April 29, 2004

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)

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.

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)${\stackrel{˜}{Q}}_{B}:={Q}_{B}+\stackrel{̂}{\Phi }\phantom{\rule{thinmathspace}{0ex}},$

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

By appropriately commuting ${\stackrel{˜}{Q}}_{B}$ with the ghost modes the conformal generators ${\stackrel{˜}{L}}_{m}^{\mathrm{tot}}$ 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 $\stackrel{̂}{\Phi }$, 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 $\stackrel{̂}{\Phi }$ 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:

(2)${Q}_{B}\mid \Phi 〉+\mid \Phi \star \Phi 〉=0\phantom{\rule{thinmathspace}{0ex}},$

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 $\stackrel{̂}{\Phi }$, the operator of star-multiplication by $\Phi$ defined by

(3)$\stackrel{̂}{\Phi }\mid \Psi 〉:=\mid \Phi \star \Psi 〉$

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

(4)${\left({Q}_{B}+\stackrel{̂}{\Phi }\right)}^{2}=0$

because

(5)$\left({Q}_{B}+\stackrel{̂}{\Phi }\right)\circ \left({Q}_{B}+\stackrel{̂}{\Phi }\right)\circ ={\underbrace{{Q}_{B}\circ {Q}_{B}\circ }}_{=0}+{\underbrace{{Q}_{B}\circ \Phi \star +\Phi \star {Q}_{B}\circ }}_{=\left({Q}_{B}\Phi \right)\star }+{\underbrace{\Phi \star \Phi \star }}_{=\left(\Phi \star \Phi \right)\star }\phantom{\rule{thinmathspace}{0ex}}.$

(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 $\stackrel{̂}{\Phi }$ to yield a nilpotent, unit ghost number deformation

(6)${\stackrel{˜}{Q}}_{B}={Q}_{B}+\stackrel{̂}{\Phi }$

of the original BRST operator.

But there remains the question why ${\stackrel{˜}{Q}}_{B}$, 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

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 $S\left(\Psi \right)$, then it is possible to construct an operator ${\stackrel{̂}{Q}}_{B}$ in terms of ${\Psi }_{\mathrm{cl}}$, acting on a subspace of the Hilbert space of combined matter-ghost CFT, such that $\left({\stackrel{̂}{Q}}_{B}{\right)}^{2}=0$. ${\stackrel{̂}{Q}}_{B}$ 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

(7)${Q}_{B}\left(\Phi +\psi \right)+\left(\Phi +\psi \right)\star \left(\Phi +\psi \right)=0$

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

(8)${Q}_{B}\mid \psi 〉+\mid \Phi \star \psi 〉+\mid \psi \star \Phi 〉=0\phantom{\rule{thinmathspace}{0ex}}.$

This is manifestly a deformation of the equation of motion

(9)${Q}_{B}\mid \psi 〉=0$

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

(10)${\stackrel{˜}{Q}}_{B}={Q}_{B}+\left\{\stackrel{̂}{\Phi },\cdot \right\}$

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

(11)$\left\{\left({Q}_{B}+\stackrel{̂}{\Phi }\right),\stackrel{̂}{\psi }\right\}=0\phantom{\rule{thinmathspace}{0ex}},$

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

(12)$\delta \Phi ={Q}_{B}\Lambda +\Phi \star \Lambda -\Lambda \star \Phi$

and that in operator language this reads equivalenty

(13)$\stackrel{̂}{\delta \Phi }=\left[\left({Q}_{B}+\stackrel{̂}{\Phi }\right),\stackrel{̂}{\Lambda }\right]=0$

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 $\star$ 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

(14)$\Phi \star \psi \ne ±\Psi \star \Phi$

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 ${\stackrel{˜}{Q}}_{B}={Q}_{B}+\stackrel{̂}{\Phi }$ 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

(15)$\left(d+\omega {\right)}^{2}=\left(d\omega \right){\delta }^{a}{}_{b}+{\omega }^{a}{}_{c}\wedge {\omega }^{c}{}_{b}\phantom{\rule{thinmathspace}{0ex}}.$

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)$\int \mathrm{tr}{\omega }^{a}{}_{c}\wedge {\gamma }^{c}{}_{b}=±\int \mathrm{tr}{\gamma }^{a}{}_{c}\wedge {\omega }^{c}{}_{b}\phantom{\rule{thinmathspace}{0ex}}.$

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

(17)$\int \Phi \star \Psi =±\int \Psi \star \Phi \phantom{\rule{thinmathspace}{0ex}}.$

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

(18)${\stackrel{˜}{Q}}_{B}={Q}_{B}+\stackrel{̂}{\Phi }\sim d+\omega \phantom{\rule{thinmathspace}{0ex}}.$

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 ${\stackrel{˜}{Q}}_{B}$ 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? 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 ${\stackrel{˜}{Q}}_{B}$ 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 ${\stackrel{˜}{Q}}_{B}$ as a similarity transform of ${Q}_{B}$? For example, a first guess would be

(1)${\stackrel{˜}{Q}}_{B}={e}^{f\left(\stackrel{̂}{\Phi }\right)}{Q}_{B}{e}^{-f\left(\stackrel{̂}{\Phi }\right)},$

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 ${\stackrel{˜}{Q}}_{B}$ 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?

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 ${d}_{A}=d+\omega$ 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)$\omega \to {g}^{-1}\omega g+{g}^{-1}\left(dg\right)\phantom{\rule{thinmathspace}{0ex}}.$

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?

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)$\left({Q}_{B}\Phi \right)\cdot \Psi =\left(-1{\right)}^{1+\mid \Phi \mid }\Phi \cdot {Q}_{B}\Psi .$

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)${Q}_{B}\stackrel{?}{=}d±{d}^{†}$

?? 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 $\Phi$, $\Psi$ be like 0-forms or 1-forms? For example, the action in Equation (1)

(3)$S=\Phi \cdot {Q}_{B}\Phi +\frac{2}{3}g{\Phi }^{3}$

looks like

(4)${S}_{\mathrm{CS}}={\int }_{M}\left(A\wedge \mathrm{dA}+\frac{2}{3}gA\wedge A\wedge A\right)$

which makes me think that $\Phi$ 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)$A\cdot B={\int }_{M}A\wedge B,$

but really more like a Hermitian version of this

(6)$A\cdot B={\int }_{M}\stackrel{˜}{A}\wedge B,$

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

(7)$A\cdot B\stackrel{?}{=}\left(-1{\right)}^{\mid A\mid \mid B\mid }\stackrel{˜}{B\cdot A}.$

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

(8)$d\left(A\wedge B\right)=\left(\mathrm{dA}\right)\wedge B+\left(-1{\right)}^{\mid A\mid }A\wedge \mathrm{dB}$

so that

(9)${\int }_{M}d\left(A\wedge B\right)=\left(\mathrm{dA}\right)\cdot B+\left(-1{\right)}^{\mid A\mid }A\cdot \mathrm{dB}.$

Then if

(10)${\int }_{M}d\left(A\wedge B\right)={\int }_{\partial M}A\wedge B=0,$

then

(11)$\left(\mathrm{dA}\right)\cdot B=\left(-1{\right)}^{1+\mid A\mid }A\cdot \mathrm{dB},$

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? 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)$S=\int \Phi \star {Q}_{B}\varphi +\frac{2}{3}\int \Phi \star \Phi \star \Phi$

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

(2)$\Phi \to \Phi +\psi$

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

(3)${\stackrel{˜}{Q}}_{B}={Q}_{B}+\left\{\Phi \star ,\cdot \right\}\phantom{\rule{thinmathspace}{0ex}}.$

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)${S}_{0}\to \int \Phi \star {Q}_{B}\Phi +2\int \psi \star {Q}_{B}\Phi +\int \psi \star {Q}_{B}\psi +\frac{2}{3}\int \Phi \star \Phi \star \Phi +$
(5)$\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}+2\int \psi \star \Phi \star \Phi ++\int \psi \left\{\Phi ,\psi \right\}\phantom{\rule{thinmathspace}{0ex}}.$

Now use the equations of motion

(6)${Q}_{B}\Phi +\Phi \star \Phi =0$

to obtain the advertised result:

(7)$\cdots ={S}_{0}+\int \psi \star \stackrel{˜}{Q}\psi +\int \psi \star \psi \star \psi \phantom{\rule{thinmathspace}{0ex}},$

where ${S}_{0}$ is the (constant, not to be varied) action of the ‘background field’ $\Phi$ and the remaining action is that for $\psi$ with the background described by the deformed BRST operator ${\stackrel{˜}{Q}}_{B}=\stackrel{˜}{Q}+\left\{\Phi \star ,\cdot \right\}$.

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

(8)${Q}_{B}=\left\{\Phi \star ,\cdot \right\}$

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)${S}_{\mathrm{backgroundfree}}=\frac{2}{3}\int \Phi \star \Phi \star \Phi \phantom{\rule{thinmathspace}{0ex}}.$

Very nice.

One more maybe interesting observation:

It turns out that the $\Phi$ 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?

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)${Q}_{B}\to {A}^{-1}{Q}_{B}A\phantom{\rule{thinmathspace}{0ex}}.$
Posted by: Urs Schreiber on May 1, 2004 10:21 PM | Permalink | Reply to this

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 ${Q}_{B}$ (= exterior derivative) can be rewritten as the supercommutator with a certain unit ghost number string field (= 1-form) ${\Phi }_{0}$ 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 $\rho$, 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)$d\omega =\left[\rho ,\omega \right]\phantom{\rule{thinmathspace}{0ex}}.$

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)${\Phi }_{0}\sim \rho \phantom{\rule{thinmathspace}{0ex}}.$

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 $\rho$.

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

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 ${L}_{±1},{L}_{0}$?). 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 $A\cdot B=\int A\wedge B$. 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:

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)$S=\frac{1}{2}\left[{Q}_{B}\Phi ,{Q}_{B}\Phi \right],$

where $\left[,\right]$ 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 $A\cdot B=\int A\wedge B$. 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 $\Phi \cdot {Q}_{B}\Phi$ depends explicitly on the flat (d = 26) space-time metric ${\eta }_{\mu \nu }=\mathrm{diag}\left(-,+,...,+\right)$, 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 $\left(\infty -p\right)$-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 $\left(3-p\right)$-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

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)${Q}_{B}\Phi =0$

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

(2)$\int \Phi \star {Q}_{B}\Phi \phantom{\rule{thinmathspace}{0ex}},$

where $\star$ is some product with respect to which ${Q}_{B}$ is graded Leibniz.

Next you need to add some interaction term, something like

(3)$\int \Phi \star \Phi \star \cdots \star \Phi \phantom{\rule{thinmathspace}{0ex}}.$

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 $\int \Phi \star {Q}_{B}\Phi$ and $\int \Phi \star \Phi \star \Phi$. 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 $\Phi =0$. 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 $\Phi$, 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

Good morning :)

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

Should I take this as a challenge? :)

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

(1)${Q}_{B}\Phi =0$

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)${d}^{†}\Phi =0.$

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

(5)$\Phi =\mathrm{dA}$

so that

(6)$d\Phi =0.$

Then we could have the action

(7)$S=\frac{1}{2}{\int }_{M}\Phi {\star }_{\mathrm{Hodge}}\Phi$

giving the equation of motion

(8)${d}^{†}\Phi =0$

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

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)${Q}_{B}\Phi =0$

as before, but that furthermore the grading (‘ghost number’) is such that both ${Q}_{B}$ as well as admissable $\Phi$ 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)${S}_{\mathrm{YM}}=〈{H}^{2}\mid {H}^{2}〉\phantom{\rule{thinmathspace}{0ex}},$

where $H$ is the holonomy 1-form and $〈\cdot \mid \cdot 〉$ 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)${S}_{\mathrm{CS}}=\frac{1}{{ϵ}^{3}}\int \left(H-d\right){H}^{2}\phantom{\rule{thinmathspace}{0ex}}.$

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

it is easy to see that we have

(5)$\cdots =\int A\phantom{\rule{thinmathspace}{0ex}}\mathrm{dA}+\int \mathrm{AAA}+order{ϵ}^{2}\phantom{\rule{thinmathspace}{0ex}}.$

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

It would be nice to use the fact that $d\omega =\left[\rho ,\omega \right]$ 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)${S}_{?}=\int \mathrm{AAA}\phantom{\rule{thinmathspace}{0ex}},$

but due to lattice effects its equations of motion are not quite $\mathrm{AA}=0$ (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 $\Lambda$ under $\star$-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)$\left[A,\omega \right]=d\omega \phantom{\rule{thinmathspace}{0ex}}.$

- 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

Sorry, there was a typo: I meant

(1)${S}_{\mathrm{CS}}=\int \left(H-\rho \right){H}^{2}\phantom{\rule{thinmathspace}{0ex}},$

where $\rho$ 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

Just curious, isn’t

(1)$A=H-\rho$

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

(2)${S}_{\mathrm{CS}}={\int }_{M}A{H}^{2}$

?

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

Yes, the holonomy 1-form starts with

(1)$H=\rho +ϵA+order{ϵ}^{2}$

by definition of holonomy along an edge.

So, $H-\rho =ϵA+order{ϵ}^{2}$.

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

Here is a crazy thought (imagine that) :)

Begin with

(1)$S={\int }_{M}{A}^{3}$

then deform $A$ via $\rho$ so that you have

(2)$S\prime ={\int }_{M}\left(A+\rho {\right)}^{3}={\int }_{M}\left(A+\rho \right){H}^{2}={S}_{\mathrm{CS}}+{\int }_{M}\rho {H}^{2}.$

What is that last term? :)

Eric

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

Re: background of backgrounds

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

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

Then

(1)${\int }_{M}{A}^{3}\to {\int }_{M}A\left(A+\rho {\right)}^{2}={\int }_{M}A{H}^{2}={S}_{\mathrm{CS}}$

:)

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

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)$S={\int }_{M}\left(H-\rho \right){H}^{2},$

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

(2)$S={\int }_{M}\left(H-\rho {\right)}^{3}$

:)

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? 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.

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 $S\left({R}^{2n}\right)$ 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?

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)$\left(A*B{\right)}^{†}={B}^{†}*{A}^{†}.$

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)$〈A\mid B〉=\frac{1}{2}\int \left({A}^{†}*B+{B}^{†}*A\right)\mathrm{dV}$

for some positive 0-form $\mathrm{dV}$.

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 ${F}^{\mu \nu }{F}_{\mu \nu }$.

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)$S=\frac{1}{2}〈F\mid F〉=\frac{1}{2}\int {F}^{†}F\mathrm{dV},$

where

(4)$F=Q\left(A\right)+A*A.$

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?

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)${Q}^{†}\mid \psi 〉=0$

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 $Q\mid \psi 〉=0$, 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)$〈Q\Lambda \mid Q\Lambda 〉$

which might come to mind.

However, maybe one could in principle use the action

(3)$〈{Q}^{†}{\kappa }_{2}\mid {Q}^{†}{\kappa }_{2}〉$

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

(4)$Q{Q}^{†}\mid {\kappa }_{2}〉=0$

i.e.

(5)$Q\mid \psi 〉=0$

if we identify the 1-form $\psi$ with ${Q}^{†}{\kappa }_{2}$.

Maybe this could make sense, but that’s too difficult a question for me to decide! :-) We should ask Mr. Witten…

Posted by: Urs Schreiber on May 11, 2004 11:35 AM | Permalink | Reply to this

Re: CFTs from OSFT?

Neat :)

We study some reparametrization invariant theories in context of the BRST-co-BRST quantization method. The method imposes restrictions on the possible gauge fixing conditions and leads to well defined inner product states through a gauge regularisation procedure. Two explicit examples are also treated in detail.

