CFTs from OSFT?

April 29, 2004

Posted by urs

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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 $S'$ defined on $BCFT''$ 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 $S$ , $S'$ written using two different BCFT, $BCFT'$ 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 $Φ_{0} = 0$ , but apparently only naively so. To me it would be intreresting if similar results could be obtained for more general classical solutions $Φ_{0}$ .

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

(1)

{\tilde{Q}}_{B} : = Q_{B} + \hat{Φ},

where $\hat{Φ}$ 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 ${\tilde{L}}_{m}^{tot}$ 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 $\hat{Φ}$ , 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 $\hat{Φ}$ are precisely the operators of left plus right star-multiplication by the string field $Φ$ 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 $Φ$ of ghost number one:

(2)

Q_{B} ∣ Φ 〉 + ∣ Φ ⋆ Φ 〉 = 0,

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 $\hat{Φ}$ , the operator of star-multiplication by $Φ$ defined by

(3)

\hat{Φ} ∣ Ψ 〉 : = ∣ Φ ⋆ Ψ 〉

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

(4)

{(Q_{B} + \hat{Φ})}^{2} = 0

because

(5)

(Q_{B} + \hat{Φ}) \circ (Q_{B} + \hat{Φ}) \circ = {\underset{︸}{Q_{B} \circ Q_{B} \circ}}_{= 0} + {\underset{︸}{Q_{B} \circ Φ ⋆ + Φ ⋆ Q_{B} \circ}}_{= (Q_{B} Φ) ⋆} + {\underset{︸}{Φ ⋆ Φ ⋆}}_{= (Φ ⋆ Φ) ⋆} .

(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 $Φ$ is of ghost number 1 and that the star-product is associative.)

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

(6)

{\tilde{Q}}_{B} = Q_{B} + \hat{Φ}

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 $Ψ_{cl}$ is a solution of the classical equations of motion derived from the action $S (Ψ)$ , then it is possible to construct an operator ${\hat{Q}}_{B}$ in terms of $Ψ_{cl}$ , 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 $Ψ_{cl}$ .

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 $Φ$ by adding a tiny correction field $ψ$ to $Φ$ , which we want to interpret as the string field due to the single test string.

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

(7)

Q_{B} (Φ + ψ) + (Φ + ψ) ⋆ (Φ + ψ) = 0

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

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

(8)

Q_{B} ∣ ψ 〉 + ∣ Φ ⋆ ψ 〉 + ∣ ψ ⋆ Φ 〉 = 0 .

This is manifestly a deformation of the equation of motion

(9)

Q_{B} ∣ ψ 〉 = 0

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

(10)

{\tilde{Q}}_{B} = Q_{B} + {\hat{Φ}, \cdot}

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

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

(11)

{(Q_{B} + \hat{Φ}), \hat{ψ}} = 0,

where the braces denote the anticommutator, as usual.

Recalling that a gauge transformation $Φ \to Φ + δ Φ$ in string field theory is (for $Λ$ a string field of ghost number 0) of the form

(12)

δ Φ = Q_{B} Λ + Φ ⋆ Λ - Λ ⋆ Φ

and that in operator language this reads equivalenty

(13)

\hat{δ Φ} = [(Q_{B} + \hat{Φ}), \hat{Λ}] = 0

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 $⋆$ product to the ordinary wedge ( $\land$ ) 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

(14)

Φ ⋆ ψ \neq \pm Ψ ⋆ Φ

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 ${\tilde{Q}}_{B} = Q_{B} + \hat{Φ}$ to a gauge covariant exterior derivative $d + ω$ suggests that we need to interpret $Φ$ not simply as a 1-form, but as a - connection!

That is, $Φ$ would correspond to a Lie-algebra valued 1-form and the $⋆$ -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

(15)

(d + ω)^{2} = (d ω) δ^{a}_{b} + ω^{a}_{c} \land ω^{c}_{b} .

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

(16)

\int tr ω^{a}_{c} \land γ^{c}_{b} = \pm \int tr γ^{a}_{c} \land ω^{c}_{b} .

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

(17)

\int Φ ⋆ Ψ = \pm \int Ψ ⋆ Φ .

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

(18)

{\tilde{Q}}_{B} = Q_{B} + \hat{Φ} \sim d + ω .

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 $Φ$ 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

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Re: CFTs from OSFT?

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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

(1)

{\tilde{Q}}_{B} = e^{f (\hat{Φ})} Q_{B} e^{- f (\hat{Φ})},

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

Posted by: Eric on April 30, 2004 1:34 PM | Permalink | Reply to this

Re: CFTs from OSFT?

