CFTs from OSFT? [Archive] - Physics Forums

Urs Schreiber

Apr29-04, 02:04 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>In string field theory the BRST operator Q is well known to be morally an\nexterior derivative on the star-product algebra of string fields. For a\ngiven solution \\Phi to the OSFT equations of motion\n\nQ \\Psi + \\Phi \\star \\Phi = 0\n\nit seems that the operator\n\nQ + \\Phi \\star\n\ncan be interpreted similarly as a gauge covariant exterior derivative, with\n\\Phi the _gauge connection_. In order to see what I mean in detail please\nhave a look at\n\nhttp://golem.ph.utexas.edu/string/archives/000356.html ,\n\nwhere formulas etc. are given.\n\nThere I try to argue that indeed\n\n\\tilde Q = Q + \\Phi \\star\n\nis a nilpotent operator of ghost number 1 (if \\Phi is physical) and that\n\\tilde Q is indeed the worldsheet BRST operator for single strings which\npropagate in the _background_ described by \\Phi.\n\nThis would be nice, since it would give a direct way to find worldsheet CFTs\nby deforming with respect to certain background fields.\n\nActually, the ideas sketched at the above link are pretty simple, so that I\nsuspect that they are either well known - or wrong. ;-) Comments are\nappreciated!\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>In string field theory the BRST operator Q is well known to be morally an
exterior derivative on the star-product algebra of string fields. For a
given solution \Phi to the OSFT equations of motion

Q \Psi + \Phi \star \Phi =

it seems that the operator

Q + \Phi \star

can be interpreted similarly as a gauge covariant exterior derivative, with
\Phi the _gauge connection_. In order to see what I mean in detail please
have a look at

http://golem.ph.utexas.edu/string/archives/000356.html ,

where formulas etc. are given.

There I try to argue that indeed

\tilde Q = Q + \Phi \star

is a nilpotent operator of ghost number 1 (if \Phi is physical) and that
\tilde Q is indeed the worldsheet BRST operator for single strings which
propagate in the _background_ described by \Phi.

This would be nice, since it would give a direct way to find worldsheet CFTs
by deforming with respect to certain background fields.

Actually, the ideas sketched at the above link are pretty simple, so that I
suspect that they are either well known - or wrong. ;-) Comments are
appreciated!

Lubos Motl

Apr29-04, 02:47 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Thu, 29 Apr 2004, Urs Schreiber wrote:\n\n> Actually, the ideas sketched at the above link are pretty simple, so that I\n> suspect that they are either well known - or wrong. ;-) Comments are\n> appreciated!\n\nI think that you nearly re-discovered several things. The first one is the\nreason why the cubic string field theory is called "Chern-Simons-like"\nstring field theory. It\'s because the action is formally reproducing the\nChern-Simons action, with wedge replaced by star, A replaced by Phi, and\nthe exterior derivative "d" replaced by a slightly more complicated "Q".\n\nThe second is the relation of Witten\'s cubic SFT and noncommutative\ngeometry, which was actually Witten\'s original motivation in\n\nNONCOMMUTATIVE GEOMETRY AND STRING FIELD THEORY.\nBy Edward Witten (Princeton U.),. Print-86-0083 (PRINCETON), Oct 1985.\n70pp.\nPublished in Nucl.Phys.B268:253,1986\n\nand this basic idea is behind many things, e.g. Gross and Taylor\'s "Split\nstring field theory"\n\nhttp://arxiv.org/abs/hep-th/0105059\nhttp://arxiv.org/abs/hep-th/0106036\n\nthat proposes the D-branes to be analogous to the GMS projectors, where\nthe matrix indices are taken to be the halves of the open string.\n\nThe third idea that your slightly vague comment seems to be related to is\nbackground-independent string field theory - an idea that allows you to\nstart with a purely cubic action (the interaction term A*A*A only), and\ngenerate the quadratic term involving "Q" as a vacuum condensate of A. A\nclosed string version of this was discussed e.g. by the Kyoto group (HIKKO)\n\nhttp://ccdb3fs.kek.jp/cgi-bin/img_index?8606274\n\nLet me choose a few random references about an alternative interpretation\nof "background independent SFT"\n\nhttp://arxiv.org/abs/hep-th/9208027\nhttp://arxiv.org/abs/hep-th/9303143\n\nIf you meant something different and more specific, then I apologize that\nI have not understood it.\n\nBest wishes\nLubos\n___________________________________ ___________________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)\n^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Thu, 29 Apr 2004, Urs Schreiber wrote:

> Actually, the ideas sketched at the above link are pretty simple, so that I
> suspect that they are either well known - or wrong. ;-) Comments are
> appreciated!

I think that you nearly re-discovered several things. The first one is the
reason why the cubic string field theory is called "Chern-Simons-like"
string field theory. It's because the action is formally reproducing the
Chern-Simons action, with wedge replaced by star, A replaced by \Phi, and
the exterior derivative "d" replaced by a slightly more complicated "Q".

The second is the relation of Witten's cubic SFT and noncommutative
geometry, which was actually Witten's original motivation in

NONCOMMUTATIVE GEOMETRY AND STRING FIELD THEORY.
By Edward Witten (Princeton U.),. Print-86-0083 (PRINCETON), Oct 1985.
70pp.
Published in Nucl.Phys.B268:253,1986

and this basic idea is behind many things, e.g. Gross and Taylor's "Split
string field theory"

http://arxiv.org/abs/http://www.arxiv.org/abs/hep-th/0105059
http://arxiv.org/abs/http://www.arxiv.org/abs/hep-th/0106036

that proposes the D-branes to be analogous to the GMS projectors, where
the matrix indices are taken to be the halves of the open string.

The third idea that your slightly vague comment seems to be related to is
background-independent string field theory - an idea that allows you to
start with a purely cubic action (the interaction term A*A*A only), and
generate the quadratic term involving "Q" as a vacuum condensate of A. A
closed string version of this was discussed e.g. by the Kyoto group (HIKKO)

http://ccdb3fs.kek.jp/cgi-bin/img_index?8606274

Let me choose a few random references about an alternative interpretation
of "background independent SFT"

http://arxiv.org/abs/http://www.arxiv.org/abs/hep-th/9208027
http://arxiv.org/abs/http://www.arxiv.org/abs/hep-th/9303143

If you meant something different and more specific, then I apologize that
I have not understood it.

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)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Urs Schreiber

Apr30-04, 09:34 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"Lubos Motl" <motl@feynman.harvard.edu> schrieb im Newsbeitrag\nnews:Pine.LNX.4.31.0404291436370.1412 0-100000@feynman.harvard.edu...\n\n\n> closed string version of this was discussed e.g. by the Kyoto group\n(HIKKO)\n>\n> http://ccdb3fs.kek.jp/cgi-bin/img_index?8606274\n\nHi Lubos -\n\nmany thanks for the many very valuable links!\n\nIndeed, I realize that the mechanism which I talked about, namely that a\n"background" string field \\Phi gives rise to a deformation of the BRST\noperator for the perturbations \\psi in the new field Phi \\to \\Phi + \\psi is\nexactly the mechanism discussed in the above paper. In order to explicitly\ndemonstrate this one just has to rewrite what I wrote in terms of equations\nof motion in terms of the Lagrangian. I have typed the formulas showing this\nhere:\n\nhttp://golem.ph.utexas.edu/string/archives/000356.html#c000998 .\n\nOf course what I did not see is that this way one can make the flat space\nBRST operator even vanish, or, from the other point of view, get ordinary\nstring field theory from just the cubic interaction vertex. That\'s extremely\nnice.\n\nBTW, there seems to be a relation to the IIB matrix model: Hata shows in the\nabove mentioned paper that the background field which gives the flat space\nBRST operator is a background of infinitesimally small strings. The\ninteraction of a single string with a background of tiny strings looks like\na kinetic term.\n\nBut something very similar is true when deriving the equations of motion of\nclosed string field theory (including the kinetic term) from the dynamics of\nWilson loops in the IIB Matrix Model. When I reviewed this calculation at\n\nhttp://golem.ph.utexas.edu/string/archives/000314.html\n\nI remarked (following the original authros, of course) that the kinetic term\nthere too comes from the split/join interaction with an infinitesimally\nsmall piece of string.\n\nBTW, in a recent thread we seemed to disagree about what happens to the\nworldsheet translation generator\n\nT - \\bar T\n\nwhen background fields are turned on. I argued that this term has to remain\nthe same no matter what, while the other terms change their form, but maybe\nI expressed myself badly. If that\'s the case I\'d like to point to the\nparagraph at the top of page 5 of the above mentioned paper, which states\nprecisely what I had in mind.\n\nFor the record, there it says:\n\n"[...] the BRS operator [Q_B] and hence the kinetic term \\Phi \\cdot Q_B \\Phi\nin (1) depend explicitly on the flat background metric \\eta_{\\mu\\nu}. On the\nother hand, the \\star-product and the interaction term \\Phi^3 are completely\nindependent of the space-time metric [...] and so is the constraint [\\int\n\\exp(-i \\theta (L - \\bar L)) d\\theta]".\n\n\nI also had a look at the papers by Witten and Shatshvili hat you kindly\nmentioned. They contain very abstract formulatins of string field theory. I\nam not sure that I fully got the point of these papers... :-)\n\nBTW, since we can think of string field theory as a theory in the space of\nworld-sheet theories of the string, there is a natural concern:\n\nEvery classical solution \\Phi to the string field equations of motion\ndescribes a worldsheet theory. But is this always conformally invariant? How\ndo we know? What if it is not?\n\nMaybe the classical solutions to the string field equations correspond to\nconformally invariant worldsheet theories, while the quantum solutions to\nstring field theory would contain "fluctuations" about the conformal point\nin the space of 2d field theories?\n\nMany thanks for your help!\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Lubos Motl" <motl@feynman.harvard.edu> schrieb im Newsbeitrag
news:Pine.LNX.4.31.0404291436370.14120-100000@feynman.harvard.edu...

> closed string version of this was discussed e.g. by the Kyoto group
(HIKKO)
>
> http://ccdb3fs.kek.jp/cgi-bin/img_index?8606274

Hi Lubos -

many thanks for the many very valuable links!

Indeed, I realize that the mechanism which I talked about, namely that a
"background" string field \Phi gives rise to a deformation of the BRST
operator for the perturbations \psi in the new field \Phi \to \Phi + \psi is
exactly the mechanism discussed in the above paper. In order to explicitly
demonstrate this one just has to rewrite what I wrote in terms of equations
of motion in terms of the Lagrangian. I have typed the formulas showing this
here:

http://golem.ph.utexas.edu/string/archives/000356.html#c000998 .

Of course what I did not see is that this way one can make the flat space
BRST operator even vanish, or, from the other point of view, get ordinary
string field theory from just the cubic interaction vertex. That's extremely
nice.

BTW, there seems to be a relation to the IIB matrix model: Hata shows in the
above mentioned paper that the background field which gives the flat space
BRST operator is a background of infinitesimally small strings. The
interaction of a single string with a background of tiny strings looks like
a kinetic term.

But something very similar is true when deriving the equations of motion of
closed string field theory (including the kinetic term) from the dynamics of
Wilson loops in the IIB Matrix Model. When I reviewed this calculation at

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

I remarked (following the original authros, of course) that the kinetic term
there too comes from the split/join interaction with an infinitesimally
small piece of string.

