Kluson on marginal deformations

Urs Schreiber

<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>I am currently looking at\n\nJ. Kluson\nExact Solutions in SFT and Marginal Deformation in BCFT\nhttp://arxiv.org/abs/hep-th/0303199\n\nand I realize that I don\'t quite follow the reasoning in the second line of\nequation (2.36), where it is claimed to be shown how the star-anticommutator\nwith a certain string field can be re-expressed as the action of an operator\nin the string\'s Hilbert space.\n\nIt seems to me that the author uses the fact that the operator K in that\npaper is a derivation with respect to the star product, as in equation\n(2.20) of the companion paper\nhttp://xxx.uni-augsburg.de/abs/hep-th/0209255, but I don\'t see why he can\nassume this.\n\nIf anyone does, please let me know.\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>I am currently looking at

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

and I realize that I don't quite follow the reasoning in the second line of
equation (2.36), where it is claimed to be shown how the star-anticommutator
with a certain string field can be re-expressed as the action of an operator
in the string's Hilbert space.

It seems to me that the author uses the fact that the operator K in that
paper is a derivation with respect to the star product, as in equation
(2.20) of the companion paper
http://xxx.uni-augsburg.de/abs/http:...ep-th/0209255, but I don't see why he can
assume this.

If anyone does, please let me know.

Urs Schreiber

<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:2h71biF9r2k2U1-100000@uni-berlin.de...\n> I am currently looking at\n>\n> J. Kluson\n> Exact Solutions in SFT and Marginal Deformation in BCFT\n> http://arxiv.org/abs/hep-th/0303199\n>\n> and I realize that I don\'t quite follow the reasoning in the second line\nof\n> equation (2.36), where it is claimed to be shown how the\nstar-anticommutator\n> with a certain string field can be re-expressed as the action of an\noperator\n> in the string\'s Hilbert space.\n>\n> It seems to me that the author uses the fact that the operator K in that\n> paper is a derivation with respect to the star product, as in equation\n> (2.20) of the companion paper\n> http://xxx.uni-augsburg.de/abs/hep-th/0209255, but I don\'t see why he can\n> assume this.\n>\n> If anyone does, please let me know.\n\nThe answer to this question is the derivation in equation (4.33) of\n\nI. Kishimoto & K. Ohmori\nCFT Description of Identity String Field: Toward Deriavation of the VSFT\nAction\nhep-th/0112169\n\nwhich demonstrates (explicitly for the BRST current, but implicitly for any\nh=1 chiral field) that for W(z) any chiral field of conformal weight h = 1,\nfor C_L/R the left/right halfs of the unit cricle in the complex plane and\nfor the definition\n\nW_{L/R} = \\int_{C_{L/R}} \\frac{dz}{2\\pi i} W(z)\n\nwe have a \'partial integration rule\' for the string field star product:\n\n(W_L A) * B = - (-1)^{|W||A|} A * (W_R B) .\n\nThat\'s extremely nice, because it allows one (and in particular allowed J.\nKluson) to answer the question which I asked before here in this group,\nnamely:\n\n"How can we re-express the operation of graded star commutation with a\ncertain string field in terms of the direct explicit action of operators on\nthe string\'s Hilbert space?"\n\nSeems to me to be a basic question when dealing with different string field\nbackgrounds, due to the fact that the deformed BRST operator is of the form\n\n\\tilde Q = Q + [\\Psi, \\cdot]\n\nbut apparently first answers have been found only recently by Kluson.\n\nHe shows, using the above formula, that for Psi-s of the form\n\nPsi = W_R (I),\n\nwhere "I" is the identity string field, we have\n\n[\\Psi, A] = - W(A) .\n\nThat\'s very nice. It answers a couple of questions which I tried to discuss\nhere recently.\n\nIn fact, together with section 4.1 of\n\nA. Recknagel & V. Schomerus\nBoundary Deformation Theory and Moduli Spaces of D-branes\nhep-th/9811237\n\none can pretty much directly associate pure gauge SFT solutions to\ntranslational symmetries of the background spacetime with the branes being\nmoved around.\n\nOne would expect that such translations are generated by the associated\ncurrent\n\nW \\sim \\partial X^\\mu\n\nand might worry how this fits in with the fact that the SFT solution\ninvolved W_R(I) instead of W itself. But it seems to me that the answer is\nhidden in the answer to another question which I asked recently\n\nhttp://groups.google.de/groups?selm=Pine.LNX.4.31.0405140632340.12546-100000%40feynman.harvard.edu\n\nconcerning the nature of the projector states in SFT.\n\nNamely according to equation (338) of\n\nW. Taylor & B. Zwiebach\nD-branes, Tachyons, and String Field Theory,\nhep-th/0311017\n\nthe identity field is nothing but the ordinary SL(2,C) invariant worldsheet\nvacuum plus a spurious state, so that\n\nW_R(psi)\n\nfor W ~ \\partial X^\\mu\n\nstarts out with the expected term plus further irrelevant stuff, roughly.\n\nIf anyone is interested in more details on what I wrote above I have\nexpanded my simple private notes on background perturbations in OSFT. The\ncurrent version can be found at\n\nhttp://www-stud.uni-essen.de/~sb0264/p8.pdf\n\nwith section 2.2. discussing J. Kluson\'s construction. I\'d enjoy hearing\nfurther comments.\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:2h71biF9r2k2U1-100000@uni-berlin.de...
> I am currently looking at
>
> J. Kluson
> Exact Solutions in SFT and Marginal Deformation in BCFT
> http://arxiv.org/abs/http://www.arxi...hep-th/0303199
>
> and I realize that I don't quite follow the reasoning in the second line
of
> equation (2.36), where it is claimed to be shown how the
star-anticommutator
> with a certain string field can be re-expressed as the action of an
operator
> in the string's Hilbert space.
>
> It seems to me that the author uses the fact that the operator K in that
> paper is a derivation with respect to the star product, as in equation
> (2.20) of the companion paper
> http://xxx.uni-augsburg.de/abs/http:...ep-th/0209255, but I don't see why he can
> assume this.
>
> If anyone does, please let me know.

