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 which is defined on [a given] BCFT, we obtain after suitable redefinition of the fluctuation modes the SFT action defined on 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 , written using two different 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 , but apparently only naively so. To me it would be intreresting if similar results could be obtained for more general classical solutions .
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.
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 for a given background, like for instance flat Minkowski space, one may consider the operator
(1)
where is some operator such that nilpotency is preserved.
By appropriately commuting with the ghost modes the conformal generators 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 , 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 are precisely the operators of left plus right star-multiplication by the string field 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 of ghost number one:
(2)
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 , the operator of star-multiplication by defined by
(3)
then, due to the associativity of the star product this can equivalently be rewritten as an operator equation
(4)
because
(5)
(Here it has been used that is an odd graded (with respect to ghost number) derivation on the star-product algebra of string fields, that is of ghost number 1 and that the star-product is associative.)
It hence follows that the equations of motion of the string field are precisely the necessary and sufficient condition for the operator to yield a nilpotent, unit ghost number deformation
(6)
of the original BRST operator.
But there remains the question why , 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 is a solution of the classical equations of motion derived from the action , then it is possible to construct an operator in terms of , acting on a subspace of the Hilbert space of combined matter-ghost CFT, such that . 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 .
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 by adding a tiny correction field to , which we want to interpret as the string field due to the single test string.
The question then is: What is the condition on so that the total field is still a solution to the equations of motion of string field theory. That is, given , one needs to solve
(7)
for . But since is supposed to be just a tiny perturbation of the filed it must be sufficient to work to first order in . This is equivalent to neglecting any self-interaction of the string described by and only considering its interaction with the ‘background’ field - just as in the first quantized theory of single strings.
But to first order and using the fact that is supposed to be a solution all by itself the above equation says that
(8)
This is manifestly a deformation of the equation of motion
(9)
of the string described by the state in the original background. Hence it is consistent to interpret
(10)
as the new worldsheet BRST operator which corresponds to the new background described by .
If we again switch to operator notation the above can equivalently be rewritten as
(11)
where the braces denote the anticommutator, as usual.
Recalling that a gauge transformation in string field theory is (for a string field of ghost number 0) of the form
(12)
and that in operator language this reads equivalenty
(13)
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 plays the role of the exterior derivative, the ghost correspond to differential form creators, the -ghosts to form annihilators and the product to the ordinary 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
(14)
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 to a gauge covariant exterior derivative suggests that we need to interpret not simply as a 1-form, but as a - connection!
That is, 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)
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)
But precisely this is what does happen in open string field theory in the formal integral. There we have
(17)
All this suggests that one should think of the deformed BRST operator as morally a gauge covariant exterior derivative:
(18)
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 to string field theory and computing the explicit realization of 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?
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 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 as a similarity transform of ? For example, a first guess would be
(1)
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 via a deformation of the inner product and leave 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
Re: CFTs from OSFT?
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 .
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)
you assume that is a classical solution and perturb about it by substituting
(2)
that then one obtains string field theory in terms of but for the modified BRST operator
(3)
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)
(5)
Now use the equations of motion
(6)
to obtain the advertised result:
(7)
where is the (constant, not to be varied) action of the ‘background field’ and the remaining action is that for with the background described by the deformed BRST operator .
Of course the key additional insight of the above paper is that there is a which alone gives the flat space BRST operator, so that
(8)
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)
Very nice.
One more maybe interesting observation:
It turns out that the 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
BRST operator as string field anticommutator
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 (= exterior derivative) can be rewritten as the supercommutator with a certain unit ghost number string field (= 1-form) 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 , 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)
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)
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 .
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…
Re: CFTs from OSFT?
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 algebra in string field theory
A. Parodi
In string field theory there is a fundamental object, the algebra of string field states , that must be understood better from a mathematical point of view. In particular we are interested in finding, if possible, a structure over it, or possibly over a subalgebra . In this paper we define a * operation on , and then using a particular description of Witten’s star product, the Moyal’s star product, we find an appropriate pre algebra 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 algebra that can be completed to a algebra.
Eric
Re: CFTs from OSFT?