BTW, in a recent thread we seemed to disagree about what happens to the
worldsheet translation generator

T - \bar T

when background fields are turned on. I argued that this term has to remain
the same no matter what, while the other terms change their form, but maybe
I expressed myself badly. If that's the case I'd like to point to the
paragraph at the top of page 5 of the above mentioned paper, which states
precisely what I had in mind.

For the record, there it says:

"[...] the BRS operator [Q_B] and hence the kinetic term \Phi \cdot Q_B \Phi
in (1) depend explicitly on the flat background metric \eta_{\mu\nu}. On the
other hand, the \star-product and the interaction term \Phi^3 are completely
independent of the space-time metric [...] and so is the constraint [\int\exp(-i \theta (L - \bar L)) d\theta]".

I also had a look at the papers by Witten and Shatshvili hat you kindly
mentioned. They contain very abstract formulatins of string field theory. I
am not sure that I fully got the point of these papers... :-)

BTW, since we can think of string field theory as a theory in the space of
world-sheet theories of the string, there is a natural concern:

Every classical solution \Phi to the string field equations of motion
describes a worldsheet theory. But is this always conformally invariant? How
do we know? What if it is not?

Maybe the classical solutions to the string field equations correspond to
conformally invariant worldsheet theories, while the quantum solutions to
string field theory would contain "fluctuations" about the conformal point
in the space of 2d field theories?

Many thanks for your help!

Urs Schreiber

May3-04, 02:27 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"Urs Schreiber" <Urs.Schreiber@uni-essen.de> schrieb im Newsbeitrag\nnews:Pine.LNX.4.31.0404300933580.1845 4-100000@feynman.harvard.edu...\n\n> Maybe the classical solutions to the string field equations correspond to\n> conformally invariant worldsheet theories, while the quantum solutions to\n> string field theory would contain "fluctuations" about the conformal point\n> in the space of 2d field theories?\n\nMaybe I am beginning to understand how to decide if a given string field\nbackground induces a conformal dynamics on the worldsheet:\n\nWhen computing the worldsheet BRST operator for the new background from the\nstring field solutions one has to check whether it has the same trilinear\nghost structure as usual. If it does then nilpotency should imply that the\nnew Virasoro generators still satisfy the Virasoro algebra and hence define\na conformal worldsheet theory.\n\nUsing equation (2.34) of Ohmori\'s review as well as my formula\nhttp://golem.ph.utexas.edu/string/archives/000356.html\nfor the new BRST operator it is easy to see that this condition is at least\nsatisfied for tachyon and gauge field backgrounds, as expected. The only\npotentitally problematic term in that equation is that proportional to\nbeta_1. Does that vanish for any reason?\n\nI have LaTeXified the general idea that I am talking about here in a brief\nnote. I\'d be grateful for critical comments on that file, which can be found\nat\n\nhttp://www-stud.uni-essen.de/~sb0264/p8.pdf .\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Urs Schreiber" <Urs.Schreiber@uni-essen.de> schrieb im Newsbeitrag
news:Pine.LNX.4.31.0404300933580.18454-100000@feynman.harvard.edu...

> Maybe the classical solutions to the string field equations correspond to
> conformally invariant worldsheet theories, while the quantum solutions to
> string field theory would contain "fluctuations" about the conformal point
> in the space of 2d field theories?

Maybe I am beginning to understand how to decide if a given string field
background induces a conformal dynamics on the worldsheet:

When computing the worldsheet BRST operator for the new background from the
string field solutions one has to check whether it has the same trilinear
ghost structure as usual. If it does then nilpotency should imply that the
new Virasoro generators still satisfy the Virasoro algebra and hence define
a conformal worldsheet theory.

Using equation (2.34) of Ohmori's review as well as my formula
http://golem.ph.utexas.edu/string/archives/000356.html
for the new BRST operator it is easy to see that this condition is at least
satisfied for tachyon and gauge field backgrounds, as expected. The only
potentitally problematic term in that equation is that proportional to
\beta_1. Does that vanish for any reason?

I have LaTeXified the general idea that I am talking about here in a brief
note. I'd be grateful for critical comments on that file, which can be found
at

http://www-stud.uni-essen.de/~sb0264/p8.pdf .

Urs Schreiber

May7-04, 09:55 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Thu, 29 Apr 2004, Lubos Motl wrote:\n\n> NONCOMMUTATIVE GEOMETRY AND STRING FIELD THEORY.\n> By Edward Witten (Princeton U.),. Print-86-0083 (PRINCETON), Oct 1985.\n> 70pp.\n> Published in Nucl.Phys.B268:253,1986\n\nThe key issue which I am trying to understand is that mentioned below\nequation (8) on p. 256 of that paper. There it says about the object\nQ+A (where Q is the BRST operator for Minkowski background and A the\noperator of *-multiplication with the classical string field solution A):\n\n"[this] means at least roughly that (Q+A) is the nilpotent BRST generator\nof some more general conformally invariant 1+1 dimensional theory"\n\nA couple of references [17] are given, which are however unfortunately\npre-arXiv preprints. I\'d be very grateful if anyone help me find any of\nthe following texts:\n\n- D. Friedan, Univ. of Chicago preprint (August, 1985)\n\n- D. Friedan and S. Shenker, talk at Aspen Summer Institute (1984)\n\n- E. Fradkin and A. Tsytlin, Lebedev preprint (1984)\n\n- C.G. Callan, Jr. D. Friedan, E. Martinec and M. Perry, Princeton\nPreprint (1985)\n\nEven better, I\'d appreciate any pointers to more recent papers\nalong the lines of these references (should there exist any).\n\nBTW, it seems to me that more precisely not Q+A is the new BRST\noperator, as Witten says "roughly", but the operator which acts as\n\n|psi> -> Q|psi> + |A * psi> + |psi * A>.\n\nThat\'s because this operator is not only nilpotent but also\nenjoys the graded Leibnitz propery with respect to the star\nproduct as well as the generalized Stokes law, both of which\nare in general not satisfied by simply Q+A.\n\nIn order to better understand this I would like to find out\nhow the operator\n\n(delta Q)|psi> := |A * psi> + |psi * A>.\n\nlooks like in the single string\'s Hilbert space, i.e. without\nexplicitly using the star product. That should be pretty easy\nfor special cases where A contains just, say, the tachyon\nor gauge field excitation but where psi is completely\ngeneral.\n\nI know in principle which formulas to use in order to calculate this\n(e.g. as indicated in section 2.3 of Ohmori\'s review hep-th/0102085)\nbut currently I don\'t see a concise form of the solution, yet.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Thu, 29 Apr 2004, Lubos Motl wrote:

> NONCOMMUTATIVE GEOMETRY AND STRING FIELD THEORY.
> By Edward Witten (Princeton U.),. Print-86-0083 (PRINCETON), Oct 1985.
> 70pp.
> Published in Nucl.Phys.B268:253,1986

The key issue which I am trying to understand is that mentioned below
equation (8) on p. 256 of that paper. There it says about the object
Q+A (where Q is the BRST operator for Minkowski background and A the
operator of *-multiplication with the classical string field solution A):

"[this] means at least roughly that (Q+A) is the nilpotent BRST generator
of some more general conformally invariant 1+1 dimensional theory"

A couple of references [17] are given, which are however unfortunately
pre-arXiv preprints. I'd be very grateful if anyone help me find any of
the following texts:

- D. Friedan, Univ. of Chicago preprint (August, 1985)

- D. Friedan and S. Shenker, talk at Aspen Summer Institute (1984)

- E. Fradkin and A. Tsytlin, Lebedev preprint (1984)

- C.G. Callan, Jr. D. Friedan, E. Martinec and M. Perry, Princeton
Preprint (1985)

Even better, I'd appreciate any pointers to more recent papers
along the lines of these references (should there exist any).

BTW, it seems to me that more precisely not Q+A is the new BRST
operator, as Witten says "roughly", but the operator which acts as

|\psi> -> Q|\psi> + |A * \psi> + |\psi * A>.

That's because this operator is not only nilpotent but also
enjoys the graded Leibnitz propery with respect to the star
product as well as the generalized Stokes law, both of which
are in general not satisfied by simply Q+A.

In order to better understand this I would like to find out
how the operator

(\delta Q)|\psi> := |A * \psi> + |\psi * A>.

looks like in the single string's Hilbert space, i.e. without
explicitly using the star product. That should be pretty easy
for special cases where A contains just, say, the tachyon
or gauge field excitation but where \psi is completely
general.

I know in principle which formulas to use in order to calculate this
(e.g. as indicated in section 2.3 of Ohmori's review http://www.arxiv.org/abs/hep-th/0102085)
but currently I don't see a concise form of the solution, yet.

Urs Schreiber

May7-04, 11:49 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nOn Fri, 7 May 2004, Urs Schreiber wrote:\n\n> |psi> -> Q|psi> + |A * psi> + |psi * A>.\n\nApologies for replying again to my own post. I just noted that\nthis is indeed the relation found and used in the modern\nliterature, e.g. in equation (5.4) of\n\nI. Kishimoto & K. Ohmori\n"CFT Description of Identity String Field: Toward Derivation of the VSFT\nAction"\nhep-th/0112169\n\nI know that the derivation of this relation is not a big deal at all,\neven though it\'s kind of neat. I find it interesting because my central\nquestion is still:\n\nWhat do we know about the worldsheet theories associated with such\ndeformed BRST operators Q\' defined as\n\nQ\'|psi> := Q|psi> + |A*psi> + |psi*A>\n\nfor some classical solution A of the original OSFT?\n\nIn the specific context of the above paper I guess that this should be\na subtle issue, since the classical solution A discussed there is argued\nto be the true tachyon vacuum, where the D25 brane has decayed.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Fri, 7 May 2004, Urs Schreiber wrote:

> |\psi> -> Q|\psi> + |A * \psi> + |\psi * A>.

Apologies for replying again to my own post. I just noted that
this is indeed the relation found and used in the modern
literature, e.g. in equation (5.4) of

I. Kishimoto & K. Ohmori
"CFT Description of Identity String Field: Toward Derivation of the VSFT
Action"
http://www.arxiv.org/abs/hep-th/0112169

I know that the derivation of this relation is not a big deal at all,
even though it's kind of neat. I find it interesting because my central
question is still:

What do we know about the worldsheet theories associated with such
deformed BRST operators Q' defined as

Q'|\psi> := Q|\psi> + |A*\psi> + |\psi*A>

for some classical solution A of the original OSFT?

In the specific context of the above paper I guess that this should be
a subtle issue, since the classical solution A discussed there is argued
to be the true tachyon vacuum, where the D25 brane has decayed.