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Hi Eric -

the first thing to note is that the exterior differention that plays a role here is quite a different one than what we discussed before. Here we are talking about the BRST operator, which contains all the constraints and can be regarded as the exterior derivative on the Virasoro gauge group, while there we talked about the constraints themselves and how then can be interpreted as exterior derivartives on configuration space (loop space).

In the context of open string field theory (OSFT) there is no Hodge inner product and no coderivative to the BRST operator. That’s conceptually because the action of string field theory is of the form of Chern-Simons theory, a topological field theory. (Lubos very much emphasized this in his latest post on spr. In fact I knew that, but had never seen people explicitly treat the string field as a nontrivial connection before. Probably my fault. :-)

The similarity transformations that you mention should correspond to gauge transformations of the string fields.

Think about the analogy with covariant exterior derivatives $d_{A} = d + ω$ with respect to some gauge group. There the similarity transformation that you have in mind is just the gauge transformation

(1)

d_{A} \to g^{- 1} d_{A} g

for some group element $g$ which sends the connection to

(2)

ω \to g^{- 1} ω g + g^{- 1} (d g) .

Concerning the fact that you don’t like the exterior derivative to be deformed: I know where you come from, but keep in mind that if we explicitly represent the exterior derivative in terms of form creators and orthonormal vector fields, that then a change of basis does also change the form of the explitcit representation of the operator. I think that’s what’s going on here.

Finally, please not that the relation of the BRST operator to exterior differention etc. is well known and all. My point was that we can get worldsheet BRST operators from string field background fields and that the new BRST operators obtained this way are formally the original operator but with respect to another formal ‘gauge connection’, the string field.

But apparently that was known already 20 years ago! :-)

Posted by: Urs Schreiber on April 30, 2004 5:27 PM | Permalink | PGP Sig | Reply to this

Re: CFTs from OSFT?

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Good morning! :)

In the context of open string field theory (OSFT) there is no Hodge inner product and no coderivative to the BRST operator. That’s conceptually because the action of string field theory is of the form of Chern-Simons theory, a topological field theory. (Lubos very much emphasized this in his latest post on spr. In fact I knew that, but had never seen people explicitly treat the string field as a nontrivial connection before. Probably my fault. :-)

If OSFT is a topological field theory, then how is it supposed to describe physics? Or is it not supposed to describe physics? Now I don’t have a clue about what you mean by “deformations” of the background. Usually, when I think of a “background”, I think of some geometric background. Are you talking about a topological background or something? I am beginning to think that “open string field theory” is a misnomer and it should instead be called “topological open string field theory” (TOSFT). Roughly speaking, I think of Chern-Simons as a kind of topological Yang-Mills theory. Analogously, what if OSFT should really be a topological version of some more physical “geometrical open string field theory” (GOSFT)? In a GOSFT, I would expect to see Hodge stars and adjoints of exterior derivatives. Does such a thoery exist? Hata claims that he doesn’t want a GOSFT because

“the separation of the action into the kinetic and interaction terms is necessarily perturbation theoretical.”

Wait a second, I should probably finish reading Hata’s paper before writing this, but I am writing as I read, so I may start answering my own questions :) On the top of page 7, Hata has the expression

(1)

(Q_{B} Φ) \cdot Ψ = (- 1)^{1 + ∣ Φ ∣} Φ \cdot Q_{B} Ψ .

Doesn’t this mean that $Q_{B}$ is self-adjoint (up to a constant). That would kind of ruin the interpretation of $Q_{B}$ as an exterior derivative, wouldn’t it? I’ll admit that I don’t really understand the $\cdot$ product though :)

Wow :) The discussion on page 7 and 8 is beautiful :) I am officially a Hata fan (or whoever wrote this section) :)

I was going to say that I don’t understand his figures because they depict closed strings and I thought we were dealing with open strings :) Better late than never, I catch that he is talking about closed strings :) For a discrete space (not loop space), our version of this picture, i.e. integrating an infinitessimal closed loop around a string to get $d$ , we’d have something similar but it would be more like little line segments instead of little closed loops.

Ok. The discussion on page 9-12 is just a bunch of uninteresting calculations (not unnnecassary, it’s just that I’m not interested in seeing them), but the conclusions on page 13 are again fascinating :)

OK. I’m left with a couple of questions. The first one, I already mentioned, is the fact that $Q_{B}$ doesn’t really look like an analog of the exterior derivative to me anymore. Could it be that it is instead something like a Dirac operator

(2)

Q_{B} \overset{?}{=} d \pm d^{†}

?? I bring this up again because $Q_{B}$ is self-adjoint (up to a constant) with respect to $\cdot$ .