In this paper he says

We have however, another very powerful quantization method, namely BRST-quantization . This method has the advantage that together with a co-BRST operator it leads to a well defined inner product space , , .

Eric

Posted by: Eric on May 11, 2004 9:49 PM | Permalink | Reply to this

Re: CFTs from OSFT?

Good morning! :)

Leaving interactions aside, we need to come up with the equations of motion $Q\mid \psi 〉$, 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

(1)$〈\mathrm{QA}\mid \mathrm{QA}〉$

which might come to mind.

However, maybe one could in principle use the action

(2)$〈{Q}^{†}{\kappa }_{2}\mid {Q}^{†}{\kappa }_{2}〉$

where ${\kappa }_{2}$ is a 2-form.

I was beginning to grow fond of this idea, but the fact that it looks a little weird motivated me to think of a more natural (?) generalization. What if we started with an “inhomomogenous form”

(3)$A=\sum _{i}{A}_{i},$

where ${A}_{i}$ is of degree $i$. Then defined a “Dirac operator”

(4)$D=Q+{Q}^{†}$

then we could write down a seemingly natural action

(5)$S=〈\mathrm{DA}\mid \mathrm{DA}〉$

with equation of motion

(6)${D}^{†}DA=0,$

which is equivalent to

(7)$\left(Q{Q}^{†}+{Q}^{†}Q\right)A=0.$

If I’m not mistaken, this would be equivalent to

(8)$\mathrm{QA}={Q}^{†}A=0.$

This looks familiar from some of those BRST/co-BRST papers I’ve been skimming over.

Maybe this could make sense, but that’s too difficult a question for me to decide! :-) We should ask Mr. Witten…

Did you see that he replied to a question on s.p.s.? Maybe it is not such a crazy idea to ask him after all :)

Eric

Posted by: Eric on May 12, 2004 2:43 PM | Permalink | Reply to this

Re: CFTs from OSFT?

Hm, good point. This might be a good idea.

In order to work our inner product must be a true sclar product, i.e. positive definite. That’s because this is the condition from which it follows that harmonic elements are closed and coclosed, due to

(1)${D}^{2}\mid \psi 〉=0⇒〈\psi \mid {D}^{2}\psi 〉=0⇒〈D\psi \mid D\psi 〉=0⇔D\mid \psi 〉=0\phantom{\rule{thinmathspace}{0ex}},$

where the last equivalence holds if the inner product is positive definite.

Yes, I can imagine that this could work. After all, the inner product that we are after is essentially that of (matrix-valued) differential forms on the Virasoro group, which should have Riemannian signature, naturally.

Hm, somebody must have thought about this… Could you find anything on BRST/coBRST applied to the worldsheet theory of strings?

Posted by: Urs Schreiber on May 12, 2004 3:09 PM | Permalink | PGP Sig | Reply to this

Re: CFTs from OSFT?

Hm, good point. This might be a good idea.

In order to work our inner product must be a true sclar product, i.e. positive definite. That’s because this is the condition from which it follows that harmonic elements are closed and coclosed, due to

(1)${D}^{2}\mid \psi 〉=0⇒〈\psi \mid {D}^{2}\psi 〉=0⇒〈D\psi \mid D\psi 〉=0⇔D\mid \psi 〉=0\phantom{\rule{thinmathspace}{0ex}},$

where the last equivalence holds if the inner product is positive definite.

Yes, I can imagine that this could work. After all, the inner product that we are after is essentially that of (matrix-valued) differential forms on the Virasoro group, which should have Riemannian signature, naturally.

Hm, somebody must have thought about this… Could you find anything on BRST/coBRST applied to the worldsheet theory of strings?

Posted by: Urs Schreiber on May 12, 2004 3:18 PM | Permalink | PGP Sig | Reply to this

Re: CFTs from OSFT?

Hello :)

Hm, good point. This might be a good idea.

Inconceivable! :)

In order to work our inner product must be a true sclar product, i.e. positive definite.

I see what you mean, but is positive definite what we’re after or is non-degenerate good enough?

Hm, somebody must have thought about this… Could you find anything on BRST/coBRST applied to the worldsheet theory of strings?

Well, I did not knowingly plagerize the idea (I know that is not what you are suggesting), but it has happened in the past when I think I came up with an idea that I had actually seen months (years) earlier :) To the best of my knowledge I did not recognize anything like this in the papers I’ve looked at. Keep in mind though, that I hardly ever read these papers in any detail. I simply skim them to get the main idea. In the BRST/coBRST papers I skimmed, I do recall seeing things like

(1)$Q\mid \mathrm{ph}〉={Q}^{†}\mid \mathrm{ph}〉=0,$

but nothing like the action

(2)$S=〈\mathrm{DA}\mid \mathrm{DA}〉$

that would give rise to those equations.

If this idea could be made to work, I would like it much better than the Chern-Simons version.

I am starting to see a hierarchy emerge. Maybe it is my imagination :) You start with an abstract differential calculus that leads to some theory. This theory often admits a BRST procedure, but this BRST procedure itself involves a “higher” abstract differential calculus. This calculus may lead to some “higher” theory. This higher theory may itself admit a “higher” BRST procedure (a BRST approach to study BRST theories :))

Where will it end?!?!

Maybe I am losing my mind :)

Eric

Posted by: Eric on May 12, 2004 3:46 PM | Permalink | Reply to this

Re: CFTs from OSFT?

Hello!

Hm, somebody must have thought about this… Could you find anything on BRST/coBRST applied to the worldsheet theory of strings?

I wasn’t sure whether this was a homework assignment or just a question based on my knowledge at the time :)

With the help of google, I found three papers that mention both “co BRST” and “string” :)

Notoph Gauge Theory as Hodge Theory
R. P. Malik

In the framework of an extended BRST formalism, it is shown that the four $\left(3+1\right)$-dimensional (4D) free Abelian 2-form (notoph) gauge theory presents an example of a tractable field theoretical model for the Hodge theory.

The above paper doesn’t really seem to be very relevant to our discussion though.

BRST Field Theory of Relativistic Particles
J.W. van Holten

A generalization of BRST field theory is presented, based on wave operators for the fields constructed out of, but different from the BRST operator. We discuss their quantization, gauge fixing and the derivation of propagators. We show, that the generalized theories are relevant to relativistic particle theories in the Brink-Di Vecchia-Howe-Polyakov (BDHP) formulation, and argue that the same phenomenon holds in string theories. In particular it is shown, that the naive BRST formulation of the BDHP theory leads to trivial quantum field theories with vanishing correlation functions.

Just after Equation (67) in the above paper he makes a remark that seems to suggest that the Chern-Simons like action is somehow “unusual”

Note, that in general the interacting theory does not take the form of a Cherns- Simons like theory, as is the case of the open string.

Finally,

Aspects of BRST Quantization
J.W. van Holten

BRST-methods provide elegant and powerful tools for the construction and analysis of constrained systems, including models of particles, strings and fields. These lectures provide an elementary introduction to the ideas, illustrated with some important physical applications.

I’ve spent the most time with this last paper. I was using it to learn some of the basics of BRST stuff. This paper also seems the most relevant. It does consider the adjoint of the BRST operator and even the “BRST Laplacian” in the section on BRST field theory. However, the action is of the form

(1)$S=〈A\mid \mathrm{QA}〉$

where $Q$ must be self adjoint w.r.t. $〈\cdot \mid \cdot 〉$, which is different than the one I’m thinking about.

On the other hand, if $A$ is an “inhomogeneous form” and $D=Q+{Q}^{†}$, then we could consider a Dirac-Kaehler like action

(2)$S=〈A\mid \mathrm{DA}〉.$

Perhaps we could consider a combination of the two. Then we would have both boson and fermion like contributions. Maybe we could even get a supersymmetry of supersymmetries :) There goes the hierarchical idea again :)

Eric

Posted by: Eric on May 12, 2004 9:24 PM | Permalink | Reply to this

Re: CFTs from OSFT?

I wasn’t sure whether this was a homework assignment or just a question based on my knowledge at the time :)

No, I was simply calling for help! :-)

I am currently so overwhelmed with work that I didn’t find the time to make a web search myself. But at the same time I found our discussion interesting and didn’t want to stop it by not responding.

I was wondering if anyone had ever considered an inner product on the string’s Hilbert space with respect to which the adjoint of the BRST operator yielded an object that could be addressed as a co-BRST operator. That would be interesting.

Apparently, judging from the references that you found, this hasn’t been done. The ‘BRST Field Theory of Relativistic Particles’ by van Holten does mention coBRST operators, but these don’t arise as adjoints of the original BRST operator in this discussion, as far as I can see.

Concerning your hierarchical ideas: I know what you mean - but at the moment I cannot see how to make any of this even slightly precise. :-)

There has always been a vague formal relation between supersymmetry and BRST formalism, and in some context (twisted) susy charges do become BRST operators and vice versa. However, that’s again something different from what I mentioned before, which related to the fact that the BRST operator for the superstring is an exterior derivative on the super-gauge group of the string, while on the other hand some odd-graded vector fields on this group can themselves be interpreted as (deformed) exterior derivatives (on loop space).

Phew - if I had more time I’d maybe be able to provide more than just a bunch of buzzwords! ;-) But better times are in sight…

Please keep letting me know about any ideas and progress that you are making. I’ll be happy to respond. As soon as the to do drawer in my desk is evacuated I’ll hope to be able to give more constructive comments.

Posted by: Urs Schreiber on May 13, 2004 10:25 AM | Permalink | PGP Sig | Reply to this

Re: CFTs from OSFT?

Good morning! :)

I was wondering if anyone had ever considered an inner product on the string’s Hilbert space with respect to which the adjoint of the BRST operator yielded an object that could be addressed as a co-BRST operator. That would be interesting.

See, one of the biggest problems I’m facing now is that I still have no idea about what BRST quantization really is :) I know that it looks like an abstract differential calculus and that is about it :)

Last night, I stumbled on this paper,

Local BRST cohomology in gauge theories
Glenn Barnich, Friedemann Brandt, Marc Henneaux

The general solution of the anomaly consistency condition (Wess-Zumino equation) has been found recently for Yang-Mills gauge theory. The general form of the counterterms arising in the renormalization of gauge invariant operators (Kluberg-Stern and Zuber conjecture) and in gauge theories of the Yang-Mills type with non power counting renormalizable couplings has also been worked out in any number of spacetime dimensions. This Physics Report is devoted to reviewing in a self-contained manner these results and their proofs. This involves computing cohomology groups of the differential introduced by Becchi, Rouet, Stora and Tyutin, with the sources of the BRST variations of the fields (“antifields”) included in the problem. Applications of this computation to other physical questions (classical deformations of the action, conservation laws) are also considered. The general algebraic techniques developed in the Report can be applied to other gauge theories, for which relevant references are given.

which seems to have everything you ever wanted to know about BRST (and more!) :)

I don’t remember where exactly I saw it (the down side of skimming 20 papers simultaneously), but I remember seeing the statement last night (paraphrased)

… a choice of gauge determines an inner product on the BRST phase (?) space …

The statement doesn’t make a whole lot of sense to me now so I may be remembering it incorrectly, but this made me think of something else. I also vaguely remember seeing somewhere that

(1)${Q}^{†}\mid \varphi 〉=0$

is a choice of gauge. It almost seems to me like you can make the claim that ANY choice of gauge amounts to choosing a ${Q}^{†}$ operator. If you are given $Q$ and ${Q}^{†}$, I wonder if there is a way to extract $†$? This would be neat because this would make the paraphrased statement above equivalent to saying that fixing the gauge determines $†$, which may then be used to define an inner product, i.e.

(2)$\mathrm{gauge}↔\mathrm{BRST}\mathrm{inner}\mathrm{product}$

Anyway, I am beginning to see that this BRST stuff is even more complicated than I originally thought :) There are fields, ghosts, antifields, antighosts, and each one of these has their own grading! :)

I would really like to try to understand this stuff more. I’ll keep working at it. I still want to understand what the meaning of “degree” and “dimension” are. For example, in what sense is the CS-like SFT action “three dimensional”? Is there a “highest” degree form?

My brain hurts! :)

Eric

Posted by: Eric on May 13, 2004 2:42 PM | Permalink | Reply to this

Re: CFTs from OSFT?

I still have no idea about what BRST quantization really is

Ok. But did you see my first attempt at an explanation here?

The basic idea is that one wants to translate gauge invariance into cohomological terms. That’s natural, because in systems with gauge invariance we have the notion that objects must be invariant under gauge tranformations (‘closed’) but are defined only up to gauge transformations (up to ‘exact’ elements).

If you look at what I wrote in the above mentioned entry you’ll see that this is quite directly translated into something like deRham cohomology on the gauge group. Physical states are 1-forms on the gauge group which are closed and hence they are defined only up to an exact piece.

I also vaguely remember seeing somewhere that ${Q}^{†}\mid \varphi 〉=0$ is a choice of gauge.

Yes, that’s what I said here! :-)

It is precisely the same idea as in deRham cohomology: Given any closed form $d\omega =0$ there is a choice of gauge $\omega \to \omega +d\lambda$ which can be fixed, in particular, by demanding that ${d}^{†}\omega =0$, i.e. by demanding $\omega$ to be harmonic.

You seem to say that you got the impression that in BRST theory for every choice of gauge one can find an scalar product such that ${Q}^{†}\mid \varphi 〉=0$ enforces this particular gauge. That would be intersting, I don’t know if it is true in general. Probably there is at least some fine print.

Hm, thinking of the free particle as in that van Holten paper, this actually looks dubious to me. But I haven’t systematically thought about it…

(Right now I am busy writing up some ‘lecture notes’ on introductory CFT for our string theory seminar… Our next meeting will then be concerned with the BRST quantization of the string. Hopefully I can provide you with some ‘lecture notes’ on that, too.)

Posted by: Urs Schreiber on May 13, 2004 4:03 PM | Permalink | PGP Sig | Reply to this

Re: CFTs from OSFT?

Good evening :)

I am feeling extremely guilty. This entire evening, I did absolutely no work and sat in front of the TV all night. True couch potato style :)

Just a quick note before I’m off to bed…

The basic idea is that one wants to translate gauge invariance into cohomological terms. That’s natural, because in systems with gauge invariance we have the notion that objects must be invariant under gauge tranformations (‘closed’) but are defined only up to gauge transformations (up to ‘exact’ elements).

I think I pretty much understand the idea at this level of detail, but I’m still after the really part, i.e. I want to understand what it really is. Any idea that is as powerful as BRST seems to be must have some deep meaning beyond the textbook stuff :)

You seem to say that you got the impression that in BRST theory for every choice of gauge one can find an scalar product such that ${Q}^{†}\mid \varphi 〉=0$ enforces this particular gauge. That would be intersting, I don’t know if it is true in general. Probably there is at least some fine print.

Yes! Wouldn’t that be poetic? Usually, gauge choices involve the metric, but what if a gauge choice determined a special metric? :)

I’ll take the point particle as a challenge, but that will have to wait until morning.

Good night!
Eric

Posted by: Eric on May 14, 2004 5:11 AM | Permalink | Reply to this

Re: CFTs from OSFT?

Ok. In the “Aspects of BRST Quantization” paper, where he discusses the relativistic scalar particle, i.e. Section 4.1, it seems like things would be more natural if he thinks of a new $G$-modified inner product, i.e.

(1)$\left(\Phi ,\Psi {\right)}_{G}=\left(\Phi ,G\Psi \right),$

where $\left(\cdot ,\cdot \right)$ is the inner product given in Equation 4.3. If $G$ is Hermitian, then it seems to me (I could be wrong) that the requirements he imposes amounts to saying that $\Omega$ is Hermitian with respect to the $G$-modified inner product.