Hello,
I am presently trying to teach myself a little bit about the basics of BRST quantization. I am hoping to better understand some of the issues we’ve been discussing here. Up to this point, I didn’t really care what was. All I needed to know was that it is a nilpotent derivation and that is already enough to say a lot about it :)
In Hata’s paper, he has a “topological” inner product which is analogous to
(1)
so that the exterior derivative is self-adjoint (up to a constant).
Now I am reading (skimming more like it) (among other things)
Quantum BRST properties of reparametrization invariant theories
Robert Marnelius, Niclas Sandstrom
and it says
In the present paper we shall make use of a precise operator formulation of BRST
quantization on inner product spaces based on the BFV scheme. It allows for a detailed treatment of the quantum theory and provides for algorithms for the physical states. Any operator BRST quantization requires [10]
(2)
where is any physical state, and where is the odd, hermitian BRST charge operator which lives in a ghost extended framework.
I wonder if it is a general property of the BRST charge operator that it needs to be Hermitian if there is an inner product laying around. In Hata’s paper, is self-adjoint with respect to the topological inner product. This is one corroborating data point :)
So now I am wondering if the YM-like SFT theory I was contemplating can be made to fit into this framework. If we begin with the abstract differential calculus and define an adjoint such that
(3)
then we can define an inner product
(4)
using . So the question is, “Is Q Hermitian?” I think it will be. Since simply reverses the orientation of the string, then is just a string so we should still have
(5)
It seems like will be Hermitian (up to a constant) as long as it satisfies
(6)
because then we’d have for
(7)
so that
(8)
Is this possible? Perhaps we can ask Witten to make an appearance :)
Cheers,
Eric
Re: CFTs from OSFT?
Hello!
I said
These orthogonal projectors seem to be just what the doctor ordered! :)
to which you replied
Er, sorry, which orthogonal projectors are you referring to, precisely? Do you mean the , , fields on pp. 54 of hep-th/0111208?
Hm, you want to think of these as delta-function somehow. Interesting. Have you seen anywhere any statements as to the completeness of such projectors?
I was referring to what YOU referred ME to in a paper I referred YOU to!! :) I can tell you are over worked. I know how you feel ;)
Equation (4.13) of Kawano & Okuyama’s paper states
(1)
This expression troubles me a little bit because it seems to suggest (maybe I’m wrong) that the algebra of “0-forms” is commutative, but I thought * was not commutative even for “0-forms”.
It might be interesting to look at the commutative part (the center?) of .
Eric
Re: CFTs from OSFT?
One more thought, then I’ll really sleep :)
It seems that 0-forms can be expressed in right- or left-component forms
(1)
where
(2)
and
(3)
Due to noncommutativity, in general we have
(4)
Hmm… but then we have
(5)
for some , then
(6)
In other words (I know this is simple algebra :))
(7)
That is kind of neat :)
With hind sight, I can write down the obvious relation
(8)
so that
(9)
:)
Then if
(10)
we have
(11)
which makes the relation to matrix multiplication pretty obvious. But then this suggests that multiplication on the left by projects the left half of the string and multiplication by on the right projects the right half of the string.
This is probably just rewriting what is already well known, but I like this presentation. It is very transparent :)
Good night! :)
Eric
Minimal Principle and Deformations
Good morning! :)
I hope you had a good weekend. I survived my third marathon and I’m somehow not feeling too much pain this morning :)
As the real world started approaching after some time off, I obviously wanted to turn my thoughts back toward physics to distract myself from reality :)
Recently, I hit on something I think is pretty neat but might be trivially obvious to you. This is the relation between deformations and the minimal principle.
You have managed to convince me that replacing
(1)
is equivalent to a deformation of the background.
However, this looks eerily like the replacement
(2)
to account for interactions.
Now I’m wondering if there is some relation between deformations and the minimal principle.
If there is anything to this idea, then the converse should apply. Is there any sense in which you can think of the replacement
(3)
as a deformation of the background of the QED action
(4)
where ?
Eric
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 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 as a similarity transform of ? 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 via a deformation of the inner product and leave 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