The second question is about the nature of “dimension” here. If we are really supposed to think of these field quantities as analogues of exterior calculus, then what is the meaning of degree? Would $Φ$ , $Ψ$ be like 0-forms or 1-forms? For example, the action in Equation (1)

(3)

S = Φ \cdot Q_{B} Φ + \frac{2}{3} g Φ^{3}

looks like

(4)

S_{CS} = \int_{M} (A \land dA + \frac{2}{3} g A \land A \land A)

which makes me think that $Φ$ should be like a 1-form. The latter is a 3d Chern-Simons action, the former looks like it is also a “3 dimensional theory”, but the word “dimensional” must have a different meaning in this context because he says that it is defined for 26 dimensions. The two concepts of “dimension” obviously cannot be the same. What is going on? :)

But this brings me back to my confusion regarding Hata’s $\cdot$ -product. It almost looks like it is a “topological” inner product analogous to

(5)

A \cdot B = \int_{M} A \land B,

but really more like a Hermitian version of this

(6)

A \cdot B = \int_{M} \tilde{A} \land B,

where $\tilde{A}$ is the complex conjugate of $A$ . Is it true that

(7)

A \cdot B \overset{?}{=} (- 1)^{∣ A ∣ ∣ B ∣} \tilde{B \cdot A} .

This would restore the interpretation of $Q_{B}$ as an exterior derivative because

(8)

d (A \land B) = (dA) \land B + (- 1)^{∣ A ∣} A \land dB

so that

(9)

\int_{M} d (A \land B) = (dA) \cdot B + (- 1)^{∣ A ∣} A \cdot dB .

Then if

(10)

\int_{M} d (A \land B) = \int_{\partial M} A \land B = 0,

then

(11)

(dA) \cdot B = (- 1)^{1 + ∣ A ∣} A \cdot dB,

which is precisely the expression Hata has at the top of page 7. This restores the interpretation of $Q_{B}$ as the exterior derivative, which is nice, but it calls into question the interpretation of $\cdot$ as a geometrical inner product, which also calls into question his claim that the action in Equation (1) really depends explicitly on the background metric. After all, it looks like it is just a generalized Chern-Simons theory. How can that depend on a metric?

There seems to be one of two (or more) possibilites. Either $\cdot$ is a geometrical inner product meaning $Q_{B}$ is not really an exterior derivative and is rather something more like a Dirac operator or $Q_{B}$ is an exterior derivative meaning that $\cdot$ is not a geometrical inner product. Oh yeah, I can’t leave out the third possibility. Maybe I don’t know what I’m talking about :) That is the most likely case :)

Neat paper :)

Eric

Posted by: Eric on May 1, 2004 4:34 PM | Permalink | Reply to this

Re: CFTs from OSFT?

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Over on sci.physics.strings Lubos Motl was so kind to point me to

Hiroyuki Hata: Pregeometrical String Field Theory: Creation of Space-Time and Motion (1986(!))

where precisely the mechanism by which I argue above that new BRST operators are obtained from string fields is used to show that in fact this way even the ‘original’ flat space BRST operator can be obtained from the trivial operator $Q_{B} = 0$ .

Heh, great, so my idea is right - but about 20 years old…

Darn, I knew that something called background free open string field theory exists, but for some reason I never looked at a paper describing it…

Many thanks to Lubos for this link!

In order to see that what I wrote above is essentially the same as done on p. 8 of the above paper, reformulate what I wrote as operator equations in terms of Lagrangian formalism:

I argued that when in the string field action

(1)

S = \int Φ ⋆ Q_{B} ϕ + \frac{2}{3} \int Φ ⋆ Φ ⋆ Φ

you assume that $Φ$ is a classical solution and perturb about it by substituting

(2)

Φ \to Φ + ψ

that then one obtains string field theory in terms of $ψ$ but for the modified BRST operator

(3)

{\tilde{Q}}_{B} = Q_{B} + {Φ ⋆, \cdot} .

I argued this in terms of equations of motion, but in order to see that it is the same idea as in the above paper look at it in terms of the Lagrangian: The substitution yields:

(4)

S_{0} \to \int Φ ⋆ Q_{B} Φ + 2 \int ψ ⋆ Q_{B} Φ + \int ψ ⋆ Q_{B} ψ + \frac{2}{3} \int Φ ⋆ Φ ⋆ Φ +

(5)

+ 2 \int ψ ⋆ Φ ⋆ Φ + + \int ψ {Φ, ψ} .

Now use the equations of motion

(6)

Q_{B} Φ + Φ ⋆ Φ = 0

to obtain the advertised result:

(7)

\dots = S_{0} + \int ψ ⋆ \tilde{Q} ψ + \int ψ ⋆ ψ ⋆ ψ,

where $S_{0}$ is the (constant, not to be varied) action of the ‘background field’ $Φ$ and the remaining action is that for $ψ$ with the background described by the deformed BRST operator ${\tilde{Q}}_{B} = \tilde{Q} + {Φ ⋆, \cdot}$ .