(2)$\left(\Omega \Phi ,\Psi {\right)}_{G}=\left(\Phi ,{\Omega }^{†}G\Psi \right)=\left(\Phi ,G\Omega \Psi \right)=\left(\Phi ,\Omega \Psi {\right)}_{G}.$

so that

(3)${\Omega }^{{†}_{G}}=\Omega .$

In this case, a natural choice for $G$ seems to be

(4)$G=bc\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}G=cb.$

Gotta run!
Eric

Posted by: Eric on May 14, 2004 5:40 AM | Permalink | Reply to this

BRST/coBRST games

Your proposed choices of $G$ don’t seem to satisfy van Holten’s (4.7). Note that he uses $G=b$ (right above) (4.14), which is not a nice deformation operator for the scalar product, since it is neither self-adjoint nor invertible

But what you originally had in mind was something maybe ‘nicer’ than (4.6), namely (ignoring the source term for the moment)

(1)${S}_{G}=\left(\left(\Omega +{\Omega }^{†}\right)\Psi ,\left(\Omega +{\Omega }^{†}\right)\Psi \right)\phantom{\rule{thinmathspace}{0ex}}.$

Varying this gives essentially the free part of (4.12)

(2)$\left(\Omega +{\Omega }^{†}{\right)}^{2}\psi =0$

but without the need of the ‘fudge operator’ $G$.

One might have to think how the sources have to be incorporated into this in order for (4.18) to remain true.

Posted by: Urs Schreiber on May 14, 2004 1:21 PM | Permalink | PGP Sig | Reply to this

Re: BRST/coBRST games

Incorporating the sources is easy, for instance by adding $\left(\left(\Omega +{\Omega }^{†}\right)\Psi ,J\right)$ to the action.

But of course what I wrote above takes us outside the scope of van Holten’s considerations, since the action which I wrote down does not enjoy the gauge invariance (4.10) simply because the gauge is explicitly fixed by the appearance of ${\Omega }^{†}$ in the action.

But thisis not really a problem. The real problem for these ideas to work out is to find a sensible positive definite scalar product along the lines of (4.3) for the string with respect to which we get a usable ${\Omega }^{†}={Q}^{†}$.

Posted by: Urs Schreiber on May 14, 2004 1:29 PM | Permalink | PGP Sig | Reply to this

Re: BRST/coBRST games

Hello again :)

But thisis not really a problem. The real problem for these ideas to work out is to find a sensible positive definite scalar product along the lines of (4.3) for the string with respect to which we get a usable ${\Omega }^{†}={Q}^{†}$.

Given an abstract differential calculus $\left(𝒜,Q\right)$, how many different adjoints can there be? If you find one adjoint operation, would all others be related by a similarity transformation?

A plan of attack might be to simply make use of Parodi’s $†$ and define an inner product from that. Since he seems to have a ${C}^{*}$-algebra (at least he’s on his way to one), then this would mean that the inner product was already positive definite, wouldn’t it?

The thing to check would be whether

(1)$\int {A}^{†}*A\ge 0.$

I don’t know, but I’d guess the chances for this to be true were pretty high :)

Eric

Posted by: Eric on May 14, 2004 3:58 PM | Permalink | Reply to this

Re: BRST/coBRST games

Oh, right, I forgot. You are talking about hep-th/0302177? I’ll have a look at it. Does his star operation invert the ghost number?

Posted by: Urs Schreiber on May 14, 2004 4:07 PM | Permalink | PGP Sig | Reply to this

Re: BRST/coBRST games

I don’t quite see the point of Parodi’s hep-th/0302188. He seems to be just reviewing well known constructions, such as equation (43). His point is that he checks that this operation really turns the star algebra into a star algebra ;-), I mean turns the Witten-star algebra of string fields into a ${C}^{*}$-algebra. Not too surprising and also not quite satisfctory from our point of view, since it seems to me that all of Parodi’s constructions are restricted to the matter sector, while in order to get that scalar product that you are looking for we nee to consider the ghost sector.

Actually, I think what we’d need is the ghost analog of what is done in section 4.5 of

Aref’eva & Belov & Giryavets & Koshelev & Medvedev: Noncommutative Field Theories and (Super) String Field Theories (2001).

You’ll get the schematic idea of that section even without having read the previous material.

Note that (4.148) is essentially a definition, think of $\mid {A}_{n}〉$ and $\mid {A}_{n}^{†}〉$ at ordinary bosonic creation/annihilation operators, of $\mid \Xi 〉$ as the corresponding vacuum (really more like $\mid \Xi 〉\sim {\sum }_{n}\mid {A}_{n}〉\mid {A}_{n}^{†}〉$ or the like) and then all the formulas on p. 61 are obvious.

- From equation (4.164) the same is repeated, just in a different representation.

If this has an analog in the ghost sector we’d probably be a step closer to what you proposed to look for. Now, the next section 4.7 of that paper is entitled Ghost sector, but does not quite repeat the above constructions. It should be pretty easy to see if and how this works, though.

Posted by: Urs Schreiber on May 14, 2004 5:41 PM | Permalink | PGP Sig | Reply to this

Re: BRST/coBRST games

I don’t quite see the point of Parodi’s hep-th/0302188. He seems to be just reviewing well known constructions, such as equation (43).

The only thing I really got out of Parodi’s paper is probably something already well known, but I thought it was neat that there was an adjoint operation on the Witten *-algebra at all, i.e. the only thing I needed to know was that we had some operation $†$ satisfying

(1)$\left(A*B{\right)}^{†}={B}^{†}*{A}^{†}.$

I thought that was pretty neat, but maybe I gave too much credit :)

Not too surprising and also not quite satisfctory from our point of view, since it seems to me that all of Parodi’s constructions are restricted to the matter sector, while in order to get that scalar product that you are looking for we nee to consider the ghost sector.

Ok. You lost me :) What is “matter sector” and what is “ghost sector”? :)

Maybe I am being overly simplistic, but all I think we need for the YM-like SFT action

(2)$S=\frac{1}{2}〈D\Psi \mid D\Psi 〉=\frac{1}{2}\int \left(D\Psi {\right)}^{†}D\Psi$

is a $†$ satisfying

(3)$\left(A*B{\right)}^{†}={B}^{†}*{A}^{†}$

and

(4)$\int {A}^{†}*A=0\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}A=0,$

i.e. $\mid \mid \cdot \mid \mid$ is non degenerate.

Which step is the hard part? :)

Eric

Posted by: Eric on May 14, 2004 6:23 PM | Permalink | Reply to this

Re: BRST/coBRST games

Which step is the hard part? :)

Let me clarify the matter/ghost issue: In the context of string theory the worldsheet embedding fields are sometimes called ‘matter fields’, because even though they are really coordinates, they have an action like bosonic fields wrt the 2d worldsheet theory. The ghosts are further fields, which are introduced to get the Fadeev-Popov determinant and hence the BRST formalism. They have no real physical interprteation, hence the term ‘ghost’.

But you don’t have to worry about Fadeev-Popov determinants and stuff.

For our purposes, it is just crucial that the ‘matter sector’ encodes the physical configuration of the string, i.e. parameterizes the configuration space. The ‘ghost sector’ on the other hand consists of the fields $c$ and $b$ which, as I said, we should, can, and will think of as creators and annihilators of differential forms.

Now, any given string field has some ‘matter oscillations’, namely excitations of the string’s coordinate fields. These you must think of as functions, 0-forms, somehow.

The string field will also have some ghost excitations. These you must think of as form creators-annihilators which turn the string field into a $p-\mathrm{form}$, where $p$ is equal to the so-called ‘ghost number’.

So string fields can be thought of as p-forms (really p-form creation/annihilation operators, but anyway) and the star product as the wedge product. Really, these must be thought of as something like matrix valued p-forms.

This means that the Witten star product can, roughly, be decomposed into that part which multiplies the ghost/p-form degrees of freedom, and one part that multiplies the 0-form degrees of freedom.

Think of some ordinary $p$-forms

(1)$\alpha =A\left(x,y\right)\mathrm{dx}$
(2)$\beta =B\left(x,y\right)\mathrm{dy}$

on a 2d manifold with $A$ and $B$ Lie-algebra valued functions (0-forms) on that manifold.

The wedge product of $\alpha$ and $\beta$

(3)$\alpha \wedge \beta$

involves the wedge product of $\mathrm{dx}$ with dy and also the matrix product of $A$ with $B$:

(4)$\alpha \wedge \beta =\left(A\cdot B\right)\left(\mathrm{dx}\wedge \mathrm{dy}\right)\phantom{\rule{thinmathspace}{0ex}}.$

In the SFT context we would address the matrix product as being in the ‘matter sector’ and the $\mathrm{dx}\wedge \mathrm{dy}$ product as being in the ‘ghost sector’, roughly.

Now my point was that Parodi considers only the ${C}^{*}$ operation on the matter sector, i.e. with respect to what is analogous to the above matrix multiplication.

What we would really like to have, though, is an adjointness relation on the differential form part, such that

(5)$\left(\mathrm{dx}\wedge {\stackrel{˙}{\right)}}^{†}=\mathrm{dx}\to \cdot$

form creators are turned into form annihilators.

It is probably not that hard, but I am afraid I won’t solve this riddle in what remains of the day at my longitude! :-)

Posted by: Urs Schreiber on May 14, 2004 6:53 PM | Permalink | PGP Sig | Reply to this

Re: BRST/coBRST games

Ah ha! Yes! Thank you for your patience. I think I am beginning to understand a few things finally :)

I am pretty confident now that we can come up with a $†$ that handles the matter and ghost sectors correctly :) For example, if

(1)$\alpha =Ac$

where $A$ is a matter field and $c$ is a ghost field, then we can simply do something like

(2)${\alpha }^{†}={A}^{†}{c}^{†}={A}^{†}b,$

where ${A}^{†}$ is something like (probably exactly like) Parodi’s ${A}^{*}$.

Piece of cake! :)

I was pulled away for a meeting so I sent my last post after you sent this one, but I hadn’t seen this one before sending mine. Make sense? :)

In any case, it seems you finally have managed to drill the idea into my head. Thanks! :)

Now, before I let you actually be tempted into thinking that I am starting to understand this stuff, let me ask some more questions :)

In the case of the scalar particle, I only saw one ghost creator $c$. Does this mean that there are no “2-forms” for a scalar particle? In other words, is c^2 = 0?

The ghosts are further fields, which are introduced to get the Fadeev-Popov determinant and hence the BRST formalism. They have no real physical interprteation, hence the term ‘ghost’.

I bet if we looked hard enough we could find a physical interpretation :) I don’t believe in magic :)

Ciao!
Eric

Posted by: Eric on May 14, 2004 8:53 PM | Permalink | Reply to this

Re: BRST/coBRST games

${A}^{†}{c}^{†}={A}^{†}b$

[…]

Piece of cake! :)

Yup, that’s tempting, isn’t it! :-) But I am afraid it’s not that simple, because we still need $\left(a*b{\right)}^{†}={b}^{†}*{a}^{†}$, which won’t work with the above prescription.

That’s why I thought we might try to copy the content of section 4.5 of that Aref’eva et al. paper to the ghost sector, because there we have creators/annihilators truly with respect to the star product.

I only saw one ghost creator $c$

Sorry for being imprecise here. $c$ is a local worldsheet field, which depends on the worldsheet position $c=c\left(z\right)$. In terms of its Fourier modes we have the infinite set $\left\{{c}_{n}{\right\}}_{n\in Z}$ of form creators, one for each mode ${L}_{n}$ of the Virasoro constraints, which you should think of as the ‘tangent vectors’ on the gauge group.

The formal analogy is

(1)${e}_{a}={e}_{a}{}^{\mu }{\partial }_{\mu }↔{L}_{n}$
(2)$\left({\mathrm{dx}}^{\mu }\wedge \cdot \right)↔{c}_{n}$
(3)$\left({\mathrm{dx}}^{\mu }←\cdot \right)↔{b}_{n}\phantom{\rule{thinmathspace}{0ex}}.$

Hopefully today I’ll find some time to think about what I have in mind concerning the above section 4.5.

Posted by: Urs Schreiber on May 17, 2004 11:23 AM | Permalink | PGP Sig | Reply to this

Re: BRST/coBRST games

This papers seems good. You probably have already seen it because I found it by simply tracking the references in Aref’eva’s paper.

Open String Fields As Matrices
Teruhiko Kawano and Kazumi Okuyama

We present a new representation of the string vertices of the cubic open string field theory. By using this three-string vertex, we attempt to identify open string fields as huge-sized matrices by following Witten’s idea. By using these huge matrices, we obtain some results about the construction of partial isometries in the algebra of open string fields.

Posted by: Eric on May 17, 2004 3:31 PM | Permalink | Reply to this

Re: BRST/coBRST games

Yes, the matrix representation is cool! I haven’t looked at the paper you cited, but the matrix rep is for instance how section 4 of that Aref’eva et al hep-th/0111208 begins.

You will surely like the relation to string discretization, as for instance in equation (4.14) of that paper! :-)

I tried to sit down to study that section in detail today, but unfortunately I didn’t find the time. We had a guest, Prof. Niemeyer from Würzburg who gave a talk on dark energy and the accelerating universe and we spend large parts of the day chatting with him about cosmology and things like that. Now I have to run again, having an appointment with some friends of mine. Seems like the gods of research have decided to keep me from doing productive work these days… ;-)

Posted by: Urs Schreiber on May 17, 2004 6:34 PM | Permalink | PGP Sig | Reply to this

Re: BRST/coBRST games

Hi Urs! :)

You will surely like the relation to string discretization, as for instance in equation (4.14) of that paper! :-)

Nice! :)

I’m probably over simplifying things but if you hand me Equation (4.13) and a graded derivation $d$ (without even telling me what the variables mean), I am more than tempted to construct a lattice theory (of string fields?) where each ${P}_{n}$ is a node of some abstract graph. I’d construct “1-forms” generated by elements of the form

(1)${P}_{\mathrm{mn}}={P}_{m}*{\mathrm{dP}}_{n}*{P}_{n}.$

Then 2-forms

(2)${P}_{\mathrm{lmn}}={P}_{\mathrm{lm}}*{P}_{\mathrm{mn}}={P}_{l}*{\mathrm{dP}}_{m}*{P}_{m}*{\mathrm{dP}}_{n}*{P}_{n}$

etc, etc :)

We could define the gluing form

(3)$\rho =\sum _{m,n}{P}_{\mathrm{mn}}$

etc, etc.

It would be too good to be true if

(4)$d\Psi =\left[\rho ,\Psi {\right]}_{*}=\rho *\Psi -\Psi *\rho .$

(at least for 0-forms) :)

Then if we could work out a nice $†$ satisfying

(5)$\mid {A}^{†}\mid =-\mid A\mid$

and

(6)$\left(A*B{\right)}^{†}={B}^{†}*{A}^{†}$

then it seems we could COMPLETELY reproduce EVERYTHING we did in our notes! :)

The sums to $\infty$ may cause headaches, but I’d be tempted to truncate them.

Is this a crazy idea? :)

These orthogonal projectors seem to be just what the doctor ordered! :)

Seems like the gods of research have decided to keep me from doing productive work these days… ;-)

You just described the story of my life :)

Eric

PS: It could be that we are onto something. It seems the orthogonal projectors were only found in August 2000 (where I see it has already been cited 157 times) :)

Posted by: Eric on May 17, 2004 7:25 PM | Permalink | Reply to this

Re: BRST/coBRST games

Er, sorry, which orthogonal projectors are you referring to, precisely? Do you mean the $\mid 𝒪〉,\mid I〉,\mid \Xi 〉$ 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?

