### CFTs from OSFT?

#### Posted by urs

#### Update 19 May 2004

I have finally found a paper which pretty much precisely discusses what I was looking for here, namely a relation between classical solutions of string field theory and deformations of the worldsheet (boundary-) conformal field theory. It’s

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

(see also this entry)

and it discusses how OSFT actions expanded about two different classical solutions correspond to two worldsheet BCFTs in the case where the latter are related by marginal deformations. In the words of the author of the above paper (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 $ which are related by marginal deformation, are in fact two SFT actions expanded around different classical solutions.

In equation (2.31) the deformed BRST operator is given, which is what I discuss in the entry below, but then it is shown in (3.8) that this operator can indeed be related to a (B)CFT with marginal deformation.

One subtlety of this paper is that the classical SFT solutions which are considered are *large* but *pure* gauge and hence naively equivalent to the trivial solution ${\Phi}_{0}=0$, but apparently only naively so. To me it would be intreresting if similar results could be obtained for more general classical solutions ${\Phi}_{0}$.

#### Update 3rd May 2004

I have now some LaTeXified notes.

Here is a rather simple — indeed almost trivial — observation concerning open string field theory (OSFT) and deformations of CFTs, which I find interesting, but which I haven’t seen discussed anywhere in the literature. That might of course be just due to my insufficient knowledge of the literature, in which case somebody please give me some pointers!

#### Update 7th May 2004

I have by now found some literature where this (admittedly very simple but interesting) observation actually appears, e.g.

- equation (2.48) of

which originates inAshjoke Sen & Barton Zwiebach: A proof of local background independence of classical closed string field theory (1993)

Ashoke Sen: Equations of motion in non-polynomial closed string field theory and conformal invariance of two dimensional field theories (1990)

- section 5 of
Isao Kishimoto & Kazuki Ohmori: CFT Description of Identity String Field: Toward Derivation of the VSFT Action (2001)

- and p. 8 of
I. Aref’eva, D. Belov, A. Giryavets, A. Koshelev and P. Medvedev: Noncommutative Field Theories and (Super) String Field Theories (2002)

Here goes:

There have been some studies (few, though) of worldsheet CFTs for various backgrounds in terms of *deformed BRST operators*. *I.e.*, starting from the BRST operator ${Q}_{B}$ for a given background, like for instance flat Minkowski space, one may consider the operator

where $\hat{\Phi}$ is some operator such that nilpotency ${\tilde{Q}}_{B}^{2}=0$ is preserved.

By appropriately commuting ${\tilde{Q}}_{B}$ with the ghost modes the conformal generators ${\tilde{L}}_{m}^{\mathrm{tot}}$ of a new CFT in a new background are obtained (the new background might of course be gauge eqivalent to the original one).

See for instance

Mitsuhiro Kato: Physical Spectra in String Theories — BRST Operators and Similarity Transformations (1995)

and

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

One problem is to understand the operators $\hat{\Phi}$, how they have to be chosen and how they encode the information of the new background.

Here I want to show, in the context of open bosonic strings, that the consistent operators $\hat{\Phi}$ are precisely the operators of *left plus right star-multiplication by the string field $\Phi $ which describes the new background in the context of open string field theory*.

In order to motivate this consider the (classical) equation of motion of cubic open bosonic string field theory for a string field $\Phi $ of ghost number one:

where for simplicity of notation the string field has been rescaled by a constant factor.

(I am using the notation as for instance in section 2 of

Kazuki Ohmori: A Review on Tachyon Condensation in Open String Field Theories (2001).)

If we now introduce $\hat{\Phi}$, the *operator of star-multiplication by $\Phi $* defined by

then, due to the associativity of the star product this can equivalently be rewritten as an operator equation

because

(Here it has been used that ${Q}_{B}$ is an odd graded (with respect to ghost number) derivation on the star-product algebra of string fields, that $\Phi $ is of ghost number 1 and that the star-product is associative.)

It hence follows that the equations of motion of the string field $\Phi $ are precisely the necessary and sufficient condition for the operator $\hat{\Phi}$ to yield a nilpotent, unit ghost number deformation

of the original BRST operator.

But there remains the question why ${\tilde{Q}}_{B}$, while nilpotent, can really be interpreted as a BRST operator of some sensible CFT. (Surely not every nilpotent operator on the string Hilbert space can be identified as a BRST operator!) The reason seems to be the following:

#### Update 21 May 2004

I have found out by now that what I was trying to argue here has already been found long ago in papers on background independence of string field theory. For instance on p.2 of

Ashoke Sen: Equations of motion in non-polynomial closed string field theory and conformal invariance of two dimensional field theories (1990)

it says:

In this paper we show that if ${\Psi}_{\mathrm{cl}}$ is a solution of the classical equations of motion derived from the action $S(\Psi )$, then it is possible to construct an operator ${\hat{Q}}_{B}$ in terms of ${\Psi}_{\mathrm{cl}}$, acting on a subspace of the Hilbert space of combined matter-ghost CFT, such that $({\hat{Q}}_{B}{)}^{2}=0$. ${\hat{Q}}_{B}$ may be interpreted as the BRST charge of the two dimensional field theory describing the propagation of the string in the presence of the background field ${\Psi}_{\mathrm{cl}}$.

