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LQG, strings and the IKKT matrix model

  1. Nov 5, 2003 #1


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    It is an old observation that both LQG and string theory are fundamentally built from 1-dimensional objects. This is no coincidence. In general, every gauge theory can be formulated in terms of Wilson loops and these can often be described by "a" theory of strings. This was discussed in detail in an old paper by John Baez
    Strings, loops, knots, and gauge fields.

    The natural question is: Which gauge theory has Wilson loops (or more generally: network states) that behave exactly like the fundamental strings of string theory. Surprisingly, an answer has been proposed already quite a while ago in the 90s by Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya: A review of their work is
    hep-th/9908038 IIB Matrix Model .

    I am trying to discuss this idea in the newsgoup sci.physics.research. But maybe here in this forum it will be interesting, too. This is what I originally wrote:

    <quote from s.p.r (here is the full thread) >
    "John Baez" <baez@galaxy.ucr.edu> schrieb im Newsbeitrag
    > In article <Pine.LNX.4.31.0309270749590.32151-100000@feynman.harvard.edu>,
    > Lubos Motl <motl@feynman.harvard.edu> wrote:

    > >Once an infinite number of incorrect terms is removed and
    > >the theory [LQG] is made equivalent to string theory, then - of course -
    > >will obtain a consistent theory of gravity.
    > Yeah, yeah. Even if you're right and we're all mixed up
    > and need to change the model drastically and when we do it
    > turns into something like string theory, we'll still have
    > something interesting, namely a manifestly background-free
    > formulation of string theory as a state sum model. I would
    > not mind this at all.

    I find it really interesting that the following looks like a way to "change
    the model" and indeed turn it into string theory:

    So let's suppose we want to formulate our fundamental theory in terms of
    functional states on a space of gauge connections A_mu taking values in some
    Lie algebra. We want to be really background free. In LQG one does away with
    background _fields_ on spacetime, but one still does need a manifold to set
    up the theory. Let's do away with the assumption of a
    (topological/differentiable) manifold, too.

    Without a manifold it makes no longer sense to have the A_mu(x) be functions
    of coordinates. Therefore let's assume, being very naive, that the
    connection is _independent_ of any coordinates.

    As we learn from LQG, functions (states) on the space of connections
    are spanned by generalized Wilson lines ("networks", graphs - I'll avoid the
    word "spin" for the moment). In ordinary LQG these are embedded into a
    manifold. But since we have just done away with this manifold we now have to
    evaluate our connections on abstract networks that are not embedded into any
    a priori structure. The natural way to do that is to equip the network that
    comes with a given state (function on the space of connections) with a
    D-tuple valued (piecewise defined) 1-form k and compute the holonomy of

    Sum_mu k^mu A_mu

    along the edges of the network, intertwining at the vertices as desired and
    finally tracing over the result.

    In other words, in this manifold-independent formulation of "Ashtekar
    geometry" a network state is given not just by a coloring of edges {e} by
    representations {r} (and coloring of vertices by intertwiners {i}) but
    instead by a coloring by representations _and_ D-tuple valued 1-forms k.
    (The information that was previously contained in the coordinates of a given
    edge has now moved into the extra piece of data k.) So a basis of states
    should now be of the form

    { psi_{e,r,i,k} }

    where each element psi_{e,r,i,k} is associated with an abstract
    combinatorial graph e colored by r,i, and k.

    This basis spans the kinematical Hilbert space. Now we need dynamics.
    Personally I feel that in LQG kinematics is very beautiful but that as soon
    as the ordinary dynamics enters the game things become rather awkward. A
    fundamental theory is not supposed to look awkward, so let's slighly modify
    the ordinary LQG dynamics. Instead of using the action of B^F theory on the
    space of connections A we'd rather use the simplest action quadratic in the

    S = Tr F^2 = Tr [A,A]^2 .

    In other words, the slight modification of LQG that I am proposing here is a
    theory whose configurations are given by _constant_ gauge connections A and
    whose observables (correlation functions) are

    <psi_{e1,r1,i1,k1} psi_{e2,r2,i2,k2}...psi_{en,rn,in,kn}>
    int DA psi_{e1,r1,i1,k1}(A) ...psi_{en,rn,in,kn}(A) exp(-S(A))

    with S and psi_{...} as defined above. The two major modifacations as
    compared to ordinary LQG are the absence of the manifold background and the
    switching from an action linear in the curvature to the simplest one
    quadratic in the curvature.

    The point of all this is the following: In 1996 the authors N. Ishibashi, H.
    Kawai, Y. Kitazawa and A. Tsuchiya have proved for us (see hep-th/9908038
    and references given there) that

    if we identify Wilson lines (network edges) in the above theory with
    fundamental strings

    and if we use U(N>>>1) as the gauge group then the above action for the
    connection induces on these Wilson line, which are now also regarded as
    functionals (states) on the configuration space of the fundamental string,
    the equations of motion of string field theory!


    I spent some time at the "Strings meet loops" symposium trying to find LQG
    people who would find this as interesting as I do. That's because my
    impression is that maybe the true value of this IKKT model is not properly
    appreciated in the string community, which might have to do with its radical
    background independence. Lubos mentioned that timelike T-duality which does
    away with time (!) looks suspicious. But maybe it is just what we need, I
    wonder. In any case, when looked at it from the proper perspective (as I
    have tried to demonstrate above) the IKKT model looks much more like LQG
    than like string theory. Of course, looked at it from another perspective it
    completely looks like string theory. Great.

    Luckily, I found several very friendly and open minded LQGists who did find
    this interesting. I am looking forward to hearing what they come up with
    when studying this in detail.
    <end quote>
  2. jcsd
  3. Nov 5, 2003 #2


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    Welcome to the PF Urs. Your idea looks extremely interesting and I hope some workers pick up on it. Wish I could myself, but I'm way to low on the knowledge ladder to qualify.
  4. Nov 5, 2003 #3


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    I agree. welcome Urs!
    At first sight the part of your idea that involves doing away with the manifold reminds me of the paper by Laurent Freidel and Etera Livine called "Spin Networks for Non-Compact Groups"

    http://arxiv.org/hep-th/0205268 [Broken]

    they find they can throw away the underlying manifold (where originally the connections lived) and use a collection of oriented graphs &Gamma;
    and define a space "discrete connections" on each graph


    and then define a measure, and an L2 hilbertspace, on each of
    these A&Gamma; spaces of discrete connections
    one such hilbertspace for each graph

    (but these graphs do not have to be embedded in a specific manifold,
    or, if they are, one must consider diffeomorphism equivalence classes of graphs---seems better to let them be abstract)

    finally they put together all these hilbertspaces (one for each &Gamma;) into one big one---reminiscent of fock space
    and this then is the quantum states of geometry

    I'm giving you only an unreliable first impression because I'm just now trying to read the paper, but it also (like your idea) seems
    to get away from the original manifold

    would like to hear more about what you propose. happy if you post
    here since easier for me to read and reply to than SPR is
    Last edited by a moderator: May 1, 2017
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