Actually, this addresses a confusion that I have: If we use the split string language and the matrix representation of string fields it seems that any infinite matrix can be interpreted as the matter part of a string field and that in particular the identity string field exists with the usual properties.

But this seems to contradict the cautionary remarks for instance in

I. Kishimoto & K. Ohmori: CFT Description of Identity String Field: Toward Deriation of the VSFT Action (2001).

I must be missing some fine print. But that’s no surprise, given that I currently spend more time citing the OSFT literature than actually reading it. ;-)

Posted by: Urs Schreiber on May 18, 2004 10:56 AM | Permalink | PGP Sig | Reply to this

Re: BRST/coBRST games

Hello!

This is my second attempt. I had written quite a bit and before I could post it, it disappeared! :)

Does his star operation invert the ghost number?

I missed this little post of yours :)

I am pretty sure his star does not invert the ghost number. Now that I think of it, this was causing me some confusion and I think I mentioned that because $†$ simply reverses orientation, then the degree $\mid {A}^{†}\mid =\mid A\mid$, but you would expect

(1)$\mid {A}^{†}\mid =-\mid A\mid$

so that

(2)$\mid {A}^{†}*A\mid =0.$

It seems like we need something that both reverses the orientation AND inverts the ghost number. Is that hard to do? :)

Is the situation with string fields analogous to that of ordinary abstract calculus where you have an algebra of 0-forms and a derivation that together generate the forms of higher degree (I’m assuming degree = ghost number without really understanding the RHS of the equality :)) ?

Ok. Good. I think I see the flaw in Parodi’s *. Now we need to fix it :)

Eric

Posted by: Eric on May 14, 2004 8:27 PM | Permalink | Reply to this

Chern-Simons vs Yang-Mills

Hi Urs :)

I know you are extremely busy and I feel like I may be bugging you with questions. That is not my intention! :) I am totally clueless about SFT and I’m trying to understand the motivations a bit better. This is not a totally selfish desire on my part because I hope that thinking about these things might help steer your research as well. You have so much more talent than I ever will, so it is probably silly to think I might help you, but I can’t help trying! :) I’ll submit this question just for the record and when/if you ever get the energy to say something, it’ll be here. If you don’t, that is totally fine too :)

If someone wanted to lecture me about a theory that is supposed to tell me how nature really is and he/she started by writing down an action that looks like

(1)$S=\int \Phi *Q\Phi +\frac{2}{3}\int \Phi *\Phi *\Phi ,$

the first thing that would happen in my mind would be a bunch of red flags and alarms screaming,

“Hey, wait a minute?!?! Why should nature be described by a Chern-Simons-like theory???”

Let me state for the record, for whatever it is worth, all of my instincts tell me that walking down this road is a dead end.

Over the past several years there seems to have been a renaissance of SFT. This seems to be related to efforts in VSFT that led to solutions in the matter sector that are in the form of projectors. It is my understanding (which isn’t very deep) that in VSFT, it has been conjectured that solutions to the equation of motion

(2)$Q\Phi +\Phi *\Phi =0$

can be decomposed into ghost and matter parts

(3)$\Phi ={\Phi }_{g}\otimes {\Phi }_{m}.$

Apparently, the BRST operator $Q$ is pure ghost, so the matter part of this equation reduces to

(4)${\Phi }_{m}=\kappa {\Phi }_{m}*{\Phi }_{m},$

which says that the solutions in the matter sector are projectors with respect to the Witten *-product. I’m pretty sure that solutions to this equation have been found only in the last few years. I do think this is a profound result, but I also think that this result transcends any particular action. Historically, the discovery was motivated by considerations of the Chern-Simons-like action, but it could very well have been motivated by anything else. As far as I’m concerned, it is a historical accident :)

Then, while fishing for alternatives to the Chern-Simons-like action, we stumbled on the idea to postulate

(5)$S=\int \left(D\Phi {\right)}^{†}*D\Phi ,$

where

(6)$D=Q+{Q}^{†}$

for some, yet undefined, $†$-operation and $\Phi$ is now to be thought of as an inhomogeneous form, or some kind of string field spinor :)

This would seem to me to be a very promising avenue to pursue. Of course there are tons of details to work out, but that is the fun part :)

Is this still something that interests you, or are there good reasons to give up that I am not yet aware of?

The obvious challenge is to define a $†$ operation satisfying

(7)$\left(A*B{\right)}^{†}={B}^{†}*{A}^{†}$

for string fields consisting of both ghost and matter parts. However, one encouraging thing I take from VSFT is that perhaps the string fields we will be interested in may be decomposed as

(8)$\Phi ={\Phi }_{m}\otimes {\Phi }_{g}$

as well. This would seem to simplify things tremendously. Then we could write things like

(9)$\Phi *\Psi =\left({\Phi }_{m}*{\Psi }_{m}\right)\otimes \left({\Phi }_{g}\otimes {\Psi }_{g}\right)$

and our task would be reduced to finding a $†$-operation in the ghost and matter sectors separately because we’d have

(10)$\left(\Phi *\Psi {\right)}^{†}=\left({\Phi }_{m}*{\Psi }_{m}{\right)}^{†}\otimes \left({\Phi }_{g}\otimes {\Psi }_{g}{\right)}^{†}.$

Best wishes,
Eric

Posted by: Eric on May 25, 2004 7:13 PM | Permalink | Reply to this

Re: Chern-Simons vs Yang-Mills

Hi Eric -

I am about to go to bed, but I certainly find this discussion here interesting, so I’ll drop a quick note:

Yes, a priori it is kind of surprising that CS theory shows up, maybe. But as it often works, after thinking about it for a while it tends to begin to look like the most reasonably thing there is.

But I also agree that you do have a point concerning the BRST/coBRST formalism. We talked about that the abstract form of the action that you are proposing certainly makes sense for the kinetic term. I.e.

(1)${S}_{\mathrm{kin}}=〈\Psi \mid \left(Q+{Q}^{†}{\right)}^{2}\mid \Psi 〉$

yields the gauge fixed and free equations of motion

(2)$Q\mid \Psi 〉=0$
(3)${Q}^{†}\mid \Psi 〉=0$

if the inner product is positive definite.

This should work for the simple examples like the free particle that we discussed in the context of that van Holten paper.

I don’t know if this has any chance of working for string field theory (though I also don’t know any compelling reason why it should not, except that the big shots haven’t considered it ;-) but if it does there must be, at least in this abstract context, a way to get the correct interaction terms.

Do you see any nice way to get these? The naive choice

(4)${S}_{\mathrm{int}}=〈\Psi \star \Psi \mid \Psi \star \Psi 〉$

for instance does not seem to work, but that does not mean anything.

One should really try to properly understand the nature of the space that $Q$ is the exterior derivative on. (I had seen a comment in this direction somehwere, but I forget where…) That should clarify how the Hodge inner product associated with it would look like. Currently I still don’t understand the star product in the ghost sector enough to make an educated guess about what your $〈\cdot \mid \cdot 〉$ should really be in SFT. (But I think I am making some progress in understanding SFT ;-)

So the bottom line is that I agree that you have an interesting point, but that it would be really interesting to understand the details of it, and that I still have not achieved.

As you write yourself, the problem can be split into matter and ghost sector. The matter sector is pretty much under control. The precise nature and details of the star product in the ghost sector is still not completely clear to me (I mean morally, apart from the bare definition).

While we are struggling to better understand SFT one simple exercise is certainly already well within reach:

One should regard ordinary CS theory and see how that can be ‘reformulated’ such that it fits into your framework, i.e. how an action would look like that has the form which you are imagining but at the same time the same classical equations of motion as the gauge fixed CS theory. Does such an action exist?

Posted by: Urs Schreiber on May 25, 2004 10:31 PM | Permalink | Reply to this

Re: Chern-Simons vs Yang-Mills

Good evening :)

I am about to hit the sack soon too.

Yes, a priori it is kind of surprising that CS theory shows up, maybe. But as it often works, after thinking about it for a while it tends to begin to look like the most reasonably thing there is.

Although I am getting old in physics years, I try to stay as open minded as possible so I am certainly willing to consider the possibility that a CS-like action for SFT is the way to go. However, if it is the way to go we should better have a very good answer to the question,

Why does it look like “3d” Chern-Simons theory? What is the meaning of “3d” here?

I’m not equipped to answer this question (yet). I can only ask it :) This is certainly an important question that I would try to answer before moving on.

Do you see any nice way to get these? The naive choice

(1)${S}_{\mathrm{int}}=〈\Psi *\Psi \mid \Psi *\Psi 〉$

for instance does not seem to work, but that does not mean anything.

My first reaction to this question was, “Well, what would you do in the case of ordinary Yang-Mills theory? It should be similar.” This is where I have to confess my ignorance if it wasn’t already obvious enough :) I went back and thumbed through Peskin & Schroeder and Sakurai (the only two books I have that discuss QFT) to refresh my memory. In P&S, I see the action

(2)${ℒ}_{\mathrm{QED}}={ℒ}_{\mathrm{Dirac}}+{ℒ}_{\mathrm{Maxwell}}+{ℒ}_{\mathrm{int}},$

where

(3)${ℒ}_{\mathrm{Dirac}}=\overline{\psi }\left(i\partial -m\right)\psi$
(4)${ℒ}_{\mathrm{Maxwell}}=-\frac{1}{4}\left({F}_{\mu \nu }{\right)}^{2}$

and

(5)${ℒ}_{\mathrm{int}}=-e\overline{\psi }{\gamma }^{\mu }\psi {A}_{\mu }.$

Here, the interaction term arises from the minimal principle, i.e. beginning with the Dirac term and making the replacement

(6)${\partial }_{\mu }\to {\partial }_{\mu }+ie{A}_{\mu }.$

Here is where I am stepping into unfamiliar waters :)

The above seems like essentially replacing the exterior derivative with the covariant exterior derivative

(7)$d\to d+ieA$

and correspondingly

(8)${d}^{†}\to {d}^{†}-ie{A}^{†}.$

Then defining

(9)$\partial =d-{d}^{†}$

we have

(10)$\partial \to \partial +ie\left(A+{A}^{†}\right)$

where now $A+{A}^{†}$ may be thought of as a Clifford algebra element.

If we make these replacements everywhere, we do not affect the Maxwell term.

Hey! This is kind of neat :)

(11)$A+{A}^{†}={\gamma }^{\mu }{A}_{\mu }$

I know I know. We’ve been talking about this for years, but I’ve never used it :)

Now to make the above look more suggestive, I’ll write

(12)${ℒ}_{\mathrm{Dirac}}={\psi }^{†}i\partial \psi$

and

(13)${ℒ}_{\mathrm{Maxwell}}=-\frac{1}{2}{F}^{†}F,$

where

(14)$F=\mathrm{dA}.$

Based purely on analogy, I can now write the string field action

(15)$S=\int {\Psi }^{†}*i\left(Q-{Q}^{†}\right)\Psi -\frac{1}{2}\int \left(Q\Phi {\right)}^{†}*Q\Phi ,$

where $\Psi$ and $\Phi$ are inhomogeneous string fields consisting of both matter and ghost sector parts.

Varying with respect to $\Psi$ seems to give

(16)$\left(Q-{Q}^{†}\right)\Psi =0.$

Varying with respect to $\Phi$ seems to give additionally

(17)${Q}^{†}Q\Phi =0.$

Then, the analogy suggests that we introduce interactions via the minimal principle

(18)$Q\to Q+ie\Phi$

which should probably (as your deformation stuff shows) should be

(19)$Q\to Q+ie\left[\Phi ,\cdot {\right]}_{*}.$

It seems like your ideas on deformations may amount to the minimal principle applied to string field theory. That would seem rather poetic, wouldn’t it? :)

I like this. What do you think?

Good night! :)
Eric

Posted by: Eric on May 26, 2004 5:06 AM | Permalink | Reply to this

Re: Chern-Simons vs Yang-Mills

Hi Eric -

I have to apologize for being stupid: While I am busy learning bosonic SFT I forgot to think about susy SFT once in a while and forgot that I knew very well that the action there, at least in Berkovits’ formalism, is indeed of a non-topological form, just what you are looking for, indeed.

The susy SFT action is not Chern-Simons like but Wess-Zumino-Witten (WZW) like. The WZW action is an action for functions $g\left(x\right)$ on a 2-dimensional manifold which take values in some groug $G$. The kinetic term is

(1)${S}_{\mathrm{kin}}\propto \int \mathrm{Tr}\left[\left({g}^{-1}dg\right)\wedge \star \left({g}^{-1}dg\right)\right]=〈\left({g}^{-1}dg\right)\mid \left({g}^{-1}dg\right)〉\phantom{\rule{thinmathspace}{0ex}}.$

In addition there is a topological term proportional to the integral of the 3-form $\left({g}^{-1}dg{\right)}^{3}$ over any 3-d manifold which has the original 2-d manifold as boundary.

I was reminded of this while reading the paper

J. Klusoň: Some Remarks About Berkovits’ Superstring Field Theory (2001)

which I highly recommend you to read.

Klusoň demonstrates explicitly how Berkovits’s SSFT action can be manifestly put in the above form by defining an exterior derivative in equation (3.1), and a Hodge star :-) in equation (3.15), such that (3.16) results.

Posted by: Urs Schreiber on May 26, 2004 7:05 PM | Permalink | Reply to this

Re: Chern-Simons vs Yang-Mills

Good morning! :)

[Note: Sorry if this is out of place for this thread, but I started writing and couldn’t think of a better place to move it.]

While I am busy learning bosonic SFT I forgot to think about susy SFT once in a while and forgot that I knew very well that the action there, at least in Berkovits’ formalism, is indeed of a non-topological form, just what you are looking for, indeed.

I guess this means that I might want to learn a little about susy string theory. I didn’t get much further than the Polyakov action before I lost my patience. Perhaps I will like the susy version better :) I assume that those same review papers you mention in the String Seminar thread, e.g. Szabo, have a decent discussion of susy ST? Is there any other notable reference for the susy version?

I guess this also means I might want to learn what WZW is too :)

Hmm… a 2 minute arxiv search makes me think that WZW might be something like a 0-form Yang-Mills. Is there any truth to that? What I mean is that you begin with a Lie-algebra valued 0-form $A$, construct the 1-form “curvature” $F=\mathrm{dA}$ and the action is

(1)$S=〈F\mid F〉=\int \mathrm{tr}\left(F\wedge \star F\right).$

Is there any truth to that? Or is it just any old Lie algebra-valued 1-form $A$ with

(2)$S=〈A\mid A〉=\int \mathrm{tr}\left(A\wedge \star A\right).$

If it is the former, I think I will like it more :)

I can imagine that the the ${A}^{3}$ term might arise from some minimal principle.

[Confession: I am so enamored by the idea of the minimal principle, that I would speculate that any physical theory that introduces interaction terms that do not arrise from the minimal principle is bogus.]

In any case, skimming over Kluson and Berkovits, it seems that my instinct wasn’t too far off, i.e. the Chern-Simons action is somewhat sick. At least when you want to include susy.

I’m holding out hope that the Yang-Mills like action might play a role somewhere :)

Eric

PS: How certain are we that WZW actually has anything to do with physics? If the answer to this question is not very convincing, then my complaints about the Chern-Simons like action will carry over to Berkovits’ WZW like action.

PPS: This looks like it might be interesting:

Deformations of WZW models
Stefan Forste

Posted by: Eric on May 27, 2004 3:38 PM | Permalink | Reply to this

Re: Chern-Simons vs Yang-Mills

I guess this means that I might want to learn a little about susy string theory.