Urs Schreiber

May7-04, 12:59 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Fri, 7 May 2004, Urs Schreiber wrote:\n\n> I. Kishimoto & K. Ohmori\n> "CFT Description of Identity String Field: Toward Derivation of the VSFT\n> Action"\n> hep-th/0112169\n\nA great set of lecture notes reviewing this stuff is\n\nAref\'eva, Belov, Giryavets, Koshelev, Medvedev\nNoncommutative Field Theories and (Super) String Field Theories\nhep-th/0111208\n\nDoes anyone know what the status of cubic _super_string field theory is?\nApperently the original proposals had some troubles with contact term\ndivergencies, which however could be (partially?) cured, it seems.\nWhat can, what cannot be reliably computed using open _super_string\nfield theory?\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Fri, 7 May 2004, Urs Schreiber wrote:

> I. Kishimoto & K. Ohmori
> "CFT Description of Identity String Field: Toward Derivation of the VSFT
> Action"
> http://www.arxiv.org/abs/hep-th/0112169

A great set of lecture notes reviewing this stuff is

Aref'eva, Belov, Giryavets, Koshelev, Medvedev
Noncommutative Field Theories and (Super) String Field Theories
http://www.arxiv.org/abs/hep-th/0111208

Does anyone know what the status of cubic _super_string field theory is?
Apperently the original proposals had some troubles with contact term
divergencies, which however could be (partially?) cured, it seems.
What can, what cannot be reliably computed using open _super_string
field theory?

Urs Schreiber

May19-04, 03:28 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Thu, 29 Apr 2004, Lubos Motl wrote:\n\n> On Thu, 29 Apr 2004, Urs Schreiber wrote:\n>\n> > Actually, the ideas sketched at the above link are pretty simple, so that I\n> > suspect that they are either well known - or wrong. ;-) Comments are\n> > appreciated!\n>\n> I think that you nearly re-discovered several things. The first one is the\n\nI was lucky enough to come across a recent paper today where indeed\nthe idea that classical solutions to OSFT give deformed worldsheet\ntheories is studied for a restricted class of solutions in some detail.\n\nThe paper is\n\nJ. Kluson,\nExact Solutions in SFT and Marginal Deformation in BCFT,\nhep-th/0303199 .\n\nI deduce from that that the essence of my original question isn\'t\nfully trivial and well known (but I am prepared to be convinced\nof the opposite ;-).\n\nThe above paper studies the special case where classical SFT solutions\ngive rise to deformed BRST operators which describe marginally\ndeformed boundary conformal field theories on the worldsheet.\n\nI have a more detailed discussion of this paper with links, formulas,\nsome comments and questions on the String Coffee Table at\n\nhttp://golem.ph.utexas.edu/string/archives/000366.html .\n\nMy main question at the end is this:\n\nGiven some string field Phi_0, is there an easy way to compute by hand\nthe action of the operator\n\n[Phi_0 *, .] : |Psi> |-> |Phi_0 * psi \\pm Psi * Phi_0>\n\nin terms of the action of the usual worldsheet creators/annihilators?\n\nI mean something like\n\n[Phi_0 *, .] = sum_{n} f_nmq alpha_m alpha_q + cdots .\n\n\nI know in principle how to compute the right hand side, using\nfor instance the operator representation of the SFT vertices,\nbut so far I didn\'t manage to obtain any closed form solutions\nthat are more enlightning than the bare definition\n\nPhi_0 * Psi = <Phi_0|,Psi| |V_3>\n\nand so on.\n\nThanks for any help and comments!\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Thu, 29 Apr 2004, Lubos Motl wrote:

> On Thu, 29 Apr 2004, Urs Schreiber wrote:
>
> > Actually, the ideas sketched at the above link are pretty simple, so that I
> > suspect that they are either well known - or wrong. ;-) Comments are
> > appreciated!
>
> I think that you nearly re-discovered several things. The first one is the

I was lucky enough to come across a recent paper today where indeed
the idea that classical solutions to OSFT give deformed worldsheet
theories is studied for a restricted class of solutions in some detail.

The paper is

J. Kluson,
Exact Solutions in SFT and Marginal Deformation in BCFT,
http://www.arxiv.org/abs/hep-th/0303199 .

I deduce from that that the essence of my original question isn't
fully trivial and well known (but I am prepared to be convinced
of the opposite ;-).

The above paper studies the special case where classical SFT solutions
give rise to deformed BRST operators which describe marginally
deformed boundary conformal field theories on the worldsheet.

I have a more detailed discussion of this paper with links, formulas,
some comments and questions on the String Coffee Table at

http://golem.ph.utexas.edu/string/archives/000366.html .

My main question at the end is this:

Given some string field \Phi_0, is there an easy way to compute by hand
the action of the operator

[\Phi_0 *, .[/itex]] : |\Psi> |-> |\Phi_0 * \psi \pm \Psi * \Phi_0>

in terms of the action of the usual worldsheet creators/annihilators?

I mean something like

[\Phi_0 *, .] = sum_{n} f_{nmq} \alpha_m \alpha_q + cdots .

I know in principle how to compute the right hand side, using
for instance the operator representation of the SFT vertices,
but so far I didn't manage to obtain any closed form solutions
that are more enlightning than the bare definition

[itex]\Phi_0 * \Psi = <\Phi_0|,\Psi| |V_3>

and so on.

Thanks for any help and comments!

Davide Gaiotto

May25-04, 04:16 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:<Pine.LNX.4.31.0405191512520.9116-100000@feynman.harvard.edu>...\n>\n> Given some string field Phi_0, is there an easy way to compute by hand\n> the action of the operator\n>\n> [Phi_0 *, .] : |Psi> |-> |Phi_0 * psi \\pm Psi * Phi_0>\n>\n> in terms of the action of the usual worldsheet creators/annihilators?\n>\n> I mean something like\n>\n> [Phi_0 *, .] = sum_{n} f_nmq alpha_m alpha_q + cdots .\n>\n>\n> I know in principle how to compute the right hand side, using\n> for instance the operator representation of the SFT vertices,\n> but so far I didn\'t manage to obtain any closed form solutions\n> that are more enlightning than the bare definition\n>\n> Phi_0 * Psi = <Phi_0|,Psi| |V_3>\n>\n> and so on.\n>\n> Thanks for any help and comments!\n\nHi,\nI assume you want some sort of analytic formula. ( though if you want\nto\nexperiment a bit with actual star products the simplest thing might be\nto\nwrite a program using the conservation laws from hep-th/0006240)\n\nIt\'s possible to easily write down the effect of the multiplication by\nthe\nSL(2) invarian vacuum:\n\n|0> * Psi = exp(\\sum A_n L_{-n}) exp(\\sum B_n L_n) Psi\n\nIn general multiplication by a surface state can be written as a\nspecific exponential of Virasoro operators. What are the coefficients\nA_n and B_n?\n\nGiven a function f = exp(v), where v is a vector field, the action of\nf\non the CFT hilbert space can be defined as U_f = exp(\\sum v_n L_n)\n\nWitten\'s triple product can be written as a BCFT correlator\n(Psi_1,Psi_2,Psi_3) =\n< U_{f1}Psi_1(-\\sqrt(3)) U_{f2}Psi_2(0) U_{f3}Psi_3(\\sqrt(3)\n>_{BCFT}\n\nfor three appropriate functions. the corresponding vector field is\npretty simple, I think it was something like (1+z^2) arctan(z).\nAnyway it\'s immediate that\n\n|0> * Psi = U_{f2}^{\\dagger} U_{f1} Psi = U_0 Psi\n\nTo extend this to descendents of the vacuum by the action of modes of\nsome\nfields one can use a trick with derivations.\n\nTake for example the Virasoro descendants of the vacuum.\nThe operators K_n = L_n - (-1)^n L_{-n} are derivations of the star\nproduct.\nAny descendant of the vacuum can be written as a string of K_n acting\non the vacuum.\n\nConsider K_n|0> * Psi. As K_n is a derivation, K_n|0> * Psi =\n[K_n,U_0] Psi.\n\nI expect that this way one could ger some messy analytic expressions\nfor star products in terms of oscillators.\n\nIf you are working with free bosons or fermions the traditional\nexpressions with the Neumann coefficients are probably the best. On\nthe other hand they require working with infinite matrices...\n\nMore pleasant star products can be evaluated between surface states\nwith\nsome conformal mapping tricks...\nDavide\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:<Pine.LNX.4.31.0405191512520.9116-100000@feynman.harvard.edu>...
>
> Given some string field \Phi_0, is there an easy way to compute by hand
> the action of the operator
>
> [\Phi_0 *, .] : |\Psi> |-> |\Phi_0 * \psi \pm \Psi * \Phi_0>
>
> in terms of the action of the usual worldsheet creators/annihilators?
>
> I mean something like
>
> [\Phi_0 *, .] = sum_{n} f_{nmq} \alpha_m \alpha_q + cdots .
>
>
> I know in principle how to compute the right hand side, using
> for instance the operator representation of the SFT vertices,
> but so far I didn't manage to obtain any closed form solutions
> that are more enlightning than the bare definition
>
> \Phi_0 * \Psi = <\Phi_0|,\Psi| |V_3>
>
> and so on.
>
> Thanks for any help and comments!

Hi,
I assume you want some sort of analytic formula. ( though if you want
to
experiment a bit with actual star products the simplest thing might be
to
write a program using the conservation laws from http://www.arxiv.org/abs/hep-th/0006240)

It's possible to easily write down the effect of the multiplication by
the
SL(2) invarian vacuum:

|0> * \Psi = \exp(\sum A_n L_{-n}) \exp(\sum B_n L_n) \Psi

In general multiplication by a surface state can be written as a
specific exponential of Virasoro operators. What are the coefficients
A_n and B_n?

Given a function f = \exp(v), where v is a vector field, the action of
f
on the CFT hilbert space can be defined as U_f = \exp(\sum v_n L_n)

Witten's triple product can be written as a BCFT correlator
(\Psi_1,\Psi_2,\Psi_3) =< U_{f1}\Psi_1(-\sqrt(3)) U_{f2}\Psi_2(0) U_{f3}\Psi_3(\sqrt(3)>_{BCFT}

for three appropriate functions. the corresponding vector field is
pretty simple, I think it was something like (1+z^2) arctan(z).
Anyway it's immediate that

|0> * \Psi = U_{f2}^{\dagger} U_{f1} \Psi = U_0 \Psi

To extend this to descendents of the vacuum by the action of modes of
some
fields one can use a trick with derivations.

Take for example the Virasoro descendants of the vacuum.
The operators K_n = L_n - (-1)^n L_{-n} are derivations of the star
product.
Any descendant of the vacuum can be written as a string of K_n acting
on the vacuum.

Consider K_n|0> * \Psi. As K_n is a derivation, K_n|0> * \Psi =[K_n,U_0] \Psi.

I expect that this way one could ger some messy analytic expressions
for star products in terms of oscillators.

If you are working with free bosons or fermions the traditional
expressions with the Neumann coefficients are probably the best. On
the other hand they require working with infinite matrices...