Of course the key additional insight of the above paper is that there is a $Φ$ which alone gives the flat space BRST operator, so that

(8)

Q_{B} = {Φ ⋆, \cdot}

where the bracket is the supercommutator with respect to the ghost grading.

This implies that one can obtain standard OSFT from the background free action which consists exclusively of the cubic term:

(9)

S_{backgroundfree} = \frac{2}{3} \int Φ ⋆ Φ ⋆ Φ .

Very nice.

One more maybe interesting observation:

It turns out that the $Φ$ which reproduces the flat space BRST operator and hence the usual kinetic term in the OSFT action represents a background of infinitesimally small strings. The interaction with these tiny background strings can be seen to be equivalent to a kinetic term.

What I find interesting about this is that precisely the same physical mechanism can be seen to be responsible for the proper kinetic term in the derivation of closed string field theory from the IIB Matrix model! I have reviewed the corresponding calculation here where I pointed out (close to the end of that entry) that

in one of these processes a piece of string of vanishing length is split off and produces not another string but a kinematical term

Posted by: Ur s on April 30, 2004 2:03 PM | Permalink | PGP Sig | Reply to this

Re: CFTs from OSFT?

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I should add that this allows to prove that all the gauge symmetries of OSFT do indeed correspond to conjugations of the BRST operator

(1)

Q_{B} \to A^{- 1} Q_{B} A .

Posted by: Urs Schreiber on May 1, 2004 10:21 PM | Permalink | Reply to this

BRST operator as string field anticommutator

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Unfortunately I have very little time currently (me and my girlfriend are moving to another flat and I have to paint walls, etc. :-), but one thing deserves maybe further attention:

Hato shows that the flat space BRST operator $Q_{B}$ (= exterior derivative) can be rewritten as the supercommutator with a certain unit ghost number string field (= 1-form) $Φ_{0}$ which describes a background of infinitesimally small strings.

That’s interesting, because in discrete differential (noncommutative) geometry something similar holds: There we have the gluing 1-form $ρ$ , a special 1-form obtained by adding all edges in the discrete space and for a large class of graphs we have the identity

(1)

d ω = [ρ, ω] .

This is morally pretty much the same as what happens in string field theory. Maybe there is a sense in which one can identify

(2)

Φ_{0} \sim ρ .

That would actually make a long-standing intuitive guess by Eric Forgy quite explicit, since one could perhaps identify the ‘infinitessimally small strings’ background with the set of edges that enter $ρ$ .

That’s of course wildly speculative at this point, but as soon as I find some time I’ll try to figure out if this could make sense…

Posted by: Urs Schreiber on April 30, 2004 9:37 PM | Permalink | PGP Sig | Reply to this

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Hi Eric -

I am once again working on a notebook with low battery, so this here has to be a short comment or none at all :-)

Concerning topological versus geometrical: The important point is that the string field theory action is an action on the space of string backgrounds, i.e. on the space of physically allowed spacetimes, if you wish. It is not an action in a single such spacetime. Theretore it can be topological-like and still say something about geometry.

Next, you are right that $Q_{B}$ is self-adjoint, but not with respect to what would be a Hodge inner product. As I have mentioned before, ghost number of string fields corresponds to form degree. So, as you deduced correctly, the physical string filed corresponds to a 1-form. The star product corresponds to the wedge product. Hatas $A \cdot B$ operation is, as you also deduced correctly, star/wedge product followed by integation. So in this sense $Q_{B}$ is ‘self-adjoint’.

Concerning the apparent 3-‘dimensionality’ (in a vague sense) of the OSFT action, its indeed not obvious how to interpret it (maybe it has to do with the three generators $L_{\pm 1}, L_{0}$ ?). But note that it has nothing to do with the number of spacetime dimensions. The OSFT action is an action on the space of spacetimes, roughly. Of course that space is infinite dimensional. The 3-dimensionality inherited from the Chern-Simons structure must have some other interpretation, but I am not sure which one.

I am not sure why you worry that $A \cdot B$ might depend on any background metric. You correctly find that $A \cdot B = \int A \land B$ . No metric (on the space of backgrounds!) anywhere.

And, yes, you are right that Hata talks about closed strings while I had mentioned open string field theory, mostly. But I don’t think that’s crucial for the key points of Hata’s paper.

As soon as I find the time I’ll write up something about all this.