It’s the super ghosts that make this business a little messy, sometimes. I can try to get you some decent introcution and answer some questions. But that might be a can of worms, since it also requires learning some CFT etc.

The good thing is, most of what Klusoň does is at a completely formal level so that almost no details of superstrings need to be known to appreciate the basic idea.

I am in the process of writing up some summary notes. Have a look at section 1.2, pp. 2 of the new version of my notes. There I discuss the WZW action, some of the usual gymnastics to get equations of motion and gauge transformations, describe how this generalizes to Grassmann coordinates and to Klusoň’s observation about how the BRST operator deforms when the background is shifted.

I am sure that you will like this much better than the CS version and I am eager to hear your first ideas on how to rewrite the deformation in terms of a deformation of the Hodge star! :-)

Hmm… a 2 minute arxiv search makes me think that WZW might be something like a 0-form Yang-Mills.

Not really. The two 1-forms that are multiplied in the kinetic term of the WZW model are not exact! See the precise definition in the above notes. That should answer this question.

Posted by: Urs Schreiber on May 27, 2004 4:09 PM | Permalink | PGP Sig | Reply to this

Re: BRST/coBRST games

Good morning! :)

Your proposed choices of $G$ don’t seem to satisfy van Holten’s (4.7). Note that he uses $G=b$ (right above) (4.14), which is not a nice deformation operator for the scalar product, since it is neither self-adjoint nor invertible

Oops! :) You are right (of course). My motivation for going there was to bring in the idea of modifying the inner product. If the inner product we are working with doesn’t seem to match our choice of gauge, we might be able to modify the inner product so that it does.

Eric

Posted by: Eric on May 14, 2004 3:32 PM | Permalink | Reply to this

Re: BRST/coBRST games

Yes, I quite agree in principle. Using the action which I sketched above we can change the gauge condition

(1)${\Omega }^{†}\mid \psi 〉=0$

by deforming the scalar product with some invertible and self-adjoint operator $G$ to

(2)${\Omega }^{†}G\mid \psi 〉=0\phantom{\rule{thinmathspace}{0ex}}.$

My question was: Can we get every possible gauge choice this way?

Posted by: Urs Schreiber on May 14, 2004 3:59 PM | Permalink | PGP Sig | Reply to this

Re: CFTs from OSFT?

Any idea that is as powerful as BRST seems to be must have some deep meaning beyond the textbook stuff

Possibly true. From a naive point of view BRST quantization seems to be just an elegant bookkeeping device for ‘old covariant’ quantization, as it is called in the context of string theory. However, when I look at the developments in string field theory (about which I am in the process of learning a little), it is remarkable how the BRST operator develops a life of its own, sort of.

I am talking about results such as are summarized in section 8 of the lecture notes

W. Taylor & B. Zwiebach: D-Branes, Tachyons, and String Field Theory (2003) .

For instance there is the result (or maybe well tested conjecture) that the BRST operator describing the ‘true’ open string vacuum, where the D25 brane has decayed consists of nothing but ghost terms (equation (323) in the above paper)! That sort of makes sense, since in this kind of true vacuum the open strings must have disappeared (becoming closed strings as the brane decays), so that it kind of makes sense that the ‘matter fields’ (worldsheet embedding fields, as opposed to ghost fields) disappear, but considering hwo the BRST operator was defined in the first place for the ordinary vacuum this is kind of surprising. It almost looks like any operator $Q$ on the string’s Hilbert space with the abstract properties of odd grade, nilpotency and graded Leibnitz can be interpreted as the BRST operator of some background.

Posted by: Urs Schreiber on May 14, 2004 2:08 PM | Permalink | PGP Sig | Reply to this

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 $Q$ 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)$〈A\mid B〉=\int A\wedge B$

so that the exterior derivative $d$ 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 

(2)$Q\mid \mathrm{ph}〉=0,{Q}^{2}=0,\left(2.1\right)$

where $\mid \mathrm{ph}〉$ is any physical state, and where $Q$ 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, $Q$ 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 $\left(𝒜,Q\right)$ and define an adjoint $†$ such that

(3)$\left(A*B{\right)}^{†}={B}^{†}*{A}^{†}$

then we can define an inner product

(4)$〈A\mid B〉=\frac{1}{2}\int \left({A}^{†}*B+{B}^{†}*A\right)$

using $†$. So the question is, “Is Q Hermitian?” I think it will be. Since $†$ simply reverses the orientation of the string, then ${A}^{†}$ is just a string so we should still have

(5)$Q\left({A}^{†}*B\right)=Q\left({A}^{†}\right)*B+\left(-1{\right)}^{\mid A\mid }{A}^{†}*Q\left(B\right).$

It seems like $Q$ will be Hermitian (up to a constant) as long as it satisfies

(6)$Q\left({A}^{†}\right)=\left(-1{\right)}^{\mid A\mid +1}Q\left(A{\right)}^{†}$

because then we’d have for $\mid A\mid =\mid B\mid +1$

(7)$\int Q\left({A}^{†}*B+{B}^{†}*A\right)=\left(-1{\right)}^{\mid B\mid }\left[〈Q\left(A\right)\mid B〉-〈A\mid Q\left(B\right)〉\right]=0$

so that

(8)$〈Q\left(A\right)\mid B〉=〈A\mid Q\left(B\right)〉.$

Is this possible? Perhaps we can ask Witten to make an appearance :)

Cheers,
Eric

Posted by: Eric on May 12, 2004 1:26 AM | Permalink | Reply to this

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 $\mid 𝒪〉$, $\mid I〉$, $\mid \Xi 〉$ 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)${P}_{m}*{P}_{n}={\delta }_{\mathrm{mn}}{P}_{m}\phantom{\rule{1em}{0ex}}and\phantom{\rule{1em}{0ex}}\sum _{n=0}^{\infty }{P}_{n}=1.$

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 $𝒜$.

Eric

Posted by: Eric on May 18, 2004 5:47 PM | Permalink | Reply to this

Re: CFTs from OSFT?

Oh, ok. I hadn’t looked at the original paper, only at its summary, where the ${P}_{n}$ notation was not used.

I currently find it hard to get a reasonable overview over all the techniques that are being used in SFT. I hat to say it, but any progress in my understanding of these matters has to wait - until tomorrow! ;-)

Posted by: Urs Schreiber on May 18, 2004 7:07 PM | Permalink | Reply to this

Re: CFTs from OSFT?

Hello!

I said

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”.

I’m pretty sure I was wrong :)

Just because the projectors commute, does not mean that the algebra is commutative.

If we define

(1)${f}_{n}={P}_{n}*f$

then

(2)$f=\sum _{n=0}^{\infty }{f}_{n}$

Then (although it may be redundant) we could define

(3)${e}_{n}={P}_{n}*1$

so that

(4)$1=\sum _{n=0}^{\infty }{e}_{n}.$

I wrongly assumed that we could write

(5)${f}_{n}=c*{e}_{n}$

for some constant $c$. If this were possible, then the algebra of 0-forms would be commutative. But it doesn’t have to be true, so the algebra doesn’t have to be commutative.

Sorry! :)

Most of everything else I said should be ok though. It seems we can use these ${e}_{n}$ to define

(6)${e}_{\mathrm{mn}}={e}_{m}*{\mathrm{de}}_{n}*{e}_{n}$

which would be an edge from ${e}_{m}\to {e}_{n}$. Why does this remind me of some kind of superselection rule? :) ${e}_{m}$ is a string mode and ${e}_{n}$ is another string mode and ${e}_{\mathrm{mn}}\ne 0$ implies that ${e}_{m}$ can evolve to ${e}_{n}$ in one “step”. Maybe some vague ideas we’ve had for a while are starting to materialize? :)

Eric

Posted by: Eric on May 18, 2004 7:41 PM | Permalink | Reply to this

Re: CFTs from OSFT?

Hi Eric -

I happen to be visiting an internet café before going home, so that’s why you get yet another response! :-)

Yes, all what you write looks pretty good, or let me say, suggestive. We should definitly try to do something along these lines. The next task is to actually get in control of the details, which are numerous, it seems. ;-)

The first, but not the least, problem that comes to my mind is the question whether these projectors that you are talking about are really complete, which is a prerequisite if string field’s are to be expanded the way you imagine.

This is just a question, currently I have no clue! :-)

The interesting thing is that these projectors are supposed to correspond to string field configurations which describe branes, as far as I understand. (Now if they could also correspond to (-1)-branes somehow, that would be suggestive…)

Maybe this is not that way off, since in that paper which you are referring to it says that the projectors are not normalizable. This inconsistency would actually be conistent with their interpretation as delta-localized somethings, if you know what I mean.

I will go to bed now, then get up tomorrow, go to my office and ignore everything else until I begin to understand at least a little more about the 1001 ways to think about the OSFT star product.

Posted by: Urs Schreiber on May 18, 2004 9:24 PM | Permalink | Reply to this

Re: CFTs from OSFT?

Howdy! :)

I happen to be visiting an internet café before going home, so that’s why you get yet another response! :-)

Great! :)

The next task is to actually get in control of the details, which are numerous, it seems. ;-)

Details shmetails :)

The first, but not the least, problem that comes to my mind is the question whether these projectors that you are talking about are really complete, which is a prerequisite if string field’s are to be expanded the way you imagine.

I could be missing something, but doesn’t the fact that the projectors sum to the identity and their images are disjoint mean they are complete? In other words, I would be tempted to take the two relations

(1)${P}_{m}*{P}_{n}={\delta }_{\mathrm{mn}}{P}_{n}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\sum _{n=0}^{\infty }{P}_{n}=1$

as the definition of a complete set of orthogonal projectors.

Gotta run!

Eric

Posted by: Eric on May 18, 2004 10:40 PM | Permalink | Reply to this

Re: CFTs from OSFT?

Good evening!

I spent all night skimming about 30 papers on slivers, butterflies, and other star algebra projectors :) I won’t claim to understand more than 5% of what I read, but it is definitely interesting to get mixed up with something that is not ironed out yet. It means there is potential to do something original :)

I might be jumping the gun, but it seems like there exists technology to construct a “discrete” abstract differential calculus as I outlined above. The abstract differential calculus of string fields was known way back in ‘86 (if not earlier).

However, the existence of these orthogonal projectors seems relatively new (submitted to arxiv in August 2000). If you hand me an abstract differential calculus and a complete set of orthogonal projectors, we can simply forget about where it comes from and turn the crank following almost precisely what we did in our notes (generalized to handle noncommutative 0-forms).

The missing piece seems to me to be this $†$ operation satisfying

(1)$\left(A*B{\right)}^{†}={B}^{†}*{A}^{†}$

that is not confined to the matter sector.

Once we have this, we can define an inner product and integration of string fields as in our notes.

The paper you cite saying that the projectors are not normalizable seems a bit strange to me. The first thing I would question is how they define normalizable? To define whether an element is normalizable, you need an inner product. If they already have an inner product for which the projectors are non-normalizable, then who is to say they are using the correct inner product? If their inner product is correct, then they should be able to define a $†$ satisfying the desired properties. Do they have such a thing? I haven’t seen it :)

If we could successfully come up with a $†$ operation, then we could define an inner product from this. If the projectors are not normalizable with respect to this inner product, then I would start to be concerned (but only a little bit :))

Actually, it seems to me that we can actually define the inner product so that that the projectors have unit norm… by definition! :)

Define

(2)${e}_{m}*{e}_{n}={\delta }_{m,n}{e}_{n},$
(3)$\sum _{n=0}^{\infty }{e}_{n}=1,$
(4)$〈{e}_{m}\mid {e}_{n}〉={\delta }_{m,n},$

and

(5)$\int \Psi =〈1\mid \Psi 〉.$

Then

(6)$〈\Psi \mid \Phi 〉=〈1\mid {\Psi }^{†}*\Phi 〉=\int {\Psi }^{†}*\Phi .$

We proceed to define

(7)${e}_{\mathrm{mn}}={e}_{m}*{\mathrm{Qe}}_{n}*{e}_{n},$
(8)${e}_{\mathrm{lmn}}={e}_{\mathrm{lm}}*{e}_{\mathrm{mn}}={e}_{l}*{\mathrm{Qe}}_{m}*{e}_{m}*{\mathrm{Qe}}_{n}*{e}_{n},$

etc. Then based purely on the belief that beautiful results should have some place in physics, I’d speculate that string fields of the form

(9)$\Psi =\sum _{{n}_{1}...{n}_{p}}{\Psi }^{{n}_{1}...{n}_{p}}*{e}_{{n}_{1}....{n}_{p}}$

with ${\Psi }^{{n}_{1}...{n}_{p}}\in 𝒜$, should have some special meaning :)

Good night! :)

Eric

Posted by: Eric on May 19, 2004 4:23 AM | Permalink | Reply to this

Re: CFTs from OSFT?

Seems to me that your ideas here have their counterpart in equation (3.38) of Kawano&Okuyama’s hep-th/0105129. Note that the ${\Psi }_{n,m}$ are (complex) numbers. The noncommutativity sits in the $\mid n〉〉〈〈m\mid$.

Posted by: Urs Schreiber on May 19, 2004 12:16 PM | Permalink | Reply to this

Re: CFTs from OSFT?

Good morning! :)

I should really be on my way to work, but oh well :)

Seems to me that your ideas here have their counterpart in equation (3.38) of Kawano&Okuyama’s hep-th/0105129. Note that the ${\Psi }_{n,m}$ are (complex) numbers. The noncommutativity sits in the $\mid n〉〉〈〈m\mid$.

Hmm… I’ll have to think about how the two are related. It is not obvious to me they are the same. For them to be the same, I think we would need

(1)${e}_{m}*\Psi *{e}_{n}={\Psi }_{m,n}\mid m〉〉〈〈n\mid .$

Ooops! Gotta run! :)

More later,
Eric

Posted by: Eric on May 19, 2004 1:04 PM | Permalink | Reply to this

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)$f=\sum _{n=0}^{\infty }{\stackrel{←}{f}}_{n}*{e}_{n}=\sum _{n=0}^{\infty }{e}_{n}*{\stackrel{\to }{f}}_{n},$

where

(2)${\stackrel{←}{f}}_{n}=f*{e}_{n}$

and

(3)${\stackrel{\to }{f}}_{n}={e}_{n}*f.$

Due to noncommutativity, in general we have

(4)${\stackrel{←}{f}}_{n}\ne {\stackrel{\to }{f}}_{n}.$

Hmm… but then we have

(5)${\stackrel{←}{f}}_{n}=\sum _{m=0}^{\infty }{e}_{m}*{f}_{\mathrm{mn}}$

for some ${f}_{\mathrm{mn}}\in 𝒜$, then

(6)${\stackrel{\to }{f}}_{m}=\sum _{n=0}^{\infty }{f}_{\mathrm{mn}}*{e}_{n}.$

In other words (I know this is simple algebra :))

(7)$f=\sum _{\mathrm{mn}}{e}_{m}*{f}_{\mathrm{mn}}*{e}_{n}.$

That is kind of neat :)

With hind sight, I can write down the obvious relation

(8)${f}_{\mathrm{mn}}={e}_{m}*f*{e}_{n}.$

so that

(9)$f=\sum _{\mathrm{mn}}{f}_{\mathrm{mn}}$

:)

Then if

(10)$g=\sum _{\mathrm{mn}}{g}_{\mathrm{mn}},$

we have

(11)$f*g=\sum _{\mathrm{mln}}{f}_{\mathrm{ml}}*{g}_{\mathrm{ln}},$

which makes the relation to matrix multiplication pretty obvious. But then this suggests that multiplication on the left by ${e}_{n}$ projects the left half of the string and multiplication by ${e}_{n}$ 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

Posted by: Eric on May 19, 2004 5:10 AM | Permalink | Reply to this

Details

Eric, I see what you have in mind - but this is going too fast for me!