We may consider, in the context of open bosonic string field theory, the motion of a single ‘*test string*’ in the background described by the excitatoins $\Phi $ by adding a tiny correction field $\psi $ to $\Phi $, which we want to interpret as the string field due to the single test string.

The question then is: What is the condition on $\psi $ so that the total field $\Phi +\psi $ is still a solution to the equations of motion of string field theory. That is, given $\Phi $, one needs to solve

for $\psi $. But since $\psi $ is supposed to be just a tiny perturbation of the filed $\Phi $ it must be sufficient to work to first order in $\psi $. This is equivalent to neglecting any self-interaction of the string described by $\psi $ and only considering its interaction with the ‘background’ field $\Phi $ - just as in the first quantized theory of single strings.

But to first order and using the fact that $\Phi $ is supposed to be a solution all by itself the above equation says that

This is manifestly a deformation of the equation of motion

of the string described by the state $\psi $ in the original background. Hence it is consistent to interpret

as the new worldsheet BRST operator which corresponds to the new background described by $\Phi $.

If we again switch to operator notation the above can equivalently be rewritten as

where the braces denote the anticommutator, as usual.

Recalling that a gauge transformation $\Phi \to \Phi +\delta \Phi $ in string field theory is (for $\Lambda $ a string field of ghost number 0) of the form

and that in operator language this reads equivalenty

one sees a close connection of the deformed BRST operator to *covariant exterior derivatives*.

As is very well known (for instance summarized in the table on p. 16 of the above review paper) there is a close analogy between string field theory formalism and exterior differential geometry.

The BRST operator ${Q}_{B}$ plays the role of the exterior derivative, the $c$ ghost correspond to differential form creators, the $b$-ghosts to form annihilators and the $\star $ product to the ordinary wedge ($\wedge $) product - or does it?

As noted on p.16 of the above review, the formal correspondence seems to cease to be valid with respect to the graded commutativity of the wedge product. Namely in string field theory

in general.

But the above considerations suggest an interpretation of this apparent failed correspondence, which might show that indeed the correspondence is better than maybe expected:

The formal similarity of the deformed BRST operator ${\tilde{Q}}_{B}={Q}_{B}+\hat{\Phi}$ to a *gauge covariant* exterior derivative $d+\omega $ suggests that we need to interpret $\Phi $ not simply as a 1-form, but as a - *connection*!

That is, $\Phi $ would correspond to a Lie-algebra valued 1-form and the $\star $-product would really be exterior wedge multiplication together with the Lie product, as very familiar from ordinary gauge field theory. For instance we would have expression like

In such a case it is clear that the graded commutativity of the wedge product is broken by the Lie algebra products.

Is it consistent to interpret the star product of string field theory this way? Seems to be, due to the following clue:

Under the *trace* graded commutativity should be restored. The trace should appear together with the integral as in

But precisely this is what does happen in open string field theory in the formal integral. There we have

All this suggests that one should think of the deformed BRST operator as morally a gauge covariant exterior derivative:

That looks kind of interesting to me. Perhaps it is not new (references, anyone?), but I have never seen it stated this way before. This way the theory of (super)conformal deformations of (super)conformal field theories might nicely be connected to string field theory.

In particular, it would be intersting to check the above considerations by picking some known solution $\Phi $ to string field theory and computing the explicit realization of ${\tilde{Q}}_{B}$ for this background field, maybe checking if it looks the way one would expect from, say, worldsheet Lagrangian formalism in the given background.

Posted at April 29, 2004 6:42 PM UTC
## Re: CFTs from OSFT?

Good morning :)

You seem to be making a case for the relation between open string field theory and exterior differential geometry. You have analogues to the (covariant) exterior derivative and wedge product. Is there an analogue to the Hodge star and global inner product?

I know I sound like a broken record, but I am not very comfortable deforming the exterior derivative and am trying to think of ways to move the deformation from ${\tilde{Q}}_{B}$ to its adjoint. But to be able to define adjoint, you need an inner product (well, then again if you define an adjoint, this gives an inner product. Hmm…)

This is a naive question, but is it possible to write ${\tilde{Q}}_{B}$ as a similarity transform of ${Q}_{B}$? For example, a first guess would be

where the product used to define the exponential is left undefined :) It could be operator product or maybe star product or even something else.

I’m sure that what I’m looking for is not precisely what I wrote above, but it should be something like it. Of course, the point is that if you could do this, then you could move the deformation to the adjoint of ${\tilde{Q}}_{B}$ via a deformation of the inner product and leave ${Q}_{B}$ unchanged. If you are talking about deforming a background, it seems a lot more natural to me to deform the inner product (which defines the background), then to deform a topological operator like the exterior derivative.

Just a thought for the morning :)

Eric