## Kluson on marginal deformations

<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">&nbsp;&nbsp;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" 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 $h = 1,$ for $C_L/R$ the left/right halfs of the unit cricle in the complex plane and for the definition $$W_{L/R} = \int_{C_{L/R}} \frac{dz}{2\pi i} W(z)$$ we have a 'partial integration rule' for the string field star product: $$(W_L A) * B = - (-1)^{|W||A|} A * (W_R B) .$$ 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 $$\tilde Q = Q + [\Psi, \cdot]$$ but apparently first answers have been found only recently by Kluson. He shows, using the above formula, that for $\Psi-s$ of the form $$\Psi = W_R (I),$$ where "I" is the identity string field, we have $$[\Psi, A] = - W(A) .$$ 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 $$W \sim \partial X^\mu$$ and might worry how this fits in with the fact that the SFT solution involved $W_R(I)$ 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 $$W_R(\psi)$$ for W $~ \partial X^\mu$ 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.