Let me show you how much behind I am: I am currently working through the paper you mentioned:

T. Kawano & K. Okuyama: Opne String Fields As Matrices (2001)

I like this a lot. One nice insight is that the SFT vertices in operator/oscillator language simplify immensely when written out in terms of the ‘local’ oscillators (2.6) instead of in the usual ${\alpha }_{n}^{i}$ in (2.1).

The idea of section 3.1 is also very nice. But I don’t quite get it yet: (3.6) is not the same as (3.1), due to the appearance of the step functions $\theta \left(\sigma \right)$, or is it?

Let this be the first in a long list of details that I’ll need to understand… :-)

Posted by: Urs Schreiber on May 19, 2004 11:47 AM | Permalink | Reply to this

Re: Details

Ah, the answer is giving below equation (3.24). So the text on p. 10 is a little misleading, but apart from that everything is fine.

Posted by: Urs Schreiber on May 19, 2004 12:05 PM | Permalink | Reply to this

Re: Details

The interesting thing is that these projectors are supposed to correspond to string field configurations which describe branes, as far as I understand. (Now if they could also correspond to $\left(-1\right)$-branes somehow, that would be suggestive…)

Maybe this isn’t that far off. On p. 18 of Kawano&Okuyama’s hep-th/0105129 it says indeed that the projectors that you, Eric, are considering here, correspond to instantons, i.e. objects localized in spacetime, smeared events so to say.

On the other hand, I am uncertain about the details here. From equation (4.18) of that paper it seems that this statement follows from the Gaussian dependence of the $a$-vacuum $\mid \Omega 〉$ (defined below (2.7)) on the zero-mode coordinate $\sim \int {X}^{i}\left(\sigma \right)\phantom{\rule{thinmathspace}{0ex}}d\sigma$.

Now the question is: Do we shift the index of ${P}_{i}\to {P}^{i}$ in (2.6) with the Minkwoski metric or with its Wick rotated form, the identity? Only in the latter case I would think that the projectors (4.18) are really localized in time. But according to equations (2.2)-(2.4) the authors mean to shift indices with the Minkwoski metric, as is also necessary for (2.14) etc. to make sense. Given this, I don’t understand why (4.18) should be localized in time.

Posted by: Urs Schreiber on May 19, 2004 12:38 PM | Permalink | PGP Sig | Reply to this

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)$Q\to Q+iϵ\left[\Phi ,\cdot {\right]}_{*}$

is equivalent to a deformation of the background.

However, this looks eerily like the replacement

(2)$d\to d+iϵ\left[A,\cdot \right]$

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)$d\to d+iϵ\left[A,\cdot \right]$

as a deformation of the background of the QED action

(4)$S=\int \left(\psi ,\partial \psi \right)vol-m\int \left(\psi ,\psi \right)vol-\frac{1}{2}\int \left(F,F\right)vol,$

where $\partial =d-{d}^{†}$?

Eric

Posted by: Eric on June 1, 2004 3:31 PM | Permalink | Reply to this

Re: Minimal Principle and Deformations

When you say ‘to account for interactions’ are you thinking of how a gauge field formally interacts with the fermions in the adjoint rep by way of the gauge covariant derivative of the form you indiceated?

If that’s the case then the answer is certainly: ‘Yes, that’s related.’ The shifted BRST operator of SFT is morally just a covariant derivative on string fields with the background string field giving the gauge connection.

I think that when you do this for superstring field theory, which contains fermion excitations, you can indeed talk about ‘deformations of the background of the QED action’ roughly as you have indicated. (The WZW like SFT action reproduces the YM+fermion effective field action to lowest order.) Unfortunately right now I cannot point you to any paper where the effective action after some level truncation for RNS superstring field theory is given. But it should look similar to the bosonic action in equation (2.46) of hep-th/0102085, with fermions included.

Unfortunately superstring field theory is quite a bit more technically involved than bosonic SFT, due to the presence of the superghosts, the different fermion sectors, etc. That’s why there are far less papers on SSFt than on bosonic SFT. But in order to have fermions, like in your idea, we have to consider SSFT.

(BTW, what do you mean by ‘minimal principle’? Do you mean the fact that classical solutions extremize the action?)

Posted by: Urs Schreiber on June 2, 2004 10:19 AM | Permalink | PGP Sig | Reply to this

Re: Minimal Principle and Deformations

Good morning :)

(BTW, what do you mean by ‘minimal principle’? Do you mean the fact that classical solutions extremize the action?)

Well, it is possible that I have been using this term incorrectly all these years. When I do a google search for “minimal principle”, all I find is references to myself from years ago :)

Basically, it is just the replacement of the exterior derivative with the covariant exterior derivative to account for coupling.

Ok ok. After some google surfing, I am convinced I’ve been using the term “minimal principle” incorrectly. From now on, I promise to use “minimal coupling” :)

So maybe I should restate some of my prior questions that probably seemed confusing :)

In the past, when I asked for the motivation behind the CS-like action for OSFT, you said

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)${Q}_{B}\Phi =0$

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

(2)$\int \Phi *{Q}_{B}\Phi ,$

where * is some product with respect to which ${Q}_{B}$ is graded Leibniz.

Next you need to add some interaction term, something like

(3)$\int \Phi *\Phi *\cdots *\Phi .$

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.

Instead of starting with a “free part of the action” and then trying to independently construct an interaction term, I think this minimal coupling idea should give you the coupling term automatically. So if you begin with the free part

(4)$S=\int \Phi *{Q}_{B}\Phi ,$

then minimal coupling tells you to make the replacement

(5)${Q}_{B}\to {Q}_{B}+iϵ\left[\Phi ,\cdot {\right]}_{*}$

giving

(6)$S=\int \Phi *{Q}_{B}\Phi +i2ϵ\int \Phi *\Phi *\Phi .$

This is the only form that would be consistent with minimal coupling.

Hmm… I just went back to look at the WZW-like version to see how minimal coupling would apply there, but it doesn’t seem to fit in at all. This is one reason I am skeptical about the WZW-like approach of Berkovits. What am I missing? :)

Eric

Posted by: Eric on June 2, 2004 3:41 PM | Permalink | Reply to this

Re: Minimal coupling and Deformations

Ah, minimal coupling! :-)

Minimal coupling is the right procedure to couple a gauge field to some fermion, for instance. But here we couple the ‘gauge field’ to itself. This gives a small modification in a prefactor. That’s because the main thing to take care of is gauge invariance. You have to choose the coupling in such a way that the resulting action is gauge invariant. The action S which you wrote down is not gauge invariant, but the true Chern-Simons action is. A similar statement holds true for the WZW action.

Posted by: Ur s on June 3, 2004 10:47 AM | Permalink | PGP Sig | Reply to this

Re: Minimal coupling and Deformations

Good morning! :)

Ah, minimal coupling! :-)

Sorry for the confusion :)

You have to choose the coupling in such a way that the resulting action is gauge invariant. The action $S$ which you wrote down is not gauge invariant, but the true Chern-Simons action is.

Hmm…

Aside from the fact I can set $iϵ=1/3$ so that I obtain the CS-like action, this statement makes me a little nervous. It sounds like you are saying that the numerical constant on the second term needs to be fixed in order for the action to be gauge invariant, but that numerical constant can be set to an arbitrary value just by rescaling $\Phi$. Wouldn’t that mean that the action is only gauge invariant for a particular scaling of $\Phi$? That doesn’t sound too good :)

Aside from that, I thought that Berkovits’ work basically points out that the CS-like action is sick anyway (which I was happy to hear someone else having the same opinion) :)

A similar statement holds true for the WZW action.

As far as WZW is concerned, when I look at the action, I don’t see a kinetic term (probably because I’m clueless as to what I’m looking at :)), so it is not obvious how minimal coupling will work there.

Minimal coupling is the right procedure to couple a gauge field to some fermion, for instance.

I usually don’t even think about fermions because I deal mostly with classical EM, but lately I have started thinking about them a little bit and I obviously gravitate toward the Dirac-Kaehler idea where fermions are really just inhomogeneous differential forms. Is there any well-known problem with DK that I should be aware of? If not, it seems to open a can of worms for me because I start to call into question the QED action because it seems it should be more uniform in the sense that the fermion is really just a gauge field of a different degree. These various gauge fields, it would seem, should be expressible in terms of some simple expression for which the QED action is a low energy approximation or something.

Don’t tell me. What I am looking for is string theory? :)

Eric

Posted by: Eric on June 3, 2004 2:40 PM | Permalink | Reply to this

Re: Minimal coupling and Deformations

this statement makes me a little nervous

Right! ;-) I was expressing my self really badly. You are right that there is an arbitrariness in how to choose the prefactor, by reabsorbing parts of it into the fields, or the other way round.

In any case, what one wants is an action of a gauge connection $\Phi$ such that it is invariant under the variation

(1)$\delta \Phi =d\Lambda +k\Phi \Lambda -k\Lambda \Phi$

for some constant $k$. The choice of $k$ should determine the choice of the prefactor in fron of the cubic term in the action.

The point that I had in mind is the following:

For a given $k$ in the above equation one is really considering the covariant derivative

(2)${d}_{\Phi }\psi =d\psi +k\left[\Phi ,\psi \right]$

but then the true CS action is not equal to $\int \Phi {d}_{\Phi }\Phi$ as one might expect.

As far as WZW is concerned, when I look at the action, I don’t see a kinetic term

True, in the WZW action (and correspondingly in Berkovits’ superstring field theory) the distinction between kinetic and interaction terms is not quite as clear as in the Chern-Simons/cubic bosonic SFT case. On the other hand, the equations of motion of WZW theory can be written (as in equation (3.16) of my notes) in a form very similar to the CS equations of motion.

Anyway, I think my point is that when you decide on the prefactor in the gauge transformation that you want the WZW action to be invariant under (equivalent to choosing a certain scaling of the fields) then the relative factor between the quadratic and the cubic term in the WZW action is fixed. (If you have access to it, see for instance equation (15.26) in the book ‘Conformal Field Theory’ by DiFrancesco et al.)

I obviously gravitate toward the Dirac-Kaehler idea where fermions are really just inhomogeneous differential forms. Is there any well-known problem with DK that I should be aware of?

Kähler originally wrote down the equation that carries his name for flat space. There the connection term in the Dirac operator vanishes and everything is fine: The equation describes ${2}^{\left[D/2\right]}$ ‘families’ of Dirac fermions (because on inhomogeneous forms you can represent ${2}^{D/2}$ copies of the Clifford algebra).

But when you turn on a gravitational field the Dirac-Kähler connection term will mix these ‘families’. This is a problem.

Some people have hoped that it could be turned into a virtue, because in reality we also observe several fermions which may interact -but they don’t seem to interact in a way that could be described by putting them into multiplets of the Dirac-Kähler equation.

You can find a detailed discussion of why the Dirac-Kähler operator mixes the spinors here.

But here is a question that you won’t find discussed in the literature, at least not explicitly:

What if we, in addition to gravity, turned on other fields, such that they cancel the mixing effect of the ‘gravitational’ (Levi-Civita) connection in the DK equation?

One (minor) point that I wanted to make in the papers hep-th/0311064 and hep-th/0401175 is that this is precisely what happens in string theory.

I emphasized the (in principle well-known) fact that the susy constraints of the superstring are nothing but the DK equation in loop space (which in particular reduces to a DK equation for the center of mass of the string, i.e. for its particle appearance) and that the reason that the undesirable ‘mixing’ is absent in string theory is precisely due to the fact that conformal invariance on the worldsheet puts certain restrictions on the background fields, such that the generalized DK connection preserves the above ‘families’.

For the special simple case where the string propagates on a group manifold, for which the background EOMs require that a torsion field has to be turned on, in addition to the curvature of the group, the torsion terms precisely cooperate with the LC-connection so as to preserve the DK-fermion families, so to say. In string language this is nothing but the fact that the worldsheet CFT splits into two chiral sectors.

So my claim has always been that string theory really makes the DK idea work, in this sense.

Don’t tell me. What I am looking for is string theory?

:-) Maybe you find it silly but: I often cannot help to agree with what Luboš sometimes said: If it is a good idea, then it is part of string theory. ;-)

Posted by: Urs Schreiber on June 4, 2004 1:14 PM | Permalink | PGP Sig | Reply to this

Re: Minimal Principle

Hi Urs! :)

(BTW, what do you mean by ‘minimal principle’? Do you mean the fact that classical solutions extremize the action?)

Ah hah! I knew I didn’t make up the term ‘minimal principle’, but the mere fact that you asked me this question tells me that you are lacking one very valuable book from your collection :)

If I could make only one recommendation for a book you should read, it would without a doubt be

The Dawning of Gauge Theory
Lochlainn O’Raifeartaigh

This book is a BEAUTIFUL gem :) You MUST get a copy of this book! Not only that, you must get a copy of this book and make it required reading for all of your students: present and future :)

The book is very light reading. In fact, you could probably get through the whole thing in an evening or two and it would be time well spent. This morning, I reread the first 5 chapters and I loved it even more than the first time I read it :)

The only thing is, it is definitely a cliff hanger. It leaves off right where string theory begins. It would be absolutely wonderful if O’Raifeartaigh could fill in the gap from where he left off to the present including string theory :)

Get the book and pass it on! :)

Eric

PS: Be forewarned, I have a whole new slew of ideas/questions coming :)

Posted by: Eric on June 5, 2004 8:46 PM | Permalink | Reply to this

Re: Minimal Principle

Be forewarned, I have a whole new slew of ideas/questions coming

Ok, let’s see the first one!

Posted by: Urs Schreiber on June 7, 2004 2:32 PM | Permalink | PGP Sig | Reply to this

Re: Minimal Principle

Good afternoon!

Ok, let’s see the first one!

