Yau" paper is not in issue here. I'm still trying to get the DGNV paper into my thick head.
     
  14. Mar 22, 2005 #13

    selfAdjoint

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    At long last. Here is the relevant theorem from Hitchens October 2000 paper.

    So basically just giving a 3-form with holonomy serves to define the geometry of any 7-dimensional manifold. He goes on to show that the 3-form defines the moduli space of the manifold too. Note that this is all pure math; Hitchin cross classifies it on the arxiv as Algebraic Geometry and Differentiqal Geometry.


    Now on to Dijkgraaf, Gukov, Neitzke, and Vafa (DGNV) in hep-th/0411073. They are concerned with a topological theory they call the B-model:
    They then define a wavefunction [tex]\Phi[/tex] as the tensor product of ZB and its complex conjugate. This wave function then has a well defined quantum thing called a Wigner Function, which measures its density in phase space. They show an identification of this Wigner function with the partition function of Hitchin's form theory, stongly suggesting that the two theories are identical.



    More to come--
     
    Last edited: Mar 22, 2005
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