More pleasant star products can be evaluated between surface states
with
some conformal mapping tricks...
Davide

Davide Gaiotto

May25-04, 04:16 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:<Pine.LNX.4.31.0405071136080.28514-100000@feynman.harvard.edu>...\n\n> What do we know about the worldsheet theories associated with such\n> deformed BRST operators Q\' defined as\n>\n> Q\'|psi> := Q|psi> + |A*psi> + |psi*A>\n>\n> for some classical solution A of the original OSFT?\n>\n> In the specific context of the above paper I guess that this should be\n> a subtle issue, since the classical solution A discussed there is argued\n> to be the true tachyon vacuum, where the D25 brane has decayed.\n\nThe situation I believe is more easily understood is that in which\nA is part of a family of solutions A(t) such that A(0) = 0 and\nthings are perturbative in t. In that case A(t) = t a_1 + t^2 a_2 +\n.....\n\na_1 is a marginal deformation, and the equations can be perturbatively\nsolved for the a_k (say in Siegel gauge) if and only if the\nperturbation is exactly marginal.\nIt is not difficult to convince themselves that correlators computed\naround this new vacuum should correspond to the one computed in the\noriginal BCFT deformed by the exactly marginal deformation a_1.\nI believe it\'s even possible to show that the Virasoro algebra of the\nold vacuum can be "parallel transported" somehow along the family of\nsolutions.\n\nOn the other hand if the solution is not continuously connected to the\nvacuum (like the tachyon vacuum) there is no reason for which the\ntheory should still have a Q-exact Virasoro algebra.\n\nDavide\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:<Pine.LNX.4.31.0405071136080.28514-100000@feynman.harvard.edu>...

> What do we know about the worldsheet theories associated with such
> deformed BRST operators Q' defined as
>
> Q'|\psi> := Q|\psi> + |A*\psi> + |\psi*A>
>
> for some classical solution A of the original OSFT?
>
> In the specific context of the above paper I guess that this should be
> a subtle issue, since the classical solution A discussed there is argued
> to be the true tachyon vacuum, where the D25 brane has decayed.

The situation I believe is more easily understood is that in which
A is part of a family of solutions A(t) such that A(0) = and
things are perturbative in t. In that case A(t) = t a_1 + t^2 a_2 +
.....

a_1 is a marginal deformation, and the equations can be perturbatively
solved for the a_k (say in Siegel gauge) if and only if the
perturbation is exactly marginal.
It is not difficult to convince themselves that correlators computed
around this new vacuum should correspond to the one computed in the
original BCFT deformed by the exactly marginal deformation a_1.
I believe it's even possible to show that the Virasoro algebra of the
old vacuum can be "parallel transported" somehow along the family of
solutions.

On the other hand if the solution is not continuously connected to the
vacuum (like the tachyon vacuum) there is no reason for which the
theory should still have a Q-exact Virasoro algebra.

Davide

Urs Schreiber

May25-04, 01:32 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Tue, 25 May 2004, Davide Gaiotto wrote:\n\n> [...] hep-th/0006240\n\n\nMany thanks indeed for this very helpful reference and your valuable\nremarks!\n\n\n> Anyway it\'s immediate that\n>\n> |0> * Psi = U_{f2}^{\\dagger} U_{f1} Psi = U_0 Psi\n\n\nIf I understand correctly here it is crucial that the L_n in U_{} are\npositively moded as explained below equation (6.5) of the above\npaper. And \\dagger should indicate the bpz conjugation?\n\n\n> I expect that this way one could ger some messy analytic expressions\n> for star products in terms of oscillators.\n\nApparently a nice non-messy way for doing what I was looking\nfor is the technique used by Josef Kluson in hep-th/0303199:\n\nHe makes use of the fact that when a weight 1 chiral field W(z) is split\nas\n\nW_{L/R} := \\int_{C_{L/R}} W(z)\n\nwith C_{L/R} the left/right half of the unit circle we have the identity\n\nW_R(A) * B = - (-1)^{|W||A|} A * W_L(B) .\n\nThis can be used to show that for any A which can be written as\n\nA = W_R(I)\n\nwith I the "identity" string field we have\n\n[A,B] = -W(B) ,\n\nwhere the lhs is the graded star commutator and W = W_L + W_R is the full\ncontour integral over W(z).\n\nOf course this is still a little formal. I will need to understand how\nto solve A = W_R(I) for W given A and the conditions under which a\nsolution exists.\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Tue, 25 May 2004, Davide Gaiotto wrote:

> [...] http://www.arxiv.org/abs/hep-th/0006240

Many thanks indeed for this very helpful reference and your valuable
remarks!

> Anyway it's immediate that
>
> |0> * \Psi = U_{f2}^{\dagger} U_{f1} \Psi = U_0 \Psi

If I understand correctly here it is crucial that the L_n in U_{} are
positively moded as explained below equation (6.5) of the above
paper. And \dagger should indicate the bpz conjugation?

> I expect that this way one could ger some messy analytic expressions
> for star products in terms of oscillators.

Apparently a nice non-messy way for doing what I was looking
for is the technique used by Josef Kluson in http://www.arxiv.org/abs/hep-th/0303199:

He makes use of the fact that when a weight 1 chiral field W(z) is split
as

W_{L/R} := \int_{C_{L/R}} W(z)

with C_{L/R} the left/right half of the unit circle we have the identity

W_R(A) * B = - (-1)^{|W||A|} A * W_L(B) .

This can be used to show that for any A which can be written as

A = W_R(I)

with I the "identity" string field we have

[A,B] = -W(B) ,

where the lhs is the graded star commutator and W = W_L + W_R is the full
contour integral over W(z).

Of course this is still a little formal. I will need to understand how
to solve A = W_R(I) for W given A and the conditions under which a
solution exists.

Urs Schreiber

May25-04, 01:51 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Tue, 25 May 2004, Davide Gaiotto wrote:\n\n> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:<Pine.LNX.4.31.0405071136080.28514-100000@feynman.harvard.edu>...\n>\n> > What do we know about the worldsheet theories associated with such\n> > deformed BRST operators Q\' defined as\n> >\n> > Q\'|psi> := Q|psi> + |A*psi> + |psi*A>\n> >\n> > for some classical solution A of the original OSFT?\n> >\n> > In the specific context of the above paper I guess that this should be\n> > a subtle issue, since the classical solution A discussed there is argued\n> > to be the true tachyon vacuum, where the D25 brane has decayed.\n>\n> The situation I believe is more easily understood is that in which\n> A is part of a family of solutions A(t) such that A(0) = 0 and\n> things are perturbative in t. In that case A(t) = t a_1 + t^2 a_2 +\n> ....\n>\n> a_1 is a marginal deformation, and the equations can be perturbatively\n> solved for the a_k (say in Siegel gauge) if and only if the\n> perturbation is exactly marginal.\n> It is not difficult to convince themselves that correlators computed\n> around this new vacuum should correspond to the one computed in the\n> original BCFT deformed by the exactly marginal deformation a_1.\n\n\nI assume that this is (part of) the content of the series of papers by\nAshoke Sen on background independence of string field theory from the\nearly 90s? (I haven\'t read them in detail yet.)\n\n\n> I believe it\'s even possible to show that the Virasoro algebra of the\n> old vacuum can be "parallel transported" somehow along the family of\n> solutions.\n\n\nI have seen this "connection on the space of theories" business in papers\nby Witten and Shatashvili that Lubos kindly pointe me to, as well as in\nSen&Zwiebach hep-th/9307088 - but it takes a little getting used to...\n\n\n> On the other hand if the solution is not continuously connected to the\n> vacuum (like the tachyon vacuum) there is no reason for which the\n> theory should still have a Q-exact Virasoro algebra.\n\n\nSo how does the string field describing the true tachyon vacuum look like?\n\nProbably I have seen this somewhere recently, but I must have forgotten.\nI know that the BRST operator in the true vacuum is argued to be a c-ghost\ninsertion at the string midpoint ~ c(i) - c(-i).\n\nThis should mean that the graded commutator with the string field\ndescribing the true vacuum is the difference between the ordinary BRST\noperator and that midpoint insertion operator. Hm....\n\nAnd you say the true vacuum is not continuously connected to the\nnaive D25 brane vacuum? How can that be, after all the tachyon\ncondensation is a continuous process?\n\n\nI don\'t quite understand what you mean by a Q-exact Virasoro algebra.\nIt is clear to me that in the true vacuum the BRST operator is no longer a\nlinear combination of Virasoro generators times ghosts, so there is no\nVirasoro algebra left at all, I guess (except for the one that apparently\nis supposed to somehow magically re-appear, being associated to the closed\nstrings that must be hidden in the formalism, somehow, as far as I\nunderstand).\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Tue, 25 May 2004, Davide Gaiotto wrote:

> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:<Pine.LNX.4.31.0405071136080.28514-100000@feynman.harvard.edu>...
>
> > What do we know about the worldsheet theories associated with such
> > deformed BRST operators Q' defined as
> >
> > Q'|\psi> := Q|\psi> + |A*\psi> + |\psi*A>
> >
> > for some classical solution A of the original OSFT?
> >
> > In the specific context of the above paper I guess that this should be
> > a subtle issue, since the classical solution A discussed there is argued
> > to be the true tachyon vacuum, where the D25 brane has decayed.
>
> The situation I believe is more easily understood is that in which
> A is part of a family of solutions A(t) such that A(0) = and
> things are perturbative in t. In that case A(t) = t a_1 + t^2 a_2 +
> ....
>
> a_1 is a marginal deformation, and the equations can be perturbatively
> solved for the a_k (say in Siegel gauge) if and only if the
> perturbation is exactly marginal.
> It is not difficult to convince themselves that correlators computed
> around this new vacuum should correspond to the one computed in the
> original BCFT deformed by the exactly marginal deformation a_1.

I assume that this is (part of) the content of the series of papers by
Ashoke Sen on background independence of string field theory from the
early 90s? (I haven't read them in detail yet.)

> I believe it's even possible to show that the Virasoro algebra of the
> old vacuum can be "parallel transported" somehow along the family of
> solutions.

I have seen this "connection on the space of theories" business in papers
by Witten and Shatashvili that Lubos kindly pointe me to, as well as in
Sen&Zwiebach http://www.arxiv.org/abs/hep-th/9307088 - but it takes a little getting used to...

> On the other hand if the solution is not continuously connected to the
> vacuum (like the tachyon vacuum) there is no reason for which the
> theory should still have a Q-exact Virasoro algebra.

So how does the string field describing the true tachyon vacuum look like?

Probably I have seen this somewhere recently, but I must have forgotten.
I know that the BRST operator in the true vacuum is argued to be a c-ghost
insertion at the string midpoint ~ c(i) - c(-i).

This should mean that the graded commutator with the string field
describing the true vacuum is the difference between the ordinary BRST
operator and that midpoint insertion operator. Hm....

And you say the true vacuum is not continuously connected to the
naive D25 brane vacuum? How can that be, after all the tachyon
condensation is a continuous process?

I don't quite understand what you mean by a Q-exact Virasoro algebra.
It is clear to me that in the true vacuum the BRST operator is no longer a
linear combination of Virasoro generators times ghosts, so there is no
Virasoro algebra left at all, I guess (except for the one that apparently
is supposed to somehow magically re-appear, being associated to the closed
strings that must be hidden in the formalism, somehow, as far as I
understand).

Urs Schreiber

May25-04, 02:23 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Tue, 25 May 2004, Urs Schreiber wrote:\n\n> I don\'t quite understand what you mean by a Q-exact Virasoro algebra.\n\nOh, of course I do. The Virasoro generators won\'t be anticommutators of\nthe ghosts or something with the BRST charge.\n\nBut isn\'t this only expected to happen when we are in a vacuum where the\nopen strings have disappeared? Whenever there are still open strings\naround, shouldn\'t we always have Q-exact Virasoro generators?\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Tue, 25 May 2004, Urs Schreiber wrote:

> I don't quite understand what you mean by a Q-exact Virasoro algebra.