The answer to this question is the derivation in equation (4.33) of

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

which demonstrates (explicitly for the BRST current, but implicitly for any
h=1 chiral field) that for W(z) any chiral field of conformal weight [itex]h = 1,[/itex]
for [itex]C_L/R[/itex] the left/right halfs of the unit cricle in the complex plane and
for the definition

[tex]W_{L/R} = \int_{C_{L/R}} \frac{dz}{2\pi i} W(z)[/tex]

we have a 'partial integration rule' for the string field star product:

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

That's extremely nice, because it allows one (and in particular allowed J.
Kluson) to answer the question which I asked before here in this group,
namely:

"How can we re-express the operation of graded star commutation with a
certain string field in terms of the direct explicit action of operators on
the string's Hilbert space?"

Seems to me to be a basic question when dealing with different string field
backgrounds, due to the fact that the deformed BRST operator is of the form

[tex]\tilde Q = Q + [\Psi, \cdot][/tex]

but apparently first answers have been found only recently by Kluson.

He shows, using the above formula, that for [itex]\Psi-s[/itex] of the form

[tex]\Psi = W_R (I),[/tex]

where "I" is the identity string field, we have

[tex][\Psi, A] = - W(A) .[/tex]

That's very nice. It answers a couple of questions which I tried to discuss
here recently.

In fact, together with section 4.1 of

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

one can pretty much directly associate pure gauge SFT solutions to
translational symmetries of the background spacetime with the branes being
moved around.

One would expect that such translations are generated by the associated
current

[tex]W \sim \partial X^\mu[/tex]

and might worry how this fits in with the fact that the SFT solution
involved [itex]W_R(I)[/itex] instead of W itself. But it seems to me that the answer is
hidden in the answer to another question which I asked recently

http://groups.google.de/groups?selm=...an.harvard.edu

concerning the nature of the projector states in SFT.

Namely according to equation (338) of

W. Taylor & B. Zwiebach
D-branes, Tachyons, and String Field Theory,
http://www.arxiv.org/abs/hep-th/0311017

the identity field is nothing but the ordinary SL(2,C) invariant worldsheet
vacuum plus a spurious state, so that

[tex]W_R(\psi)[/tex]

for W [itex]~ \partial X^\mu[/itex]

starts out with the expected term plus further irrelevant stuff, roughly.

If anyone is interested in more details on what I wrote above I have
expanded my simple private notes on background perturbations in OSFT. The
current version can be found at

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

with section 2.2. discussing J. Kluson's construction. I'd enjoy hearing
further comments.

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