Understanding String Field Theory and Its Role in the Study of String Theory

[Creighton Hogg]

[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

 

[Urs Schreiber]

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?[/color]

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.

 

[Lubos Motl]

On Thu, 1 Apr 2004, Urs Schreiber wrote:

> IIRC the textbook by Kaku emphasizes string field theory aspects.[/color]

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.[/color]

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/
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[Creighton Hogg]

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

 

[Urs Schreiber]

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.[/color]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.