Oh, of course I do. The Virasoro generators won't be anticommutators of
the ghosts or something with the BRST charge.

But isn't this only expected to happen when we are in a vacuum where the
open strings have disappeared? Whenever there are still open strings
around, shouldn't we always have Q-exact Virasoro generators?

Davide Gaiotto

May26-04, 02:16 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:<Pine.LNX.4.31.0405251237320.4235-100000@kaluza.harvard.edu>...\n> On Tue, 25 May 2004, Davide Gaiotto wrote:\n>\n> > [...] hep-th/0006240\n>\n>\n> Many thanks indeed for this very helpful reference and your valuable\n> remarks!\n>\n>\n> > Anyway it\'s immediate that\n> >\n> > |0> * Psi = U_{f2}^{\\dagger} U_{f1} Psi = U_0 Psi\n>\n>\n> If I understand correctly here it is crucial that the L_n in U_{} are\n> positively moded as explained below equation (6.5) of the above\n> paper. And \\dagger should indicate the bpz conjugation?\n>\n>\n> > I expect that this way one could ger some messy analytic expressions\n> > for star products in terms of oscillators.\n>\n> Apparently a nice non-messy way for doing what I was looking\n> for is the technique used by Josef Kluson in hep-th/0303199:\n>\n> He makes use of the fact that when a weight 1 chiral field W(z) is split\n> as\n>\n> W_{L/R} := \\int_{C_{L/R}} W(z)\n>\n> with C_{L/R} the left/right half of the unit circle we have the identity\n>\n> W_R(A) * B = - (-1)^{|W||A|} A * W_L(B) .\n>\n> This can be used to show that for any A which can be written as\n>\n> A = W_R(I)\n>\n> with I the "identity" string field we have\n>\n> [A,B] = -W(B) ,\n>\n> where the lhs is the graded star commutator and W = W_L + W_R is the full\n> contour integral over W(z).\n>\n> Of course this is still a little formal. I will need to understand how\n> to solve A = W_R(I) for W given A and the conditions under which a\n> solution exists.\n\nIf you want W to be a current of weight 1 there cannot be many A, as\nthere\nis a finite number of comformal dimension 1 primaries...\n\nAnyway, I\'m always uneasy about formal solutions or tricks based on\nsplitting\noperators in left and right halves ( like when people say that a shift\nof\nQ_R(I) of the string field gives a "purely cubic" OSFT).\nThese are based on a strong assumption about associativity of the star\nproduct. While it is surely true that the star product is associative\non\nelements of the BCFT Fock space, it may fail to be associative for\nmore singular or formal objects as states built out of the identity.\n\nA prototypical example is the statement that derivations of the star\nproduct\nhave to kill the identity ( a reasonable request for the identity of\nan algebra...). If d is a derivation, d A = d(A*I) = d A * I + A*d I\n= d A + A*(d I). Hence d I is zero, or at least gives zero in any star\nproduct.\n\nNow, c_0 ( the mode from the c(z) ghost) is a derivation,\nnevertheless c_0 I is definitely not zero. Should we believe that\nc_0 I * A = 0 for any A? Maybe, but sounds strange.\n\nMore seriously, I remember seeing statements about associative\nanomalies for the star product.\n\nI believe it is an open question to understand which string fields do\nactually admit meaningful star products, and which regularity\ncondition should string fields satisfy to be acceptable solutions of\nclassical OSFT.\n\nI\'m sorry for the confused statements, but it is an issue that\nconfuses me...\n\nDavide\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:<Pine.LNX.4.31.0405251237320.4235-100000@kaluza.harvard.edu>...
> On Tue, 25 May 2004, Davide Gaiotto wrote:
>
> > [...] http://www.arxiv.org/abs/hep-th/0006240
>
>
> Many thanks indeed for this very helpful reference and your valuable
> remarks!
>
>
> > Anyway it's immediate that
> >
> > |0> * \Psi = U_{f2}^{\dagger} U_{f1} \Psi = U_0 \Psi
>
>
> If I understand correctly here it is crucial that the L_n in U_{} are
> positively moded as explained below equation (6.5) of the above
> paper. And \dagger should indicate the bpz conjugation?
>
>
> > I expect that this way one could ger some messy analytic expressions
> > for star products in terms of oscillators.
>
> Apparently a nice non-messy way for doing what I was looking
> for is the technique used by Josef Kluson in http://www.arxiv.org/abs/hep-th/0303199:
>
> He makes use of the fact that when a weight 1 chiral field W(z) is split
> as
>
> W_{L/R} := \int_{C_{L/R}} W(z)
>
> with C_{L/R} the left/right half of the unit circle we have the identity
>
> W_R(A) * B = - (-1)^{|W||A|} A * W_L(B) .
>
> This can be used to show that for any A which can be written as
>
> A = W_R(I)
>
> with I the "identity" string field we have
>
> [A,B] = -W(B) ,
>
> where the lhs is the graded star commutator and W = W_L + W_R is the full
> contour integral over W(z).
>
> Of course this is still a little formal. I will need to understand how
> to solve A = W_R(I) for W given A and the conditions under which a
> solution exists.

If you want W to be a current of weight 1 there cannot be many A, as
there
is a finite number of comformal dimension 1 primaries...

Anyway, I'm always uneasy about formal solutions or tricks based on
splitting
operators in left and right halves ( like when people say that a shift
of
Q_R(I) of the string field gives a "purely cubic" OSFT).
These are based on a strong assumption about associativity of the star
product. While it is surely true that the star product is associative
on
elements of the BCFT Fock space, it may fail to be associative for
more singular or formal objects as states built out of the identity.

A prototypical example is the statement that derivations of the star
product
have to kill the identity ( a reasonable request for the identity of
an algebra...). If d is a derivation, d A = d(A*I) = d A * I + A*d I= d A + A*(d I). Hence d I is zero, or at least gives zero in any star
product.

Now, c_0 ( the mode from the c(z) ghost) is a derivation,
nevertheless c_0 I is definitely not zero. Should we believe that
c_0 I * A = for any A? Maybe, but sounds strange.

More seriously, I remember seeing statements about associative
anomalies for the star product.

I believe it is an open question to understand which string fields do
actually admit meaningful star products, and which regularity
condition should string fields satisfy to be acceptable solutions of
classical OSFT.

I'm sorry for the confused statements, but it is an issue that
confuses me...

Davide

Urs Schreiber

May26-04, 09:10 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"Davide Gaiotto" <dgaiotto@fas.harvard.edu> schrieb im Newsbeitrag\nnews:302ccf73.0405251736.71cee2c2-100000@posting.google.com...\n\n> If you want W to be a current of weight 1 there cannot be many A, as\n> there is a finite number of comformal dimension 1 primaries...\n\nMaybe I am misunderstanding something, but I\'d say (starting from a trivial\nMinkowski spacetime background) that \\oint W can in fact be for instance any\nDDF invariant, e.g. W(z) = \\partial X^i(z) exp(n k \\cdot X(z)) with k^2 = 0,\ni spacelike and n any integer, or combinations thereof, e.g. W(z) = \\partial\nX^i(z) \\prod_r A_{n_r}^{\\mu_r}, where the A_n^\\mu are DDF oscillators.\n\nThe entire argument concerning W is supposed to be a direct generalization\nof what Kishimoto&Ohmori demonstrate on p.18-19 of their hep-th/0112169 for\nthe BRST operator Q, by noting that the only property of Q used in\n(4.31)-(4.33) of that paper is that it is the integral over a unit weight\nfield and hence commutes with the (total) Virasoro generators/with conformal\ntransformations.\n\nI could imagine that the condition of _true_ marginality of W in the sense\nof section 3 of\n\nA. Recknagel & V. Schomerus\nBoundary Deformation Theory and Moduli Spaces of D-branes\nhep-th/9811237\n\nreduces the number of admissable W to a finite amount, maybe, I am lacking\nintuition/understanding of _true_ marginality. But in section 3.2 of that\npaper it is shown that every self-local marginal operator is in fact truly\nmarginal. This makes it sound like there are many of them. Indeed, from the\nphysical point of view that every continuous symmetry of the background\nshould be associated with a truly marginal operator of the CFT (cf. last\nparagraph on p.2 of that paper) it seems that there are lots of them.\n\nBut I may well be wrong. Please correct me.\n\n\n> Anyway, I\'m always uneasy about formal solutions or tricks based on\n> splitting operators in left and right halves ( like when people say that a\nshift\n> of Q_R(I) of the string field gives a "purely cubic" OSFT).\n\n\nThis remark finally made me look up the paper\n\nHorowitz & Lykken & Rohm & Strominger:\nPurely Cubic Action for String Field Theory\nPhys. Rev. Let 57 3 p.283\n\nwhich I unfortunately hadn\'t read before because I somehow erroneously\nassumed it wouldn\'t go much further than Hata\'s purely cubic construction. I\nsee now that this was a mistake, as in fact the above paper contains most of\nthe information that I was originally looking for. In particular I now see\nthat the insight that\n\n[ W_R(I) , B ] = W(B)\n\nfor any invariant W is, contrary to the impression that I got, not due to J.\nKluson, but at least 7 years older and used by the above authors. So many\nthanks for this remark!\n\nAs you say, and as the authors of that paper also emphasize, the\nconstruction there is somehwat formal which is potentially dangerous since\ncounterexamples to many naive formal properties in SFT are well known. But I\nam wondering if the more recent studies by Kishimoto and Ohmori put this\nresult on a firmer footing.\n\nI am thinking again of the discussion in\n\nI. Kishimoto & K. Ohmori,\nCFT Description of Identity String Field: Toward Derivation of the VSFT\naction\nhep-th/0112169 .\n\nIn particular in this paper it is emphasized that the "identity" field not\nnecessarily has the desired formal property on the full set of string fields\nand a precise definition of this field in terms of the 360 degree wedge\nstate/boundary state is used instead of blindly relying on the assumed\nformal properties. My understanding is that the discussion in that paper\ntherefore removes possible doubts about the admissibility of the algebraic\nmanipulations.\n\nIn particular, I am under the impression that the proof in equations (4.31)\nand (4.33) of Kishimoto&Ohmori\'s paper makes equations (12a) and (12b) in\nHorowitz&Lykken&Rohm&Strominger rigorously precise. Isn\'t that correct? That\nwould be very helpful, since (12c) is trivial and nothing more is needed in\norder to demonstrate the identity in question.\n\nBTW, another question concerning the "purely cubic" paper:\n\nAt the end the gauge group of OSFT is briefly mentioned, namely that group\ngenerated by string fields \\Lambda of 0 gauge number (in the modern ghost\nnumber convention). It says that it is "an infinite-dimensional\ngeneralization of U(N)".\n\nSo is the gauge group just U(oo), as in the matrix models? From the "note\nadded in proof" on p. 293 of\n\nE. Witten,\nNon-commutative geometry and string field theory,\nNuc. Phys. B 268 (1986) 253-294\n\nit seems like the gauge algebra of OSFT for compactifications on an even\nunimodular lattice is the "monster Lie algebra". Is that right?\n\n\n> These are based on a strong assumption about associativity of the star\n> product. While it is surely true that the star product is associative\n> on\n> elements of the BCFT Fock space, it may fail to be associative for\n> more singular or formal objects as states built out of the identity.\n\n\nThis addresses a point that seems to be elementary but which I am admittedly\nstill confused about: Kishimoto&Ohmori mention repeatedly in the above paper\na distinction between "Fock states" and more general states. But doesn\'t\nevery string field come from a space in the Hilbert space of the string and\nhence is a Fock space? How couldn\'t it be?\n\nYou mention formal objects, build from the identity. But I get the\nimpression that the "identity" field when defined as by Kishimoto&Ohmori is\nperfectly well defined and in fact, as you explained to me last time, an\nelement of the Fock space. What am I missing?\n\n> A prototypical example is the statement that derivations of the star\n> product\n> have to kill the identity ( a reasonable request for the identity of\n> an algebra...). If d is a derivation, d A = d(A*I) = d A * I + A*d I\n> = d A + A*(d I). Hence d I is zero, or at least gives zero in any star\n> product.\n>\n> Now, c_0 ( the mode from the c(z) ghost) is a derivation,\n> nevertheless c_0 I is definitely not zero. Should we believe that\n> c_0 I * A = 0 for any A? Maybe, but sounds strange.\n\nI have seen precisely this discussion somewhere recently. What is the\ncurrent status of the answer? Does anyone have an idea how this seeming\nparadox is resolved?\n\n> More seriously, I remember seeing statements about associative\n> anomalies for the star product.\n\nYes, I have seen such, too. One thing that would easily make sense to me is\nthat there may be elements of the Fock space which are not normalizable, for\ninstance, and that these have strange formal properties, due to infinities\narising (or hidden) in the calculations. If this is the case I imagine all\nthe paradoxes could be resolved by working with normalizable states only and\nthen carefully keeping track of which limits are taken. Does that sound\nreasonable?\n\n> I believe it is an open question to understand which string fields do\n> actually admit meaningful star products, and which regularity\n> condition should string fields satisfy to be acceptable solutions of\n> classical OSFT.\n\nI see.\n\n> I\'m sorry for the confused statements, but it is an issue that\n> confuses me...\n\nI am extremely grateful for your help, and this includes the pointing out of\nissues which are not clarified yet.\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Davide Gaiotto" <dgaiotto@fas.harvard.edu> schrieb im Newsbeitrag
news:302ccf73.0405251736.71cee2c2-100000@posting.google.com...