P.S. Tuesday-Wednesday I’ll be at University of Hambug giving a talk about our work on discrete differential geometry. With a bit of luck I can make the connection to Hata’s BRST operator = anticommutator with a 1-form precise until Tuesday. Too bad that I have to spend so much time working on our new flat currently…

Posted by: Urs Schreiber on May 1, 2004 10:14 PM | Permalink | PGP Sig | Reply to this

Re:

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Good afternoon! :)

While you are busy painting, I’m training for my next marathon. I just ran about 14-15 miles around the Charles River. It’s a beautiful day (which is rare in New England) :)

Concerning topological versus geometrical: The important point is that the string field theory action is an action on the space of string backgrounds, i.e. on the space of physically allowed spacetimes, if you wish. It is not an action in a single such spacetime. Theretore it can be topological-like and still say something about geometry.

Ok. If you have a space of spacetime geometries, I can believe that ANY theory you define on such a space may have something to say about geometry. The big question in my mind is, “Does it?” :)

It is still not obvious that you would want a topological field theory of spacetime geometries instead of a geometrical field theory of spacetime geometries. To ask the question Feynman warns us never to ask, why should nature be like that? :)

The 3-dimensionality inherited from the Chern-Simons structure must have some other interpretation, but I am not sure which one.

This seems like it is worth thinking about. Being the chronic skeptic I am, I wonder if there even is a good answer to this question :) Since I am such a lover of Maxwell’s equations, wouldn’t it be poetic if the correct theory was four-“dimensional” and of the form

(1)

S = \frac{1}{2} [Q_{B} Φ, Q_{B} Φ],

where $[,]$ is is some kind of Hodge inner product? :)

I am not sure why you worry that $A \cdot B$ might depend on any background metric. You correctly find that $A \cdot B = \int A \land B$ . No metric (on the space of backgrounds!) anywhere.

I wasn’t really worried about it. Hatas was :) He says on page 2:

However, the action (1) has an unpleasant feature that $Q_{B}$ in the kinetic term $Φ \cdot Q_{B} Φ$ depends explicitly on the flat (d = 26) space-time metric $η_{μ ν} = diag (-, +, . . ., +)$ , and the separation of the action into the kinetic and interaction terms is necessarily perturbation theoretical.

Oops! When I look at this statement now it seems he is saying that $Q_{B}$ depends on the metric not $\cdot$ . In haste, I must have assumed he meant $\cdot$ depends on the metric (which actually would make more sense). Looking at Equation (4), it seems that it could be $Q_{B}$ that depends on the metric. I’m guessing my confusion is arising because there are apparently two notions of “metric” here, which I wasn’t aware of before. There is a metric of the spacetime geometry and there is a metric on the space of spacetime geometries. Apparently neither $Q_{B}$ nor $\cdot$ depend on the metric of the space of spacetimes.

I don’t think I like the basic idea of this whole theory we are talking about. Do you know why? :) It doesn’t seem to have a very natural representation in the discrete theory. Since I am biased, the only possible conclusion is that it must be wrong (just kidding) :)

Another small point, since it seems the space we are dealing with is supposed to be the space of spacetimes, which is infinite dimensional, it might seem weird trying to talk about Hodge star. This would send a $p$ -form to an $(∞ - p)$ -form :) But really, the theory seems to be more like a 3 dimensional theory if you interpret it as the degree of the form that makes up the action. In this way, even on this space it seems we can define a map from $p$ -forms to $(3 - p)$ -forms that has the right to be interpretted as a kind of Hodge star.

Gotta run! (not literally this time) :)

Ciao,
Eric

Posted by: Eric on May 2, 2004 12:10 AM | Permalink | Reply to this

background of backgrounds

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It’s a beautiful day (which is rare in New England)

Sounds like a line by John Irving. ;-)

Ok. If you have a space of spacetime geometries, I can believe that ANY theory you define on such a space may have something to say about geometry. The big question in my mind is, ‘Does it?’ :)

Well, it surely describes some spacetime(s). Whether it describes ours is the big question.

Hata mentions that you get Einstein’s equations from closed string field theory. If you want to see in more detail how this works, take a look maybe at Ohmori’s review pp. 29, which however treats the open string and hence derives Yang-Mills + additional stuff.

It is still not obvious that you would want a topological field theory of spacetime geometries instead of a geometrical field theory of spacetime geometries. To ask the question Feynman warns us never to ask, why should nature be like that? :)

Oh, that’s surprisingly simple to answer: There is no other choice!

We want a second quantized theory of string, i.e. an action whose classical equations of motion are the quantum equations of motion of a single string. Since the single free string has the quantum equation of motion

(1)

Q_{B} Φ = 0

the free part of the classical action must necessarily be something like

(2)

\int Φ ⋆ Q_{B} Φ,

where $⋆$ is some product with respect to which $Q_{B}$ is graded Leibniz.

Next you need to add some interaction term, something like

(3)

\int Φ ⋆ Φ ⋆ \dots ⋆ Φ .

