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String Field Theory?

  1. Mar 31, 2004 #1
    [SOLVED] String Field Theory?

    Stupid Question of the Day:

    What advantage does string field theory have over string theory? In all
    the standard texts I own, there is very little reference made to string
    field theory yet I know it exists.

    Can anyone help me understand the motivation for it or point me to a good
    primer?

    Thanks
    ========================

    Lubos Motl (moderator):

    Let me start with an answer. I hope that people will post other answers,
    too. String field theory is a tool to formulate string theory in a
    language that is as similar to regular quantum field theory as possible.
    (But it has infinitely many component fields.) Unlike the usual rules of
    string theoretical CFT, string field theory allows one to compute
    off-shell quantities.

    On-shell quantities are the scattering amplitudes, off-shell quantities
    are e.g. the Green's functions, to simplify it a bit. String field theory
    is only simple and predictive for open strings; the closed string field
    action must be corrected by new terms at every order in the Taylor
    expansion in g. Because of the off-shell character of (open) string field
    theory, string field theory is very good to study questions such as the
    tachyon condensation - to check Sen's conjecture that the minimum of the
    tachyon potential corresponds to a total destruction of the D-brane.
    String field theory used to be believed to be useful for nonperturbative
    treatment of string theory, but evidence supporting such a far-reaching
    claim has been very limited so far.

    String field theory has played virtually no role in the second
    superstring revolution, but it was useful to study the tachyons, and I
    think that a good review is the following:

    http://arxiv.org/abs/hep-th/0311017

    Cheers,
    Lubos
     
  2. jcsd
  3. Apr 1, 2004 #2
    On Wed, 31 Mar 2004, Creighton Hogg wrote:

    > Stupid Question of the Day:
    >
    > What advantage does string field theory have over string theory? In all
    > the standard texts I own, there is very little reference made to string
    > field theory yet I know it exists.
    >
    > Can anyone help me understand the motivation for it or point me to a good
    > primer?


    IIRC the textbook by Kaku emphasizes string field theory aspects.

    Think of second quantization. The equation of motion for the single
    string is

    Q|psi> = 0

    where Q is the BRST charge. When second-quantizing this the
    state psi becomes a classical field and the quantum equation
    of motion must become a classical equation of motion. It
    is easy to find an action with this property, namely

    S0 ~ <psi|Q|psi> .

    Here the scalar product is taken in the single string's Hilbert
    space, but the coefficients of the various string oscillator
    excitations in |psi> now play the role of spacetime fields
    that become quantized by quantizing the above action.

    That's free (bosonic) string field theory.

    Now add the correct interaction. For the open bosonic string
    there is really only a single form of interaction, namely the
    splitting/joining of two strings. The difficult problem was
    to find a product operation '*' on the single string's Hilbert
    space such that

    |psi> * |phi>

    is again a state in the single string's Hilbert space and
    in particular the one obtained by joining the strings in the
    states described by |psi> and |phi>. Such a star product was
    found by Witten, who used it to construct what is called
    open bosonic cubic string field theory, where the free action
    is as above and the interaction term is the cubic vertex
    of the form

    <0| psi * psi * psi |0>

    Lubos Motl has a lot of very nice pictures illustrating this
    and related concepts in his hep-th/0403187.


    When varying the total action the equation of motion now
    is that of the free sting plus an interaction term:


    Q |psi> = c |psi * psi>,

    where c is some constant.

    The BRST charge is essentially the exterior derivative on the
    gauge group (where the ghost fields play the role of differential
    forms) and it is graded-Leibnitz with respect to '*', i.e.

    Q (psi * phi) = (Q psi) * phi + (-1)^a psi * (Q phi),

    where a is the ghost number of phi. This means that you can
    morally think of the above equation of motion as something
    analogous to

    d omega = c omega /\ omega ,

    where omega is a differential form and '/\' is the wedge
    product. This is very suggestive.


    Berkovits has, maybe inspired by this suggestiveness, proposed
    a construction somewhat similar to that sketched above but
    for the superstring. There, the string field action is similar in
    structure to that of a WZW model.

    As far as I understand it can be proven that open bosonic
    string field theory correctly reproduces the amplitudes
    of the first-quantized theory, but the same has not been
    completely checked yet for Berkovits superstring field
    theory, I think.
     