> If you want W to be a current of weight 1 there cannot be many A, as
> there is a finite number of comformal dimension 1 primaries...

Maybe I am misunderstanding something, but I'd say (starting from a trivial
Minkowski spacetime background) that \oint W can in fact be for instance any
DDF invariant, e.g. W(z) = \partial X^i(z) \exp(n k \cdot X(z)) with k^2 = 0,
i spacelike and n any integer, or combinations thereof, e.g. W(z) = \partialX^i(z) \prod_r A_{n_r}^{\mu_r}, where the A_n^\mu are DDF oscillators.

The entire argument concerning W is supposed to be a direct generalization
of what Kishimoto&Ohmori demonstrate on p.18-19 of their http://www.arxiv.org/abs/hep-th/0112169 for
the BRST operator Q, by noting that the only property of Q used in
(4.31)-(4.33) of that paper is that it is the integral over a unit weight
field and hence commutes with the (total) Virasoro generators/with conformal
transformations.

I could imagine that the condition of _true_ marginality of W in the sense
of section 3 of

A. Recknagel & V. Schomerus
Boundary Deformation Theory and Moduli Spaces of D-branes
http://www.arxiv.org/abs/hep-th/9811237

reduces the number of admissable W to a finite amount, maybe, I am lacking
intuition/understanding of _true_ marginality. But in section 3.2 of that
paper it is shown that every self-local marginal operator is in fact truly
marginal. This makes it sound like there are many of them. Indeed, from the
physical point of view that every continuous symmetry of the background
should be associated with a truly marginal operator of the CFT (cf. last
paragraph on p.2 of that paper) it seems that there are lots of them.

But I may well be wrong. Please correct me.

> Anyway, I'm always uneasy about formal solutions or tricks based on
> splitting operators in left and right halves ( like when people say that a
shift
> of Q_R(I) of the string field gives a "purely cubic" OSFT).

This remark finally made me look up the paper

Horowitz & Lykken & Rohm & Strominger:
Purely Cubic Action for String Field Theory
Phys. Rev. Let 57 3 p.283

which I unfortunately hadn't read before because I somehow erroneously
assumed it wouldn't go much further than Hata's purely cubic construction. I
see now that this was a mistake, as in fact the above paper contains most of
the information that I was originally looking for. In particular I now see
that the insight that

[ W_R(I) , B ] = W(B)

for any invariant W is, contrary to the impression that I got, not due to J.
Kluson, but at least 7 years older and used by the above authors. So many
thanks for this remark!

As you say, and as the authors of that paper also emphasize, the
construction there is somehwat formal which is potentially dangerous since
counterexamples to many naive formal properties in SFT are well known. But I
am wondering if the more recent studies by Kishimoto and Ohmori put this
result on a firmer footing.

I am thinking again of the discussion in

I. Kishimoto & K. Ohmori,
CFT Description of Identity String Field: Toward Derivation of the VSFT
action
http://www.arxiv.org/abs/hep-th/0112169 .

In particular in this paper it is emphasized that the "identity" field not
necessarily has the desired formal property on the full set of string fields
and a precise definition of this field in terms of the 360 degree wedge
state/boundary state is used instead of blindly relying on the assumed
formal properties. My understanding is that the discussion in that paper
therefore removes possible doubts about the admissibility of the algebraic
manipulations.

In particular, I am under the impression that the proof in equations (4.31)
and (4.33) of Kishimoto&Ohmori's paper makes equations (12a) and (12b) in
Horowitz&Lykken&Rohm&Strominger rigorously precise. Isn't that correct? That
would be very helpful, since (12c) is trivial and nothing more is needed in
order to demonstrate the identity in question.

BTW, another question concerning the "purely cubic" paper:

At the end the gauge group of OSFT is briefly mentioned, namely that group
generated by string fields \Lambda of gauge number (in the modern ghost
number convention). It says that it is "an infinite-dimensional
generalization of U(N)".

So is the gauge group just U(oo), as in the matrix models? From the "note
added in proof" on p. 293 of

E. Witten,
Non-commutative geometry and string field theory,
Nuc. Phys. B 268 (1986) 253-294

it seems like the gauge algebra of OSFT for compactifications on an even
unimodular lattice is the "monster Lie algebra". Is that right?

> These are based on a strong assumption about associativity of the star
> product. While it is surely true that the star product is associative
> on
> elements of the BCFT Fock space, it may fail to be associative for
> more singular or formal objects as states built out of the identity.

This addresses a point that seems to be elementary but which I am admittedly
still confused about: Kishimoto&Ohmori mention repeatedly in the above paper
a distinction between "Fock states" and more general states. But doesn't
every string field come from a space in the Hilbert space of the string and
hence is a Fock space? How couldn't it be?

You mention formal objects, build from the identity. But I get the
impression that the "identity" field when defined as by Kishimoto&Ohmori is
perfectly well defined and in fact, as you explained to me last time, an
element of the Fock space. What am I missing?

> A prototypical example is the statement that derivations of the star
> product
> have to kill the identity ( a reasonable request for the identity of
> an algebra...). If d is a derivation, d A = d(A*I) = d A * I + A*d I
> = d A + A*(d I). Hence d I is zero, or at least gives zero in any star
> product.
>
> Now, c_0 ( the mode from the c(z) ghost) is a derivation,
> nevertheless c_0 I is definitely not zero. Should we believe that
> c_0 I * A = for any A? Maybe, but sounds strange.

I have seen precisely this discussion somewhere recently. What is the
current status of the answer? Does anyone have an idea how this seeming
paradox is resolved?

> More seriously, I remember seeing statements about associative
> anomalies for the star product.

Yes, I have seen such, too. One thing that would easily make sense to me is
that there may be elements of the Fock space which are not normalizable, for
instance, and that these have strange formal properties, due to infinities
arising (or hidden) in the calculations. If this is the case I imagine all
the paradoxes could be resolved by working with normalizable states only and
then carefully keeping track of which limits are taken. Does that sound
reasonable?

> I believe it is an open question to understand which string fields do
> actually admit meaningful star products, and which regularity
> condition should string fields satisfy to be acceptable solutions of
> classical OSFT.

I see.

> I'm sorry for the confused statements, but it is an issue that
> confuses me...

I am extremely grateful for your help, and this includes the pointing out of
issues which are not clarified yet.

Urs Schreiber

May26-04, 12:11 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"Urs Schreiber" <Urs.Schreiber@uni-essen.de> schrieb im Newsbeitrag\nnews:2hjj8fFdi9a0U1-100000@uni-berlin.de...\n\n> Horowitz & Lykken & Rohm & Strominger:\n> Purely Cubic Action for String Field Theory\n> Phys. Rev. Let 57 3 p.283\n\nI forgot to ask one thing:\n\nHata argues in his original paper on purely cubic SFT\n\nhttp://ccdb3fs.kek.jp/cgi-bin/img_index?8606274\n\nthat the string field solution Phi_0 of the purely cubic action which\nemulates the Minkowski space BRST operator Q\n\nQ psi = [Phi_0, psi]\n\n(where we have the graded star commutator on the rhs for the usual\ndefinition of the star product in OSFT)\n\nis a string field which describes a background of "infinitesimally small\nstrings". I found that a philosophically very intriguing observation,\nbecause it suggests that spacetime may maybe be thought of as a network of\ntiny strings, roughly, maybe. Anyway, would a similar statement also apply\nto the\n\nPhi_0 = Q_L \\mathcal{I}\n\nthat Horowitz & Lykken & Rohm & Strominger consider in their equation (11)?\nCan one say anything about what configuration of strings this Phi_0\ndescribes?\n\nClosely related: Every projector P of the matter SFT star algebra\n\nP * P = P\n\nis supposed to describe instantons (or maybe solitons?) in spacetime, as for\ninstance mentioned on pp.17-18 of\n\nT. Kawano & K. Okuyama:\nOpen String Fields As Matrices\nhep-th/0105129 .\n\nCan one say anything about what the wedges states and in particular the\nidentity state describe physically?\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Urs Schreiber" <Urs.Schreiber@uni-essen.de> schrieb im Newsbeitrag
news:2hjj8fFdi9a0U1-100000@uni-berlin.de...

> Horowitz & Lykken & Rohm & Strominger:
> Purely Cubic Action for String Field Theory
> Phys. Rev. Let 57 3 p.283

I forgot to ask one thing:

Hata argues in his original paper on purely cubic SFT

http://ccdb3fs.kek.jp/cgi-bin/img_index?8606274

that the string field solution \Phi_0 of the purely cubic action which
emulates the Minkowski space BRST operator Q

Q \psi = [\Phi_0, \psi]

(where we have the graded star commutator on the rhs for the usual
definition of the star product in OSFT)

is a string field which describes a background of "infinitesimally small
strings". I found that a philosophically very intriguing observation,
because it suggests that spacetime may maybe be thought of as a network of
tiny strings, roughly, maybe. Anyway, would a similar statement also apply
to the

\Phi_0 = Q_L \mathcal{I}

that Horowitz & Lykken & Rohm & Strominger consider in their equation (11)?
Can one say anything about what configuration of strings this \Phi_0
describes?