Okie dokie! :)

The first thought that has been on my mind relates to the original question I asked in the beginning of this thread

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

(1)${\stackrel{˜}{Q}}_{B}={e}^{f\left(\stackrel{̂}{\Phi }\right)}{Q}_{B}{e}^{-f\left(\stackrel{̂}{\Phi }\right)},$

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

Although I didn’t really understand it at the time, I think you responded in the affirmative when you wrote

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

(2)${d}_{A}\to {g}^{-1}{d}_{A}g$

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

(3)$\omega \to {g}^{-1}\omega g+{g}^{-1}\left(\mathrm{dg}\right)$

I still don’t fully understand this, but it reminds me of what I read in Dawning of Gauge Theory. There, O’Raifeartaigh says that Schrodinger’s prescription for the minimal principle

(4)$d\to d+iϵA$

while keeping the wave function $\psi$ the same is equivalent to London’s prescription

(5)$\psi \to U\psi$

while keeping the exterior derivative $d$ the same, where

(6)$U\left(x\right)=\mathrm{exp}\left(iϵ{\int }_{{x}_{0}}^{x}A\right).$

I’m on the verge of proclaiming my preference for London’s prescription because of my belief that the exterior derivative is sacred and should not be deformed since the exterior derivative should rightly be considered as the transpose of the boundary map. It would seem silly to deform the boundary map, so why should it not seem silly to deform the exterior derivative? :) Oh well, I could get sidetracked here so let me jump to another thought…

This reminds me of a conversation we had before. In EM, we often work with phasors (no, it’s not from Star Trek!) that tend to simplify computations. Engineers tend to think that the electromagnetic field is a real (as opposed to complex) space and time dependent quantity. For time harmonic, a.k.a. sinusoidal, excitations $A\left(x,t\right)$, the phasor $\stackrel{˜}{A}\left(x\right)$ is defined as that complex quantity satisfying

(7)$A\left(x,t\right)=\Re \left({e}^{i\omega t}\stackrel{˜}{A}\left(x\right)\right),$

where $\omega$ is the angular frequency of the excitation. Although I haven’t seen it written down anywhere, I thought it would be interesting to consider differential forms as phasors for describing time harmonic fields. Now for $A$ any time harmonic $p$-form, we have what I call a $p$-phasor $\stackrel{˜}{A}$ defined by

(8)$A=\Re \left({e}^{i\omega t}\stackrel{˜}{A}\right).$

Then I spent some time and wrote up some notes to study the differential geometry of differential phasors. It was pretty fun! :)

One of the obvious things to do is to look at the exterior derivative

(9)$\mathrm{dA}=\Re \left[d\left({e}^{i\omega t}\stackrel{˜}{A}\right)\right]=\Re \left[{e}^{i\omega t}\left(d+i\omega \mathrm{dt}\right)\stackrel{˜}{A}\right)\right],$

which begs for the definition

(10)$\stackrel{˜}{d}=d+i\omega \mathrm{dt}.$

Since $\stackrel{˜}{A}$ is independent of time, the $d$ and $\stackrel{˜}{d}$ on the right-hand side may be thought of as operators in space rather than spacetime. This observation made me think of the covariant exterior derivative and made me speculate that the covariant exterior derivative was really just the projection of an exterior derivative in some larger space down to a lower dimensional space. I guess this idea isn’t so crazy after all, which is discussed in O’Raifeartaigh. Unfortunately, I can’t claim to have thought of it on my own since I did read O’Raifeartaigh several years ago even though I didn’t understand it at the time. Yet another example where I think I come up with an idea that I had actually seen years earlier in a different context :) The difference is that maybe now I am starting to understand it :)

Ok. I think this is probably pretty obvious to you and everyone else who has ever picked up a book on differential geometry, but maybe I can say it in a weird enough way that it might be interesting :)

If you have a Lie group $G$ and a manifold $M$ you can consider the manifold $G×M$. Gauge theory is basically doing the same thing to more general fields what engineers do when they study phasors! I am pretty happy to have understood this finally (sorry… I am typing while I’m learning when I should probably think things through before putting things down, but that is no fun :))

A gauge field is really just a field living on $G×M$ (at least locally, I am avoiding the unnecessary complications of more general bundles). If we project this down to $M$, it looks like a field with value in $G$. Or something :)

I’ve got to run and this note is hardly worth public viewing, but I’ll submit it anyway and try to pick up later and make some sense out of it.

The point I am after is that if a (real) field $\psi$ has a symmetry under some Lie group $G$, then the field can be expressed as

(11)$\psi =tr\left[\mathrm{exp}\left(iϵ\int A\right)\stackrel{˜}{\psi }\right],$

where the (complex) field $\stackrel{˜}{\psi }$ is what we usually deal with and

(12)$d\psi =tr\left[\mathrm{exp}\left(iϵ\int A\right)\left(d+iϵA\right)\stackrel{˜}{\psi }\right]$

so that

(13)$\stackrel{˜}{d}=d+iϵA.$

Eric

PS: It is kind of cute to note that for the case of $U\left(1\right)\cong \mathrm{SO}\left(2\right)$ we have $\Re \cong \mathrm{tr}$.

Posted by: Eric on June 7, 2004 10:51 PM | Permalink | Reply to this

Re: Minimal Principle

Good morning :)

I know that the last post didn’t really seem to have a question mark on it, but it seems I could probably use some stearing in the right direction :)

I am trying to understand why a deformation of the exterior derivative

(1)$d\to d+iϵA$

seems to work so well for introducing interactions. If we could move the deformation of $d$ to a deformation of some other structure, I’d be happy. This is why I started to embrace London’s prescription, but I must be missing something. If we replace

(2)$\psi \to U\psi ,$

then we have

(3)$d\left(U\psi \right)=\left(\mathrm{dU}\right)\wedge \psi +U\wedge d\psi =U\left[d+{U}^{-1}\left(\mathrm{dU}\right)\right]\psi$

so that

(4)$\stackrel{˜}{d}=d+{U}^{-1}\left(\mathrm{dU}\right).$

This looks encouraging, but it doesn’t seem to be enough to explain

(5)$d\to {d}_{A}=d+iϵA$

in general. The above prescription (involving $U$) seems to only account for pure gauge transformations, i.e. where $A$ is exact (or at most closed).

However, if we combine the two ideas, we get

(6)${d}_{A}\left(U\psi \right)=U\left[d+iϵ{U}^{-1}AU+{U}^{-1}\left(\mathrm{dU}\right)\right]$

which looks suspciously like what you said above

(7)$\omega \to {U}^{-1}\omega U+{U}^{-1}\left(\mathrm{dU}\right),$

where $\omega =iϵA$.

So it seems that the idea $\psi \to U\psi$ is not enough to account for the general covariant exterior derivative, but only the case when $A$ is exact.

Ok, here is a neat little calculation. I had the idea that maybe my transformation wasn’t the correct one so I wondered, what if we should treat $\psi$ as an operator and transform it accordingly as

(8)$\psi \to U\psi {U}^{-1}.$

Something kind of neat happens. Since ${U}^{-1}U=1$ we have

(9)$d\left({U}^{-1}U\right)=\left({\mathrm{dU}}^{-1}\right)U+{U}^{-1}\left(\mathrm{dU}\right)=0\to {U}^{-1}\left(\mathrm{dU}\right)=-\left({\mathrm{dU}}^{-1}\right)U$

and

(10)$d\left(U\psi {U}^{-1}\right)=U\left\{d\psi +\left[{U}^{-1}\left(\mathrm{dU}\right),\psi \right]\right\}{U}^{-1}$

where $\left[\cdot ,\cdot \right]$ is the graded commutator so that

(11)$\stackrel{˜}{d}=d+\left[{U}^{-1}\left(\mathrm{dU}\right),\cdot \right].$

It’s kind of nice to see the commutator appearing :)

Ok. A quick verification shows that

(12)${d}_{A}\left(U\psi {U}^{-1}\right)=U\left\{d+iϵ\left[{U}^{-1}\mathrm{AU},\psi \right]+\left[{U}^{-1}\left(\mathrm{dU}\right),\psi \right]\right\}{U}^{-1}$

giving

(13)$\stackrel{˜}{{d}_{A}}={d}_{\stackrel{˜}{A}},$

where

(14)$\stackrel{˜}{\omega }={U}^{-1}\omega U+{U}^{-1}\left(\mathrm{dU}\right)$

as before so everything seems to be holding together.

Ok. My notation is a little sloppy and I am too lazy to fix it, but a summary might help. If we define

(15)$\psi =U\stackrel{˜}{\psi }{U}^{-1}$

then

(16)${d}_{A}\psi =U\left(\stackrel{˜}{{d}_{A}}\stackrel{˜}{\psi }\right){U}^{-1},$

where

(17)$\stackrel{˜}{{d}_{A}}={d}_{\stackrel{˜}{A}},$
(18)$\stackrel{˜}{\omega }={U}^{-1}\omega U+{U}^{-1}\left(\mathrm{dU}\right),$

and $\omega =iϵA$.

Sorry for stumbling around like this. My goal is to better understand how gauge theory arises from string field theory and how deformations fit into the whole picture. Unfortunately, I only know abelian gauge theory very well so I’m teaching myself non-abelian gauge theory and SFT concurrently :)

Nonetheless, I do think that the relations to phasors is kind of neat. In the above, I could have easily defined

(19)$\psi =tr\left(U\stackrel{˜}{\psi }{U}^{-1}\right)$

and everything would seem to go through, where now $\psi$ is just a regular $p$-form and $\stackrel{˜}{\psi }$ is a complex $p$-phasor :)

Eric

Posted by: Eric on June 8, 2004 4:24 PM | Permalink | Reply to this

Re: Minimal Principle

Hi Eric -

yes! That is looking good. You have apparently rediscovered what it means to gauge a theory. Namely what one does is see how the (exterior) derivative deforms under a symmetry transformation, noting that it acquires a term of the form ${g}^{-1}\left(dg\right)$ and then one says: Hey, this looks like a connection! But in this form it is always a flat connection. So let’s also allow non-flat connections which don’t come from a simple gauge transformation.

This way one ends up with the covariant exterior derivative ${d}_{A}=d+A$, where $A$ is some connection 1-form taking values in the respective Lie algebra.

What you write about phasors is an example of a special $U\left(1\right)$ gauge transformation, one that only depends on the time coordinate. I don’t think that this can be interpreted as projecting something from some product space down to one of the factors, since the differential always knows where it comes from. I.e. in your formula $\stackrel{˜}{d}=d+i\omega \phantom{\rule{thinmathspace}{0ex}}\mathrm{dt}$ with $d$ purely spatial you just have the time-Fourier transform of the full exterior derivative on spacetime.

Concerning the question whether one can move the deformation from the exterior derivative to something else: This does work, as we have discussed at length, if the deformed $d$ is a similarity transform of the original one. Otherwise I don’t see how it could work.

It is important to keep in mind that in our previous discussion which was concerned with CFT deformations the deformations of the exterior derivative that we talked about were all of the form of a similarity transformation $d↦{A}^{-1}\circ d\circ A$. The reason why not all these deformations were pure gauge was that the formalism we needed in that stringy context required to deform ${d}^{†}$ by a different similarity transformation, namely the adjoint one. This way the total deformation was not pure gauge.

It is therefore really crucial to cleanly distinguish the various notions of exterior derivatives that may occur. Mind the hierarchy of exterior derivatives! :-)

For instance, I am claiming that when open strings propagate on a space filling D-brane, that turning on a background field corresponds to a deformation of the BRST operator that is not a similarity transformation, but that at the same time (when the boundary conditions at the brane are kept fix) this non-similarity deformation of the BRST operator goes along with two different similarity transformations of the exterior derivatives on loop space that enter as partial derivatives in the superstring BRST operator.

Posted by: Urs Schreiber on June 8, 2004 7:14 PM | Permalink | PGP Sig | Reply to this

Re: Minimal Principle

Urs! :)

Thanks for your response :)

Hey, this looks like a connection! But in this form it is always a flat connection. So let’s also allow non-flat connections which don’t come from a simple gauge transformation.

Ok. I am making progress it seems. By that trick I was hoping to arrive at the covariant exterior derivative, but I only arrive at it for a flat connection. The final piece for me to understand before I can move on is why a non-flat connection works so well for introducing interactions. It still seems artificial to me. The basis for my concern is that Stokes’ theorem

(1)${\int }_{ℳ}d\alpha ={\int }_{\partial ℳ}\alpha$

is too beautiful not to play a fundamental role in nature and this expression involves $d$ and not ${d}_{A}$! :)

I thought I was on the verge of a breakthrough when I saw the expression

(2)${d}_{A}=Ud{U}^{-1}=d+A$

in O’Raifeartaigh, but my recent calculations suggest that this only works if $A$ is flat. Is that correct? Too bad I don’t have the text here with me or I could do a sanity check. Maybe I’m remember things wrong.

Then again, is it possible that

(3)${d}_{U}=Ud{U}^{-1}$

is the most fundamental way to express the covariant exterior derivative anyway? That would make a lot of sense to me and would pave the way for a discrete translation as well.

Of course, if this is possible, then that would make me very happy because, as you said, we could then shift this deformation off $d$ and onto the inner product (where deformations should be anyway) :)

I don’t think that this can be interpreted as projecting something from some product space down to one of the factors, since the differential always knows where it comes from. I.e. in your formula $\stackrel{˜}{d}=d+i\omega \mathrm{dt}$ with $d$ purely spatial you just have the time-Fourier transform of the full exterior derivative on spacetime.

Yes. That is exactly my point. It really is the full exterior derivative on spacetime, but you can look at it from the viewpoint of space as an operator on space only. The 1-form $\mathrm{dt}$ then becomes something like a place-holder for a scalar function, i.e. the coefficient in front of $\mathrm{du}=i\omega \mathrm{dt}$ is thought of as a scalar function on space rather than the fourth component of a 1-form in spacetime. In this way, all fields look like they are defined on space and all operators look like they are also only defined on space. We know that they are really fields and operators on spacetime, but they can be viewed as being purely spatial. This (admittedly strange) viewpoint is what I meant by projecting to some lower dimensional space. I should quote the pertinent section from O’Raifeartaigh that made me start thinking this way (when I get it in my hands again).

It is therefore really crucial to cleanly distinguish the various notions of exterior derivatives that may occur. Mind the hierarchy of exterior derivatives!

I am! :) That is why I am bothering with all this stuff at the lower rung on the hierarchical ladder because I intend to carry it up the hierarchy with me :)

For instance, I am claiming that when open strings propagate on a space filling D-brane, that turning on a background field corresponds to a deformation of the BRST operator that is not a similarity transformation, but that at the same time (when the boundary conditions at the brane are kept fix) this non-similarity deformation of the BRST operator goes along with two different similarity transformations of the exterior derivatives on loop space that enter as partial derivatives in the superstring BRST operator.

Neat :) This is the first I’ve heard you talk about this. It sounds like you are making some progress. However, I wonder if you should heed your own warning. Mind the hierarchy! :)

It sounds encouraging that you get similarity transforms on the lower rungs (which is what you want I think), but the fact that the deformation at the higher rung is not a similarity transform sounds a little worrisome to me for some reason. Could it be that you are not working with the right form of the BRST operator?

Mind the hierarchy! :)

Eric

Posted by: Eric on June 8, 2004 8:19 PM | Permalink | Reply to this

Re: Minimal Principle

Then again, is it possible that

(1)${d}_{U}=Ud{U}^{-1}$

is the most fundamental way to express the covariant exterior derivative anyway?

This obviously cannot be correct because then

(2)${d}_{U}^{2}=0$

always, which is another way to say that ${d}_{U}$ is always flat.

Hmmm…

A guess would be to modify this to

(3)${d}_{U}=Ud{U}^{†}$

which has the nice property of making it obvious that a unitary deformation is a pure gauge transformation, i.e. if $U$ is unitary, then ${d}_{U}$ is flat. That is kind of neat I guess, but now I am grasping for straws :)

Eric

Posted by: Eric on June 8, 2004 8:50 PM | Permalink | Reply to this

Re: Minimal Principle

Good morning!

I had written a little post the other night, right before going to bed and I see now that I forgot to hit the submit button before shutting down the computer. Doh! :)

It’s probably just as well because now I see I was perhaps trying to make connections where there might not supposed to be any :)

Basically, I had commented on the trivial fact that if you define

(1)$U=\mathrm{exp}\left(W\right),$

then we have

(2)${U}^{†}=\mathrm{exp}\left({W}^{†}\right)$

and

(3)$U\mathrm{unitary}⇔W\mathrm{anti}-\mathrm{Hermitian}.$

I then made the leap that I should compare this to my guess

(4)${d}_{U}=Ud{U}^{†}$

and made the connections

(5)${d}_{U}\mathrm{flat}⇔U\mathrm{unitary}⇔W\mathrm{anti}-\mathrm{Hermitian}⇔W\mathrm{pure}\mathrm{gauge}.$

I don’t know if there is any truth to this, but it seems to make a little sense.