But only the cubic term makes sense, as is discussed on p. 19 of Ohmori. There are many ways to see this, heuristically. For instance it is clear that the single type of interaction between open strings is the trivalent graph, where two strings merge to become a single one or one string splits in two, alternatively. All other diagrams are built from this simplest one.

So then the only thing that remains to be fixed is the relative factor between $\int Φ ⋆ Q_{B} Φ$ and $\int Φ ⋆ Φ ⋆ Φ$ . But because you can always shift this factor by rescaling $A$ that’s not too important and you can include the open string coupling constant or absorb it in the string field as desired.

I wasn’t really worried about it. Hatas was

Ok, sorry. We need to distinguish between the two levels of ‘geometry’ and ‘background’ here. Hata is worried about the fact that the ordinary string field action explicitly includes properties of one of its (classical) solutions, namely the vacuum Minkowski space corresponding to $Φ = 0$ . That’s because the BRST operator enetering the kinetic term is that describing strings in Minkowski space.

So this singles out a special background of spacetime. But this does not mean that there is a metric in the space of string fields $Φ$ , so that the string field action is still topological-like as a theory on the space of spacetimes.

BTW, I am not sure that Hata is correct that by showing that the kinetic term can actually be subsumed in the cubic term the theory no longer makes explicit reference to the Minkowski solution. Seems to me that the definition of the star product still involves correlators/commutators that belong to strings in flat space. But that might be a matter of interpretation.

Posted by: Urs Schreiber on May 3, 2004 10:33 AM | Permalink | PGP Sig | Reply to this

Re: background of backgrounds

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Good morning :)

Oh, that’s surprisingly simple to answer: There is no other choice!

Should I take this as a challenge? :)

If you take $Φ$ to be a 1-form, then the equation of motion

(1)

Q_{B} Φ = 0

has more than one interpretation. For instance, this could be the equation of motion for a “scalar” wave equation if $Q_{B}$ is taken as the adjoint of some other operator $Q_{B}^{†}$ . For the time being, let me make a replacement

(2)

Q_{B} \to d^{†}

and

(3)

Q_{B}^{†} \to d .

Then your equation of motion looks like

(4)

d^{†} Φ = 0 .

Now we could define some “0-form” potential $A$ with

(5)

Φ = dA

so that

(6)

d Φ = 0 .

Then we could have the action

(7)

S = \frac{1}{2} \int_{M} Φ ⋆_{Hodge} Φ

giving the equation of motion

(8)

d^{†} Φ = 0

as desired :)

The point is, there seems to be many ways to obtain a desired equation of motion.

The interpretation of $Q_{B}$ as being adjoint to some other operator seems to make sense. I realize that there are many different notions of “geometry” here, but the fact that $Q_{B}$ depends on any one of these sorts of geometries suggests to me that it is not really the purely topological operator that an analogue of $d$ should be. $Q_{B}$ seems more suited to be an analogue of $d^{†}$ .

Eric

Posted by: Eric on May 3, 2004 3:01 PM | Permalink | Reply to this

Re: background of backgrounds

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I see what you mean. I now pull a standard trick: Instead of admitting that my claim was wrong, I change the rules of the game! ;-)

Seriously, I think there is one more constraint on the equations of motion that we are looking for which, unfortunately, rules out your idea in this particular case: The point is that the equations of motion of the single string are

(1)

Q_{B} Φ = 0

as before, but that furthermore the grading (‘ghost number’) is such that both $Q_{B}$ as well as admissable $Φ$ are of grade $+ 1$ . In your example we’d have $Q_{B}$ be an operator which loweres the grade. But it must raise it!

BTW, we had found that the action of lattice YM theory in discrete Dimakis/Müller-Hoissen-like formalism (plus inner product) has the nice form

(2)

S_{YM} = 〈 H^{2} ∣ H^{2} 〉,

where $H$ is the holonomy 1-form and $〈 \cdot ∣ \cdot 〉$ the inner product.

One might wonder what similarly is the lattice version of Chern-Simons theory. When this thought first occured to me I guessed that it would be $\int H^{3}$ . But apparently it is instead

(3)

S_{CS} = \frac{1}{ϵ^{3}} \int (H - d) H^{2} .

I believe that the term subtracted this way would actually diverge in the continuum limit $ϵ \to 0$ , so that this might be good enough reason not to include it.

The nice thing about the discrete version is that gauge invarince is manifest, which is not exactly true for the usual version of this action. Using

(4)

\frac{1}{ϵ^{2}} H^{2} = dA + AA + 𝒪 (ϵ)

it is easy to see that we have

(5)

\dots = \int A dA + \int AAA + order ϵ^{2} .

