May 19, 2004

CFTs from OSFT!

Posted by Urs Schreiber

A while ago I began to think about how deformed worldsheet CFTs could be related to deformed BRST operators obtained from classical solutions of OSFT. I knew that I was in the dark probably trying to re-invent the wheel, being ignorant of lots of results (see the discussion on s.p.s.) - but one has to start somewhere.

Now I am glad that I have finally found a recent paper where pretty much precisely the question which I was concerned with is studied. It is

[Update 21 May 2004: The general point has been made already in 1990 by Ashoke Sen in

Ashoke Sen: On the background independence of string field theory (1990).

There in the abstract it says:

Given a solution ${\Psi }_{\mathrm{cl}}$ of the classical equations of motion in either closed or open string field theory formulated around a given conformal field theory background, we can construct a new operator ${\stackrel{̂}{Q}}_{B}$ [$={\stackrel{̂}{Q}}_{B\phantom{\rule{thinmathspace}{0ex}}\mathrm{original}}+\left[{\Psi }_{\mathrm{cl}}\star ,\cdot \right]$ (my remark)] in the corresponding two dimensional field theory such that $\left({\stackrel{̂}{Q}}_{B}{\right)}^{2}=0$. It is shown that in the limit when the background field ${\Psi }_{\mathrm{cl}}$ is weak, ${\stackrel{̂}{Q}}_{B}$ can be identified to the BRST charge of a new local conformal field theory

]

J. Klusoň: Exact Solutions in SFT and Marginal Deformation in BCFT (2003)

In the introduction it says (p. 2):

Our goal is to show that when we expand [the] string field around [a] classical solution and insert it into the original SFT action $S$ which is defined on [a given] BCFT, we obtain after suitable redefinition of the fluctuation modes the SFT action ${S}^{\prime }$ defined on ${\mathrm{BCFT}}^{\prime \prime }$ that is related to the original BCFT by inserting [a] marginal deformation on the boundary of the worldsheet. […] To say differently, we will show that two SFT action $S$, ${S}^{\prime }$ written using two different BCFT, ${\mathrm{BCFT}}^{\prime \prime }$ which are related by marginal deformation, are in fact two SFT actions expanded around different classical solutions.

That’s the type of result what I was thinking about. Apparently there are old related results in

A. Sen & B. Zwiebach: A proof of local background independence of classical closed string field theory (1993)

[Update 20 May 2004: The file on the arXiv does not seem to properly compile. Here is a working pdf version. Thanks to Yuji Tachikawa!]

The main point of my previous ponderings was the, maybe not very deep but in any case maybe interesting, speculation that to a given classical solution ${\Phi }_{0}$ of OSFT the deformed BRST operator

(1)$\stackrel{˜}{Q}=Q+\left[{\Phi }_{0}\star ,\cdot \right]$

can in fact be identified as the BRST operator of a new worldsheet CFT, corresponding to the background described by ${\Phi }_{0}$.

From the responses that I received I got the impression that to some people this seems maybe obvious or even trivial. But on the other hand it is hard to find literature on any specific details on how this works in given examples.

The only work done in this direction which I knew of was the one by Ioannis Giannakis, especially

Ioannis Giannakis, Strings in Nontrivial Gravitino and Ramond-Ramond Backgrounds (2002)

which used such deformations of BRST operators for closed superstrings to guess the deformations which should follow, as I begin to understand now, from superstring SFT at classical solutions describing Ramond-sector backgrounds. (The subtleties related to worldsheet supersymmetry in Ramond backgrounds have been discussed here.)

It would be nice if things like that could really be derived from SFT, and in particular I would like to see if the deformations that I discuss in hep-th/0401175 can be derived from SSFT - but that’s still a long way to go…

For these reasons I am glad to have found the above paper by Klusoň, where at least some aspects of the relation (O)SFT $↔$ (B)CFT are discussed.