Closely related: Every projector P of the matter SFT star algebra

P * P = P

is supposed to describe instantons (or maybe solitons?) in spacetime, as for
instance mentioned on pp.17-18 of

T. Kawano & K. Okuyama:
Open String Fields As Matrices
http://www.arxiv.org/abs/hep-th/0105129 .

Can one say anything about what the wedges states and in particular the
identity state describe physically?

Lubos Motl

May27-04, 10:01 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Wed, 26 May 2004, Urs Schreiber wrote:\n\n> [The condensate that generates the quadratic term from the cubic term]\n> is a string field which describes a background of "infinitesimally small\n> strings". I found that a philosophically very intriguing observation, ...\n\nI loved it, too. The pre-geometric, purely cubic string field theory was\nthe first or second paper on string theory I ever read (of course, I could\nnot understand most of its details), and the idea that the equations of\neverything are written as A*A=0 was pretty attractive.\n\n> Anyway, would a similar statement also apply\n> to the Phi_0 = Q_L \\mathcal{I}\n> that Horowitz & Lykken & Rohm & Strominger consider in their equation (11)?\n\nNope, because they study open string field theory - unlike the Kyoto group\nthat focused on the (more problematic) closed string field theory. The\ncubic vertices differ. The usual cubic vertex in open string field theory\nresembles the logo of Mercedes-Benz. The second half of the first string\nand the first half of the second string overlap and "annihilate", while\nthe remaining pieces of the string produce the third string.\n\nAll strings in open string field theory are therefore equally long, and\nthere is nothing "infinitesimal" about them. These strings are close to\nthe identity functional if the shape of the first part of the string is\ncorrelated with the shape of the second half (like RNA folding).\n\nOn the other hand, Hata et al. (HIKKO) use a closed string vertex inspired\nfrom light cone gauge string field theory - two closed string join into\nthe third closed string much like two digits "0" can combine into "8". In\nthis picture, different strings must be allowed to have different\n"length". In the light cone gauge, this length is interpreted as a\nlight-like component of the momentum p^+. In the covariant treatment, it\nis a new unphysical parameter \\alpha - and the dependence on \\alpha must\nbe trivialized at the end. Nevertheless the moduli (or changes of the\nbackground) carry no momentum which implies that they carry p^+=0, and the\ncovariant string field theory implies an analogous statement that these\ncondensed strings are infinitesimally short (\\alpha=0).\n\n> Can one say anything about what configuration of strings this Phi_0\n> describes?\n\nWell, the equations speak for themselves. They infinitesimally short\nstrings that satisfy the condition that if you connect them with another\nstring state PHI in all possible ways (via the commutator), it is\nequivalent to the action of the BRST operator.\n\n> Closely related: Every projector P of the matter SFT star algebra\n>\n> P * P = P\n>\n> is supposed to describe instantons (or maybe solitons?) in spacetime, as for\n> instance mentioned on pp.17-18 of\n> ...\n> Can one say anything about what the wedges states and in particular the\n> identity state describe physically?\n\nWell, a possible answer might be that there is no answer because these\nstates are not elements of the original Hilbert space (they are infinitely\nbig excitations) and the identities they satisfy are rather formal, but of\ncourse, there might exist a much more exciting spacetime interpretation of\nthese "projectors". Someone may be able to give a better comment...\n______________________________________ ________________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)\n^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Wed, 26 May 2004, Urs Schreiber wrote:

> [The condensate that generates the quadratic term from the cubic term]
> is a string field which describes a background of "infinitesimally small
> strings". I found that a philosophically very intriguing observation, ...

I loved it, too. The pre-geometric, purely cubic string field theory was
the first or second paper on string theory I ever read (of course, I could
not understand most of its details), and the idea that the equations of
everything are written as A*A=0 was pretty attractive.

> Anyway, would a similar statement also apply
> to the \Phi_0 = Q_L \mathcal{I}
> that Horowitz & Lykken & Rohm & Strominger consider in their equation (11)?

Nope, because they study open string field theory - unlike the Kyoto group
that focused on the (more problematic) closed string field theory. The
cubic vertices differ. The usual cubic vertex in open string field theory
resembles the logo of Mercedes-Benz. The second half of the first string
and the first half of the second string overlap and "annihilate", while
the remaining pieces of the string produce the third string.

All strings in open string field theory are therefore equally long, and
there is nothing "infinitesimal" about them. These strings are close to
the identity functional if the shape of the first part of the string is
correlated with the shape of the second half (like RNA folding).

On the other hand, Hata et al. (HIKKO) use a closed string vertex inspired
from light cone gauge string field theory - two closed string join into
the third closed string much like two digits "" can combine into "8". In
this picture, different strings must be allowed to have different
"length". In the light cone gauge, this length is interpreted as a
light-like component of the momentum p^+. In the covariant treatment, it
is a new unphysical parameter \alpha - and the dependence on \alpha must
be trivialized at the end. Nevertheless the moduli (or changes of the
background) carry no momentum which implies that they carry p^+=0, and the
covariant string field theory implies an analogous statement that these
condensed strings are infinitesimally short (\alpha=0).

> Can one say anything about what configuration of strings this \Phi_0
> describes?

Well, the equations speak for themselves. They infinitesimally short
strings that satisfy the condition that if you connect them with another
string state \PHI in all possible ways (via the commutator), it is
equivalent to the action of the BRST operator.

> Closely related: Every projector P of the matter SFT star algebra
>
> P * P = P
>
> is supposed to describe instantons (or maybe solitons?) in spacetime, as for
> instance mentioned on pp.17-18 of
> ...
> Can one say anything about what the wedges states and in particular the
> identity state describe physically?

Well, a possible answer might be that there is no answer because these
states are not elements of the original Hilbert space (they are infinitely
big excitations) and the identities they satisfy are rather formal, but of
course, there might exist a much more exciting spacetime interpretation of
these "projectors". Someone may be able to give a better comment...
__{_______________________________________________ _____________________________}
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)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Urs Schreiber

May27-04, 01:54 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"Lubos Motl" <motl@feynman.harvard.edu> schrieb im Newsbeitrag\nnews:Pine.LNX.4.31.0405270948450.4141-100000@lamb.physics.harvard.edu...\n> On Wed, 26 May 2004, Urs Schreiber wrote:\n>\n> > [The condensate that generates the quadratic term from the cubic term]\n> > is a string field which describes a background of "infinitesimally small\n> > strings". I found that a philosophically very intriguing observation,\n....\n>\n> I loved it, too. The pre-geometric, purely cubic string field theory was\n> the first or second paper on string theory I ever read\n\nHm. This implies that I am probably only about 15 years behind of you...\n\n> (of course, I could\n> not understand most of its details), and the idea that the equations of\n> everything are written as A*A=0 was pretty attractive.\n\nYes, it\'s great. But as argued by Horowitz&Lykken&Rohm&Strominger Hata\'s\noriginal result for closed SFT does not quite work. With respect to their\nimproved construction for the open string they say (in the first paragraph\non p.258) that "a similar result may hold for the closed-string field theory\nof De Alwis and Ohta". Did anyone look at that? Do we have a background-free\nversion of closed string theory that removes the problems that remain with\nHata\'s construction?\n\nI did now have a look at the background free superstring. Apparently this\ngoes back to\n\nJosef Kluson\nProposal for Background Independent Berkovits\' Superstring Field Theory\nhep-th/0106107\n\nwhere it is noticed that by assembling some well known results, like for\ninstance the intriguing insight of\n\nAcosta & Berkovits & Chandia\nA note on the superstring BRST operator\nhep-th/9902178\n\n(which shows that the superstring BRST operator can be written in the form\n\nQ = e^{-R} o Q_0 o e^R\n\n\nwith Q_0 a pure ghost BRST-like operator and R something linear in the\nworldsheet matter supercurrent, Q_0 is background-independent, only R\nencodes the background structure)\n\none can also find a background-free version of "NSFT" .\n\nThis doesn\'t look quite as neat as the\n\nA * A = 0\n\nthat you mentioned above. Rather one finds (I assume you know all this, I am\njust enjoying talking about it and maybe getting some comments)\n\n\\eta_0 ( g^{-1} Q_0 (g) ) = 0\n\nwith\n\ng = e^A,\n\nwhere\n\nA = R_L(I)\n\nis some string field and star multiplication is implicit.\n\nIf anyone is interested in this background free superstring field theory, I\nhave typed a draft of some sketchy notes (work in progress) at\n\nhttp://www-stud.uni-essen.de/~sb0264/p8.pdf\n\nwhere in section 1.2 the above construction by Kluson is summarized and\nfurther references are given.\n\n> > Anyway, would a similar statement also apply\n> > to the Phi_0 = Q_L \\mathcal{I}\n> > that Horowitz & Lykken & Rohm & Strominger consider in their equation\n(11)?\n>\n> Nope, because they study open string field theory - unlike the Kyoto group\n> that focused on the (more problematic) closed string field theory.\n\n<snip further details>\n\nThat makes sense. Thanks for the explanation!\n\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Lubos Motl" <motl@feynman.harvard.edu> schrieb im Newsbeitrag
news:Pine.LNX.4.31.0405270948450.4141-100000@lamb.physics.harvard.edu...
> On Wed, 26 May 2004, Urs Schreiber wrote:
>
> > [The condensate that generates the quadratic term from the cubic term]
> > is a string field which describes a background of "infinitesimally small
> > strings". I found that a philosophically very intriguing observation,
....
>
> I loved it, too. The pre-geometric, purely cubic string field theory was
> the first or second paper on string theory I ever read

Hm. This implies that I am probably only about 15 years behind of you...

> (of course, I could
> not understand most of its details), and the idea that the equations of
> everything are written as A*A=0 was pretty attractive.

Yes, it's great. But as argued by Horowitz&Lykken&Rohm&Strominger Hata's
original result for closed SFT does not quite work. With respect to their
improved construction for the open string they say (in the first paragraph
on p.258) that "a similar result may hold for the closed-string field theory
of De Alwis and Ohta". Did anyone look at that? Do we have a background-free
version of closed string theory that removes the problems that remain with
Hata's construction?

I did now have a look at the background free superstring. Apparently this
goes back to

Josef Kluson
Proposal for Background Independent Berkovits' Superstring Field Theory
http://www.arxiv.org/abs/hep-th/0106107

where it is noticed that by assembling some well known results, like for
instance the intriguing insight of

Acosta & Berkovits & Chandia
A note on the superstring BRST operator
http://www.arxiv.org/abs/hep-th/9902178

(which shows that the superstring BRST operator can be written in the form

Q = e^{-R} o Q_0 o e^R

with Q_0 a pure ghost BRST-like operator and R something linear in the
worldsheet matter supercurrent, Q_0 is background-independent, only R
encodes the background structure)

one can also find a background-free version of "NSFT" .