  4. Apr 1, 2004 #3
    On Thu, 1 Apr 2004, Urs Schreiber wrote:

    > IIRC the textbook by Kaku emphasizes string field theory aspects.


    Which might be one of the reasons why nearly none emphasizes the textbook
    by Kaku. ;-) Otherwise, your explanation of SFT looks perfect to me.

    > As far as I understand it can be proven that open bosonic
    > string field theory correctly reproduces the amplitudes
    > of the first-quantized theory, but the same has not been
    > completely checked yet for Berkovits superstring field
    > theory, I think.


    I would agree. The nontrivial part of the argument occurs for the loops,
    and Barton Zwiebach showed why Witten's CFT reproduces the correct
    integral over the moduli spaces of Riemann surfaces.

    http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+A+ZWIEBACH,B+AND+TITLE+cover

    A corresponding proof in the twistorial SFT would require to cover the
    whole moduli space of Riemann surfaces, including the GL(1) Wilson lines
    around the 2h cycles, where h is the genus. No one has properly understood
    the loops in the twistor language yet (or at least: she has not published
    it yet), and it is a subtle thing, and therefore no one has also tried to
    prove that the moduli space is covered.

    The tree level calculations should be easier, and the disk diagram should
    be covered the same way like in bosonic SFT.

    Barton's proof is a very interesting technical and geometrical result. A
    scanned version is available at the web address above. Zwiebach shows that
    string field theory picks the metrics of minimal area among those that
    satisfy the constraint that nontrivial open Jordan curves are never
    shorter than pi.

    At any rate, the huge industry based on SFT was mostly connected with
    tachyons and their fate.

    Best wishes
    Lubos
    ______________________________________________________________________________
    E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
    eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
    ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
     
  5. Apr 1, 2004 #4
    I want to thank both Urs Schreiber and Lubos Motl for their explanations.
    I found them quite helpful.
     
  6. Apr 2, 2004 #5
    On Thu, 1 Apr 2004, Creighton Hogg wrote:

    > I want to thank both Urs Schreiber and Lubos Motl for their explanations.
    > I found them quite helpful.



    Maybe one should emphasize, as Lubos has already mentioned,
    that only open string field theory is understood
    to a larger extent. Closed string field theory is as yet non-existent
    as a full theory, though that doesn't prevent people from studying
    what is known.

    For instance from the transparacies at

    http://vishnu.phys.h.kyoto-u.ac.jp/~shinpei/handai.pdf

    one quickly sees that the first terms of CSFT are completely
    analogous to those of OSFT.

    One can apparently also write down 'open-closed SFT' as for
    instance in hep-th/9711100 which has vertices for all the
    ways that open and closed strings can split and join.



    But in purely open SFT if there are open strings they should
    be able to combine into closed strings. So how can we have
    OSFT without closed strings?

    Apparently the answer is that indeed somehow closed strings
    do play a role in OSFT. I am no expert on this, but I have been
    told of two mechanisms:

    - Closed strings should correspond to certain poles in the
    amplitudes of OSTF.

    - In bosonic OSFT we know that the D25 brane decays as the
    tachyon condenses. This forces all open strings, whose ends
    are attached to this brane, to close. So somehow 'after' the
    decay of the D25 branes OSFT must describe closed strings.
    But how precisely this is supposed to work is apparently not
    really known.

    (Hopefully this is about right. Otherwise somebody will
    hopefully correct me.)


    What I always found interesting is how something like
    closed string field theory can be derived in a very
    elementary and illuminating way from matrix models.

    I have once reproduced the elementary calculation which
    demonstrates that Wilson loops in completely
    dimensionally reduced Yang-Mills theory obey the
    equations of motion (or at least approximations thereof)
    of a closed string field theory at

    http://golem.ph.utexas.edu/string/archives/000314.html .

    However I don't know if this derivation has ever been
    developed any further. I am wondering how the higher terms
    in the Taylor expansion with respect to g in CSFT
    would turn up in this matrix model derivation. Maybe
    this is hidden in the subtleties with taking the continuum
    and N-> infty limit in that model?


    Recently the paper

    A. Sen, Energy Momentum Tensor and Marginal Deformations
    in Open String Field Theory, hep-th/0403200

    has appeared, which I should read, but haven't yet. Since
    I have thought quite a bit about deformations of
    (S)CFTs in hep-th/0401175 I would like to understand how
    this might carry over to OSFT.
     
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