The main restriction of Klusoň’s approach, from the above point of view, seems to be that he restricts attention to classical SFT solutions of the form

(2)${\Phi }_{0}={e}^{-W}\star Q{e}^{W}\phantom{\rule{thinmathspace}{0ex}},$

i.e. to solutions which naively seem to be gauge equivalent to the trivial solution ${\Phi }_{0}=0$. But apparently there is a subtlety related to finite gauge transformations, which can make this a non-trivial classical solution to SFT.

Anyway, as discussed on p.8 of that paper, expanding the SFT action around such a classical ‘background’ ${\Phi }_{0}$ yields the mentioned deformations of the BRST operator

(3)$\stackrel{˜}{Q}={Q}^{\prime }=Q+\left[W,Q\right]+\cdots \phantom{\rule{thinmathspace}{0ex}},$

as expected. The essential point of Klusoň’s paper is that we can now write down the SFT action in terms of ${Q}^{\prime }$ (equation (3.1)), massage it appropriately and demonstrate this way that it is precisely of the form of the SFT action that one would write down with respect to a (B)CFT obtained from marginal deformations with the operator ${e}^{W}$. In other words, the SFT action using ${Q}^{\prime }$ which describes excitations about the classical solution ${\Phi }_{0}$ is exactly what one obtains alternatively when all the correlators $〈\cdots 〉$ in the CFT-language version of the OSFT action

(4)$S\sim 〈I\circ \Psi \left(0\right)\phantom{\rule{thinmathspace}{0ex}}Q\Psi \left(0\right)〉+\frac{2}{3}〈{f}_{1}\circ \Psi \left(0\right)\phantom{\rule{thinmathspace}{0ex}}{f}_{2}\circ \Psi \left(0\right)\phantom{\rule{thinmathspace}{0ex}}{f}_{3}\circ \Psi \left(0\right)〉$

are replaced by their respective marginal deformed correlators

(5)$〈{\Psi }_{1}\left({x}_{1}\right){\Psi }_{2}\left({x}_{2}\right)\cdots 〉\to 〈{\Psi }_{1}\left({x}_{1}\right){\Psi }_{2}\left({x}_{2}\right)\cdots {〉}_{W}:=〈{e}^{W}{\Psi }_{1}\left({x}_{1}\right){e}^{W}{\Psi }_{2}\left({x}_{2}\right)\dots 〉\phantom{\rule{thinmathspace}{0ex}}.$

That’s nice, because marginal deformations of BCFTs have been studied in quite some detail, the canonical reference being apparently

A. Recknagel & V. Schomerus: Boundary deformation theory and moduli spaces of D-branes (1999).

That’s all very nice. But here is one related riddle that I have pondered all day and which seems to be simple, but which I couldn’t get a handle on yet:

Apart from the backgrounds discussed above, which correspond to marginal deformations, there is at least one further background which should be a simple testing ground for these deformations, I’d say, namely a pure gauge field background, i.e. with

(6)${\Phi }_{A}=\int {d}^{D}k\phantom{\rule{thinmathspace}{0ex}}{A}_{\mu }\left(k\right){\alpha }_{-1}^{\mu }{c}_{1}\mid k〉$

(e.g. equation (2.34) of the review hep-th/0102085).

We know that the related BCFT differs from the unperturbed one just by the charged endpoints of the string in the given gauge field. The question to me is: Is this also the result that we obtain from looking at

(7)$Q+\left[{\Phi }_{A}\star ,\cdot \right]\phantom{\rule{thinmathspace}{0ex}}?$

In order to decide this I tried to reexpress the operator $\left[{\Phi }_{A}\star ,\cdot \right]$ in the usual first quantized formalism, i.e. reexpressing the graded star-commutator with this particular ${\Phi }_{A}$ by polynomials in the ${\alpha }_{n},{c}_{n}$.

I have tried to find a closed expression for this using the machinery in

T. Kawano & K. Okuyama: Open String Fields as Matrices (2001),

which seemed to come in handy, since the SFT vertices simplify greatly in that formalism (e.g. formula (2.24) in that paper), but of course one has to deal instead with the Bogoliubov transformation (2.19), which obscures things again, at least to me at the moment. So either I am not seeing the obvious or this is harder than I expected. Hints are appreciated.

Posted at May 19, 2004 8:00 PM UTC

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