(A relative factor between the two terms can always be introduced by rescaling $A$ )

It would be nice to use the fact that $d ω = [ρ, ω]$ in order to pull a trick similar to Hata’s background free SFT to get a ‘background free’ CS theory in the sense that there is no gauge covariant derivative with respect to any fixed gauge field. (Namely the analogue of Hata’s string field argument would be to argue that $d$ is the gauge covariant derivative of the trivial connection and that we do not want to have this in the action!)

So one might look at the lattice action

(6)

S_{?} = \int AAA,

but due to lattice effects its equations of motion are not quite $AA = 0$ (as true, in the analogous sense, in Hata’s paper for the string field of the ‘pre-geometric’ action ).

Seems to me that the essence of what Hata is doing is actually the following:

There is a CS theory in the ‘continuum’ but for an immense gauge group. Namely that generated by ghost number 0-fields $Λ$ under $⋆$ -commutation. (BTW, what is this gauge group? Is it maybe $E_{10}$ ?)

Hata essentially argues that to this ‘group’ there is a connection $A$ such that

(7)

[A, ω] = d ω .

- From this point of view this claim appears rather strange. There must be some subtlety at work to make it true.

Posted by: Urs Schreiber on May 3, 2004 3:43 PM | Permalink | PGP Sig | Reply to this

Re: background of backgrounds

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Sorry, there was a typo: I meant

(1)

S_{CS} = \int (H - ρ) H^{2},

where $ρ$ is the gluing 1-form.

Posted by: Urs Schreiber on May 3, 2004 3:47 PM | Permalink | PGP Sig | Reply to this

Re: background of backgrounds

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Just curious, isn’t

(1)

A = H - ρ

so that what you write for the CS action can also be written as

(2)

S_{CS} = \int_{M} A H^{2}

This seems to be correct. Just checking :)

Then my last suggestion would actually be

(3)

S = \int_{M} A^{3}

which is Hata-like :)

Eric

Posted by: Eric on May 3, 2004 4:24 PM | Permalink | Reply to this

Re: background of backgrounds

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Yes, the holonomy 1-form starts with

(1)

H = ρ + ϵ A + order ϵ^{2}

by definition of holonomy along an edge.

So, $H - ρ = ϵ A + order ϵ^{2}$ .

I wanted to avoid $\int A^{3}$ since it is not gauge invariant in the discrete theory.

Posted by: Urs Schreiber on May 3, 2004 4:32 PM | Permalink | PGP Sig | Reply to this

Re: background of backgrounds

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Here is a crazy thought (imagine that) :)

Begin with

(1)

S = \int_{M} A^{3}

then deform $A$ via $ρ$ so that you have

(2)

S' = \int_{M} (A + ρ)^{3} = \int_{M} (A + ρ) H^{2} = S_{CS} + \int_{M} ρ H^{2} .

What is that last term? :)

Eric

Posted by: Eric on May 3, 2004 4:33 PM | Permalink | Reply to this

Re: background of backgrounds

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Exactly, that was my original idea! I think that the last term diverges (when you divide everything by $1 / ϵ^{3}$ , as it should be).

Posted by: Urs Schreiber on May 3, 2004 4:38 PM | Permalink | PGP Sig | Reply to this

Re: background of backgrounds

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It is too bad that you cannot deform 2 of the three A’s while leaving the third unchanged :)

Then

(1)

\int_{M} A^{3} \to \int_{M} A (A + ρ)^{2} = \int_{M} A H^{2} = S_{CS}

I wonder if we can use the trick of introducing a second variable, something like

(2)

S = \int_{M} B A^{2} - \frac{1}{2} \int_{M} B^{2} A

or something :) Varying $B$ gives $A = B$ , which when plugged back in gives $A^{3}$ , but then we might be able vary $A$ and $B$ separately. (note: don’t bother checking my algebra. I know that statement is probably false, but I’m just fishing for ideas :)) Then the kinetic term could arise as an off-shell effect :)

Welcome to our speculation playground :)

Eric

Posted by: Eric on May 3, 2004 4:46 PM | Permalink | Reply to this

Re: background of backgrounds

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Neat! You are looking at discrete CS. I wish I had more time to play with this! :)

The most I can do right now is throw out wild thoughts (as usual) and see if anything sticks :)

I have some catching up to do to follow why you get

(1)

S = \int_{M} (H - ρ) H^{2},

but just by looking at this, I can ask, why not

(2)

S = \int_{M} (H - ρ)^{3}

Go Urs! :)

I can be the cheering squad :)

Eric

Posted by: Eric on May 3, 2004 4:13 PM | Permalink | Reply to this

Re: CFTs from OSFT?

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Hello!

I haven’t even read it yet (I am a little overloaded with other things at the moment), but I just came across this paper, which seems to be doing something similar to what I have been asking for. It constructs a C*-algebra in SFT.