This doesn't look quite as neat as the

A * A =

that you mentioned above. Rather one finds (I assume you know all this, I am
just enjoying talking about it and maybe getting some comments)

\eta_0 ( g^{-1} Q_0 (g) ) =

with

g = e^A,

where

A = R_L(I)

is some string field and star multiplication is implicit.

If anyone is interested in this background free superstring field theory, I
have typed a draft of some sketchy notes (work in progress) at

http://www-stud.uni-essen.de/~sb0264/p8.pdf

where in section 1.2 the above construction by Kluson is summarized and
further references are given.

> > Anyway, would a similar statement also apply
> > to the \Phi_0 = Q_L \mathcal{I}
> > that Horowitz & Lykken & Rohm & Strominger consider in their equation
(11)?
>
> Nope, because they study open string field theory - unlike the Kyoto group
> that focused on the (more problematic) closed string field theory.

<snip further details>

That makes sense. Thanks for the explanation!

Urs Schreiber

May27-04, 06:01 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>"Urs Schreiber" <Urs.Schreiber@uni-essen.de> schrieb im Newsbeitrag\nnews:2hmo9eFf5mn5U1-100000@uni-berlin.de...\n\n> Josef Kluson\n> Proposal for Background Independent Berkovits\' Superstring Field Theory\n> hep-th/0106107\n\nOne interesting thing about Kluson\'s way of writing the Berkovit\'s SSFT\naction is that, while the usual form of the SSFT equations of motion\n\n\\eta_0 ( e^{-\\Phi} Q e^\\Phi) = 0\n\nlooks very different from that of bosonic SFT which reads\n\nQ \\Phi + \\Phi \\Phi = 0,\n\nusing Kluson\'s formal exterior derivative\n\nd = dx^1 \\wedge \\eta_0 + dx^2 \\wedge Q\n\nwith the commuting forms dx^1 and dx^2 one can equivalently rewrite the SSFT\nequations of motion as\n\nd A + A A = 0\n\nby setting\n\nA = - * e^{-\\Phi} (d e^{Phi})\n\nwhere * is the Hodge star in the WZW context (btw, I think Kluson\'s\ndefinition is missing a factor of i in the definition of the Hodge star and\none in front of the topological term of the WZW/Berkovits action, both\nmissing factors cancel, though, so it may be a matter of taste).\n\nThis way the SSFT equations of motion take precisely the same formal form as\nthose of bosonic cubic SFT - unless I am confused that is (I should really\nbe asleep right now).\n\nI wonder if this can be useful for anything...\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>"Urs Schreiber" <Urs.Schreiber@uni-essen.de> schrieb im Newsbeitrag
news:2hmo9eFf5mn5U1-100000@uni-berlin.de...

> Josef Kluson
> Proposal for Background Independent Berkovits' Superstring Field Theory
> http://www.arxiv.org/abs/hep-th/0106107

One interesting thing about Kluson's way of writing the Berkovit's SSFT
action is that, while the usual form of the SSFT equations of motion

\eta_0 ( e^{-\Phi} Q e^\Phi) =

looks very different from that of bosonic SFT which reads

Q \Phi + \Phi \Phi = 0,

using Kluson's formal exterior derivative

d = dx^1 \wedge \eta_0 + dx^2 \wedge Q

with the commuting forms dx^1 and dx^2 one can equivalently rewrite the SSFT
equations of motion as

d A + A A =

by setting

A = - * e^{-\Phi} (d e^{\Phi})

where * is the Hodge star in the WZW context (btw, I think Kluson's
definition is missing a factor of i in the definition of the Hodge star and
one in front of the topological term of the WZW/Berkovits action, both
missing factors cancel, though, so it may be a matter of taste).

This way the SSFT equations of motion take precisely the same formal form as
those of bosonic cubic SFT - unless I am confused that is (I should really
be asleep right now).

I wonder if this can be useful for anything...

Charlie Stromeyer Jr.

May28-04, 03:12 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:\n\n> This way the SSFT equations of motion take precisely the same formal form as\n> those of bosonic cubic SFT - unless I am confused that is (I should really\n> be asleep right now).\n>\n> I wonder if this can be useful for anything...\n\nYou are not confused and so I hope you won\'t mind if I burst one of\nyour 99 red balloons :-) by mentioning that this line of thought has\nalready been considered e.g. in papers (hep-th/0204155, 0209186 and\nother papers by these authors).\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:

> This way the SSFT equations of motion take precisely the same formal form as
> those of bosonic cubic SFT - unless I am confused that is (I should really
> be asleep right now).
>
> I wonder if this can be useful for anything...

You are not confused and so I hope you won't mind if I burst one of
your 99 red balloons :-) by mentioning that this line of thought has
already been considered e.g. in papers (http://www.arxiv.org/abs/hep-th/0204155, 0209186 and
other papers by these authors).

Urs Schreiber

May28-04, 04:02 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Fri, 28 May 2004, Charlie Stromeyer Jr. wrote:\n\n> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:\n>\n> > This way the SSFT equations of motion take precisely the same formal form as\n> > those of bosonic cubic SFT - unless I am confused that is (I should really\n> > be asleep right now).\n> >\n> > I wonder if this can be useful for anything...\n>\n> You are not confused and so I hope you won\'t mind if I burst one of\n> your 99 red balloons :-) by mentioning that this line of thought has\n> already been considered e.g. in papers (hep-th/0204155, 0209186 and\n> other papers by these authors).\n\n\nThanks immensely for these references! There is a lot of information in\nthere, and I haven\'t absorbed it all yet, but the basic idea (for those\nreading our conversation) is this:\n\nGiven some equations of motion of the form of a zero curvature confition\n\n0 = F := (d + A)^2\n\nfor some covariant exterior derivative, it helps to study the space of\n"covariantly constant" fields Psi\n\n(Q + A)Psi\n\nbecause solutions of this equation imply that F = 0. This is commonly well\nknown as the "method of Lax pairs", I think.\n\nNow the point is that the equations of motion of the open string field\ntheory are manifestly of the above form, with d = Q and A = the string\nfield.\n\nThe authors of the above papers now try to adapt this techniques to\n_super_string field theory, where the equations of motion are not\nalways/not manifestly of the "zero curvature form".\n\nIn particular in section 4 on p.8 of hep-th/0204155 they give (in\nequation (4.3)) an implementation of this idea for Berkovits\' version of\nsusy SFT. They do this by "extending" the ordinary SSFT operator to\nfunctions of some auxiliary parameter lambda and by identifying the\nexterior derivative d from above with the sum of the (extended version of)\nthe worldsheet BRST operator and the eta_0 super-ghost. The introduction\nof lambda is necessary in order to decouple two terms in the expression\nfor the respective curvature (equation (4.4)), so that one ideed gets the\nBerkovits SSFT equations of motion (4.6).\n\nThis in not quite the idea that I had in mind, though! I was also thinking\nin terms of formulating the Berkovits string in a form that it looks\napproximately like the bosonic Witten string field, so that similar formal\ntechniques might be aopplied by analogy, but the method that I had in\nmind, motivated by Kluson\'s papers, is rather based on the introduction of\ndifferentials corresponding to Q and eta_0, "meta-ghosts" so to say, than\non the introduction of an auxiliary parameter lambda.\n\nNamely my point was, as explained in my previous post in this thread, that\nusing the "meta-ghosts" as in Kluson\'s papers we can replace the ordinary\nBRST operator by the "meta-BRST operator"= exterior derivative\n\nd = dx^1 eta_0 + dx^2 Q\n\nin a way that Berkovbits\' equations of motion are precisely of the formal\nform as the bosonic eoms of bosonic cubic SFT. It would perhaps be\ninteresting to see what happens to this ansatz is fed in to the Lax pair\nkind of methods discussed by Lechtenfeld, Popov, Uhlmann and Kling in the\nabove mentioned papers and the crank is turned.\n\nWell, on a second thought it would probably yield precisely the method of\nhep-th/0204155, since essentially the role played by my "meta ghosts" in\ndistinguishing eta_0 contributions from Q-contributions is precisely that\nof the factor lambda in equation (4.3) of that paper, I think.\n\nBut I haven\'t taken the time to thtink this to the end. I really have to\nrun now.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Fri, 28 May 2004, Charlie Stromeyer Jr. wrote:

> Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:
>
> > This way the SSFT equations of motion take precisely the same formal form as
> > those of bosonic cubic SFT - unless I am confused that is (I should really
> > be asleep right now).
> >
> > I wonder if this can be useful for anything...
>
> You are not confused and so I hope you won't mind if I burst one of
> your 99 red balloons :-) by mentioning that this line of thought has
> already been considered e.g. in papers (http://www.arxiv.org/abs/hep-th/0204155, 0209186 and
> other papers by these authors).

Thanks immensely for these references! There is a lot of information in
there, and I haven't absorbed it all yet, but the basic idea (for those
reading our conversation) is this:

Given some equations of motion of the form of a zero curvature confition

= F := (d + A)^2

for some covariant exterior derivative, it helps to study the space of
"covariantly constant" fields \Psi(Q + A)\Psi

because solutions of this equation imply that F = . This is commonly well
known as the "method of Lax pairs", I think.

Now the point is that the equations of motion of the open string field
theory are manifestly of the above form, with d = Q and A = the string
field.

The authors of the above papers now try to adapt this techniques to
_super_string field theory, where the equations of motion are not
always/not manifestly of the "zero curvature form".

In particular in section 4 on p.8 of http://www.arxiv.org/abs/hep-th/0204155 they give (in
equation (4.3)) an implementation of this idea for Berkovits' version of
susy SFT. They do this by "extending" the ordinary SSFT operator to
functions of some auxiliary parameter \lambda and by identifying the
exterior derivative d from above with the sum of the (extended version of)
the worldsheet BRST operator and the \eta_0 super-ghost. The introduction
of \lambda is necessary in order to decouple two terms in the expression
for the respective curvature (equation (4.4)), so that one ideed gets the
Berkovits SSFT equations of motion (4.6).

This in not quite the idea that I had in mind, though! I was also thinking
in terms of formulating the Berkovits string in a form that it looks
approximately like the bosonic Witten string field, so that similar formal
techniques might be aopplied by analogy, but the method that I had in
mind, motivated by Kluson's papers, is rather based on the introduction of
differentials corresponding to Q and \eta_0, "meta-ghosts" so to say, than
on the introduction of an auxiliary parameter \lambda.

Namely my point was, as explained in my previous post in this thread, that
using the "meta-ghosts" as in Kluson's papers we can replace the ordinary
BRST operator by the "meta-BRST operator"= exterior derivative

d = dx^1 \eta_0 + dx^2 Q

in a way that Berkovbits' equations of motion are precisely of the formal
form as the bosonic eoms of bosonic cubic SFT. It would perhaps be
interesting to see what happens to this ansatz is fed in to the Lax pair
kind of methods discussed by Lechtenfeld, Popov, Uhlmann and Kling in the
above mentioned papers and the crank is turned.

Well, on a second thought it would probably yield precisely the method of
http://www.arxiv.org/abs/hep-th/0204155, since essentially the role played by my "meta ghosts" in
distinguishing \eta_0 contributions from Q-contributions is precisely that
of the factor \lambda in equation (4.3) of that paper, I think.

But I haven't taken the time to thtink this to the end. I really have to
run now.