Toward the construction of a $C^{*}$ algebra in string field theory
A. Parodi

In string field theory there is a fundamental object, the algebra of string field states $𝒜$ , that must be understood better from a mathematical point of view. In particular we are interested in finding, if possible, a $C^{*}$ structure over it, or possibly over a subalgebra $U \subset 𝒜$ . In this paper we define a * operation on $𝒜$ , and then using a particular description of Witten’s star product, the Moyal’s star product, we find an appropriate pre $C^{*}$ algebra $S (R^{2 n})$ on a finite dimensional manifold, where the finite dimensionality is obtained with a cutoff procedure on the string oscillators number. Then we show that using an inductive limit we obtain a pre $C^{*}$ algebra $S \subset 𝒜$ that can be completed to a $C^{*}$ algebra.

Eric

Posted by: Eric on May 5, 2004 10:01 PM | Permalink | Reply to this

Re: CFTs from OSFT?

Hmm… the *-operation is reversing orientation. Sounds familiar :)

Posted by: Eric on May 5, 2004 11:52 PM | Permalink | Reply to this

Re: CFTs from OSFT?

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In Parodi’s paper, it seems he has defined a *-operation, which I will instead denote by $†$ to make the connection to our notes more clear (not to mention that * is already being used for Moyal *), that satisfies

(1)

(A * B)^{†} = B^{†} * A^{†} .

The physical interpretation of $†$ seems simple enough. It is merely reversal of orientation.

What we learned from our notes is that if you begin with an abstract differential calculus, which it seems that $Q$ and $*$ provide for string fields, then all you need is an adjoint $A^{†}$ and we are off and running for a kind of abstract differential geometry. In other words, it seems we can define an inner product of string fields $A$ , $B$ , via

(2)

〈 A ∣ B 〉 = \frac{1}{2} \int (A^{†} * B + B^{†} * A) dV

for some positive 0-form $dV$ .

In Witten’s ‘86 paper on NCG and SFT, he says,

As there is no analogue of “raising and lowering indices” within these axioms, there is no analogue of the usual Yang-Mills action $F^{μ ν} F_{μ ν}$ .

However, this statement seems circumventible by introducing the above inner product so that the analogous Yang-Mills-like action for string fields would be

(3)

S = \frac{1}{2} 〈 F ∣ F 〉 = \frac{1}{2} \int F^{†} F dV,

where

(4)

F = Q (A) + A * A .

Introducing a $†$ operation, defining an inner product, and using this inner product to construct an action for some version of string field theory seems like a perfectly natural thing to do to me. Is there any obvious reason not to pursue this?

Eric

Posted by: Eric on May 11, 2004 5:48 AM | Permalink | Reply to this

Re: CFTs from OSFT?

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You are perfectly right:

We could introduce an inner product which would relate to the BRST operator as the Hodge inner product relates to the exterior derivative. Taking adjoints wrt to this inner product would essentially carry ghosts (form creators) $c_{n}$ to anti-ghosts (form annihilators) $b_{n}$ and leave the physical degrees of freedom alone (up to a sign, maybe). Taking the adjoint of the BRST operator $Q$ wrt to this inner product would yield $Q^{†}$ , which is known as the co-BRST operator.

I am currently short of time, but if you like you should be able to find some papers on BRST/coBRST formalism on the web. This formalism has been applied with success to several systems, even though I have never seen it in the context of worldsheet theory of strings. The basic idea is that a coBRST operator allows to fix a gauge

(1)

Q^{†} ∣ ψ 〉 = 0

analogous to picking the harmonic form component of a general closed form on a compact Riemannian manifold.

If you look for papers by a swedish physicist called Marnelius you should be able to find something on this.

It is very important, though, to realize that we are dealing with two different notion of inner product and generalized exterior derivatives here:

The BRST operator for superstrings is kind of a meta-exterior derivative, because it contains the worldsheet supercurrents. The OSFT inner product with respect to which $Q$ is ‘graded self-adjoint’ is actually the correct Hodge-like inner product wrt to these supercurrents, which are essentially exterior(co)derivatives on loop space.

But your question really is: Can we construct a sensible SFT action using the Hodge-like inner product for $Q$ ?

I don’t know! Leaving interactions aside, we need to come up with the equations of motion $Q ∣ ψ 〉 = 0$ , the theory must be solved by a closed 1-form, which however must not be exact, in general. This condition excludes the Maxwell-like action

(2)

〈 Q Λ ∣ Q Λ 〉

which might come to mind.

However, maybe one could in principle use the action

(3)

〈 Q^{†} κ_{2} ∣ Q^{†} κ_{2} 〉

where $κ_{2}$ is a 2-form. The equation of motion would be

(4)

Q Q^{†} ∣ κ_{2} 〉

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