Basically, I am trying to find another way to think of the (non-flat) covariant exterior derivative as a deformation of the exterior derivative in a similar way that you deformed the exterior derivative to give those dualities (and more) a while back. A similarity transform doesn’t cut it because then ${d}_{U}$ would always be flat. The next obvious thing (for me) to try was

(6)${d}_{U}=Ud{U}^{†},$

which seems to make a little sense when $U$ is unitary, because it seems to resonate with what you did before, but the interesting case would be when $U$ is not unitary. This would give a non-flat covariant exterior derivative and a corresponding non-gauge transformation, i.e. a true physical deformation.

I don’t know why, but something just doesn’t feel right about writing

(7)$d\to {d}_{A}=d+A.$

Algebra is neat, geometry is neat, but most profound things in physics lay where algebra and geometry overlap. The above replacement seems too algebraic without an obvious geometrically interpretation.

I had similar complaints about you deforming

(8)$d\to {d}_{X}=d+{i}_{X}$

but I forgave you when I learned that you can think of ${d}_{X}$ as the “square root” of the Lie derivative ${ℒ}_{X}$ :)

Maybe there is a similar thing here where I can also forgive you. Actually, now that I think of it, maybe there is. ${d}_{A}$ is the “square root” of the curvature!

Hmm… maybe my fears have been misplaced afterall :)

I see a pattern. We have geometric operators $F$ (curvature), $△$ (Laplace-Beltrami), and ${ℒ}_{X}$ (Lie). Something interesting seems to happen when you find their “square roots”, i.e.

(9)$F\stackrel{\text{"square root"}}{⇒}d+A$
(10)$△\stackrel{\text{"square root"}}{⇒}d+{d}^{†}\text{(Dirac-Kaehler)}$
(11)${ℒ}_{X}\stackrel{\text{"square root"}}{⇒}d+{i}_{X}$

Is that somehow profound? :)

Eric

Posted by: Eric on June 10, 2004 3:54 PM | Permalink | Reply to this

Re: Minimal Principle

Good morning!

I had written a little post the other night, right before going to bed and I see now that I forgot to hit the submit button before shutting down the computer. Doh! :)

It’s probably just as well because now I see I was perhaps trying to make connections where there might not supposed to be any :)

Basically, I had commented on the trivial fact that if you define

(1)$U=\mathrm{exp}\left(W\right),$

then we have

(2)${U}^{†}=\mathrm{exp}\left({W}^{†}\right)$

and

(3)$U\mathrm{unitary}⇔W\mathrm{anti}-\mathrm{Hermitian}.$

I then made the leap that I should compare this to my guess

(4)${d}_{U}=Ud{U}^{†}$

and made the connections

(5)${d}_{U}\mathrm{flat}⇔U\mathrm{unitary}⇔W\mathrm{anti}-\mathrm{Hermitian}⇔W\mathrm{pure}\mathrm{gauge}.$

I don’t know if there is any truth to this, but it seems to make a little sense.

Basically, I am trying to find another way to think of the (non-flat) covariant exterior derivative as a deformation of the exterior derivative in a similar way that you deformed the exterior derivative to give those dualities (and more) a while back. A similarity transform doesn’t cut it because then ${d}_{U}$ would always be flat. The next obvious thing (for me) to try was

(6)${d}_{U}=Ud{U}^{†},$

which seems to make a little sense when $U$ is unitary, because it seems to resonate with what you did before, but the interesting case would be when $U$ is not unitary. This would give a non-flat covariant exterior derivative and a corresponding non-gauge transformation, i.e. a true physical deformation.

I don’t know why, but something just doesn’t feel right about writing

(7)$d\to {d}_{A}=d+A.$

Algebra is neat, geometry is neat, but most profound things in physics lay where algebra and geometry overlap. The above replacement seems too algebraic without an obvious geometrically interpretation.

I had similar complaints about you deforming

(8)$d\to {d}_{X}=d+{i}_{X}$

but I forgave you when I learned that you can think of ${d}_{X}$ as the “square root” of the Lie derivative ${ℒ}_{X}$ :)

Maybe there is a similar thing here where I can also forgive you. Actually, now that I think of it, maybe there is. ${d}_{A}$ is the “square root” of the curvature!

Hmm… maybe my fears have been misplaced afterall :)

I see a pattern. We have geometric operators $F$ (curvature), $△$ (Laplace-Beltrami), and ${ℒ}_{X}$ (Lie). Something interesting seems to happen when you find their “square roots”, i.e.

(9)$F\stackrel{\text{"square root"}}{⇒}d+A$
(10)$△\stackrel{\text{"square root"}}{⇒}d+{d}^{†}\text{(Dirac-Kaehler)}$
(11)${ℒ}_{X}\stackrel{\text{"square root"}}{⇒}d+{i}_{X}$

Is that somehow profound? :)

Eric

Posted by: Eric on June 10, 2004 4:07 PM | Permalink | Reply to this

Re: Minimal Principle

Hi Eric -

it seems that now you are asking the question:

Given any gauge covariant exterior derivative ${d}_{A}=d+A$ with $A$ a connection 1-form, is it possible to write it as ${d}_{A}=R\circ d\circ S$, where $R$ and $S$ are some 0-forms. This would imply ${d}_{A}=d+R\left(\mathrm{dS}\right)$ and hence that the connection $A$ is expressed in terms of $R$ and $S$ as $A=R\left(\mathrm{dS}\right)$.

So probably you are asking if for any given connection 1-form there are 0-forms $R$ and $S$ such that $A=R\left(\mathrm{dS}\right)$.

I think this can be true only for special cases of $A$. Of course every 1-form can be written as a sum of exact forms multiplied by 0-forms: $A={\sum }_{i}{r}_{i}\left(d{s}_{i}\right)$. But only for special cases will all but 1 term in this sum vanish.

Actually, I have been thinking about a very similar problem recently: As we have discussed before, for the closed string turning on a background gauge field is a pure gauge transformation under which both exterior and co-exterior derivatives on loop space transform as $extd↦{U}^{-1}extdU$ and $coextd↦{U}^{-1}coextdU$. This makes sense, since the closed string does not couple to the gauge field non-trivially.

But the open string does. The two endpoints of the open string pretty much behave like two Dirac particles that couple to the gauge field by way of the minimal coupling procedure. So here pretty much precisely the issues that you raised come up. The question is: What deformation of the (co)exterior derivative on loop space corresponds to this coupling? I am not quite sure yet that I fully understand how it works, but you can have a look at what is currently section 3.6 ‘On BSCFT deformations in loop space’ in the newest version of my OSFT notes.

Posted by: Urs Schreiber on June 10, 2004 4:10 PM | Permalink | Reply to this

Re: Minimal Principle

Good morning! :)

Sorry for posting that twice. The first time, I got an internal server error and reposted without thinking to check if it actually made it through the first time :)

I’ll think about what you are saying, but I’d like to clarify something real quick.

At first, I was trying to find a deformation of $d$ that gave the non-flat covariant exterior derivative because I didn’t like the replacement

(1)$d\to d+A.$

However, towards the end, I convinced myself not to worry about this. I simply need to change the way I think about it. In other words, I should not think of

(2)$d+A$

as a deformation of $d$ any more than I should think of

(3)$d+{i}_{X}$

or

(4)$d+{d}^{†}$

as a deformation of $d$. Although it is little more than an issue of semantics, I now think I should think of $d+A$ as the “square root” of the curvature instead of a deformation of $d$. This way, I can still treat $d$ as being the sacred operator making Stokes’ theorem valid. The others are merely different operators constructed from my sacred $d$ :)

Let me point out again what I think seems “profound” :)

The three most profound geometrical operators have “square roots” that are playing crucial roles in everything you have been doing over the past several months (years :)).

(5)${F}_{A}⇒d+A$
(6)$△⇒d+{d}^{†}$
(7)${ℒ}_{A}⇒d+{A}^{†}.$

This seems profound to me :) By your reaction, I’m guessing you might not feel the same? :)

Eric

Posted by: Eric on June 10, 2004 4:41 PM | Permalink | Reply to this

Re: Minimal Principle

By your reaction, I’m guessing you might not feel the same?

Heh, no, I hadn’t reacted to that message yet at all! :-)

It was by coincidence that my last comment appeared after your own comment on the comment before. I had written it hours ago and then something happend and I didn’t get to hit ‘post’.

I do think that the relations ${F}_{A}=\left(d+A{\right)}^{2}$, $\Delta =\left(d+{d}^{†}{\right)}^{2}$, ${ℒ}_{X}=\left(d+{\iota }_{x}{\right)}^{2}$ are profound.

That has to do with supersymmetry, especially with susy in 1+1 dimensions. Depending on your point of view, supersymmetry explains why these relations are nice or, the other way round, the profoundness of these relations explains why supersymmetry is a nice idea! ;-)

More precisely, when we are dealing with operators $A$ such that $\left(d+A+{\iota }_{X}\right)$ still squares to the Lie derivative, i.e. $\left(d+A+{\iota }_{X}{\right)}^{2}={ℒ}_{X}$, then $d+A+{\iota }_{X}$ together with its adjoint are the global susy generators of an $N=1$ $D=\left(1,1\right)$ susy algebra (in $D=2$ a spinor has two components, so $N=1$ supersymmetry means there are $N×2=2$ susy generators) and their squares

(1)${ℒ}_{X}=\left(d+A+{\iota }_{X}{\right)}^{2}$
(2)$\Delta =\left\{d+A+{\iota }_{X},\left(d+A+{\iota }_{X}{\right)}^{†}\right\}$

are the generators of spatial translation (along the vector field $X$) and the generator of time translation (the Hamiltonian $H=\delta$), respectively.

I think that this simple fact is pretty far reaching.

Posted by: Urs Schreiber on June 10, 2004 5:13 PM | Permalink | PGP Sig | Reply to this

Re: Minimal Principle

Sorry, there were some typos:

I meant $D=1+1$ instead of $D=\left(1,1\right)$ and $H=\Delta$ instead of $H=\delta$.

Posted by: Urs Schreiber on June 10, 2004 5:19 PM | Permalink | PGP Sig | Reply to this

Re: Minimal Principle

Sorry, there were some typos:

I meant $D=1+1$ instead of $D=\left(1,1\right)$ and $H=\Delta$ instead of $H=\delta$.

Posted by: Urs Schreiber on June 10, 2004 5:36 PM | Permalink | PGP Sig | Reply to this

Re: Minimal Principle

Wow, I just realized that Eric is completely right with his intuition that there should be a unitary transformation even for the gauge field background! The key to enlightment is the technique of boundary state deformation as used in

Koji Hashimoto: Generalized supersymmetric boundary state (2000).

The author discusses, coming from a completely different direction, precisely the unitary transformation which describe the gauge field background and the B-field backrgound as I have discussed in hep-th/0401175 for the closed string. The key seems to be that when this unitary operator is applied to the boundary state only, this does not imply a complete isomorphism of the entire formalism and hence is gauge-non-trivial.

I am glad that I have found this paper, which seems to have been waiting for me all along… ;-)

Posted by: Urs Schreiber on June 10, 2004 9:34 PM | Permalink | PGP Sig | Reply to this

Re: Minimal Principle

Urs! :)

Wow, I just realized that Eric is completely right with his intuition that there should be a unitary transformation even for the gauge field background!

Well, if you throw enough darts at random the way I do, there is bound to be one that at least hits the dart board once in a while :) I suspect you’re being too nice, but I’ll go ahead and frame the paragraph anyway :)

I am glad that I have found this paper, which seems to have been waiting for me all along…

Well, maybe if you like it that much you might be interested in summarizing it a little bit? :) My first 2 minute pass over it didn’t reveal anything that looks like

(1)${d}_{U}=Ud{U}^{†},$

but I admit, the notation is so horrible that I can’t make much out of it. Just imagine if the alphabet consisted of only two symbols: 0 and 1. Imagine how difficult literature would be. That is how I feel about the notation string theorists use. It is like 0’s and 1’s trying to describe Shakespeare or something :)

Cheers! :)
Eric

Posted by: Eric on June 11, 2004 1:58 AM | Permalink | Reply to this

Re: Minimal Principle

Good morning!

there should be a unitary transformation even for the gauge field background!

I am beginning to think that this is hopelessly beyond my ability to comprehend, but I’ll try a little more.

First, I misunderstood what you meant by “unitary transformation.” I know that a similarity transformation is of the form

(1)$S\prime =US{U}^{-1}.$

When I guessed a transformation of the form

(2)${d}_{U}=Ud{U}^{†},$

I was wondering what this type of transformation might be called. When you mentioned “unitary transformation,” I thought, “Ah ha! That is what it’s called.” This was verified by a quick check on mathworld

Mathworld - Unitary Transformation

Now, it seems to me that what you mean by a unitary transformation is a similarity transformation via a unitary matrix so that

(3)$d\to {d}_{U}=Ud{U}^{†}=d+U\left({\mathrm{dU}}^{†}\right),$

where $U$ is unitary. Then maybe somehow with something to do with “boundary states,” you can obtain nontrivial gauge fields? :)

The paper seems to transform the states, where I’m transforming the operators, which reminds me of Heisenberg vs Schrodinger :)

I know your busy and I also know I am making even less sense than I usually do because I can never spend more than 5 minutes thinking about this stuff and the rest of my time is spent thinking of mundane stuff that dulls my brain :) Don’t feel obligated to respond anytime soon :)

Ciao!
Eric

Posted by: Eric on June 15, 2004 3:04 PM | Permalink | Reply to this

Re: Minimal Principle

Hi Eric -

sorry for being vague and unclear.

By unitary transformation I am thinking of the action of some unitary operator $U$, ${U}^{†}={U}^{-1}$. This is either on states of the form $\mid \psi 〉↦U\mid \psi 〉$ or on other operators in the form $A↦UA{U}^{†}=UA{U}^{-1}$.

The interesting insight that I was referring to is roughly the following:

We had seen that turning on a gauge field in closed string theory resulted in a unitary transformation of the constraint algebra, where the unitary operator is the Wilson line of the gauge field along the string. This was good, because for the closed string turning on a gauge field is always just an equivalent redefinition of the same physics. Formally, this is of course due to the fact that the inner product becomes $〈1\mid {U}^{†}\cdots U\mid 1〉$ where $\mid 1〉$ denotes some kind of ‘vacuum’ state and by similarly transforming the terms indicated by $\cdots$ it just remains invariant.

Now in string theory you can think of this inner product as the gluing of a closed string propagating in the form of a hemisphere from the right to the left with a closed string propagaing on the opposite hemisphere from the left to the right, roughly. The point is that one can describe open strings by ‘cutting open the closed string inne product’, i.e. by just using the ‘boundary state’ $U\mid 1〉$ on the right, but not on the left. So this implies to use an inner product of the form $〈1\mid \cdots U\mid 1〉$.

This comes from a unitary transformation of only one half instead of both halfs. And therefore this transformation is cannot be undone by a mere redefinition of the ‘$\cdots$’ terms and hence it describes a genuinely new physical situation: An open string coupled to a gauge field.

This was explicitly shown to be true by Hashimoto and recently in more detail by Maeda, Nakatsu and Oonishi. The details require some details of string theory, but the main idea is pretty much as sketched above.

Posted by: Urs Schreiber on June 16, 2004 10:50 AM | Permalink | PGP Sig | Reply to this

Re: Minimal Principle

Hi Urs,

Now in string theory you can think of this inner product as the gluing of a closed string propagating in the form of a hemisphere from the right to the left with a closed string propagaing on the opposite hemisphere from the left to the right, roughly.

Does this have anything to do with the fact that string theory is a CFT so (somehow maybe) you can map the worldsheet of a string to a hemisphere? The inner product is gluing two hemispheres together into a single sphere and this sphere somehow evaluates to a number? If so, this sounds eerily reminiscent of the inner product in spin networks. There a “closed” spin network , i.e. one without loose edges, evaluates to a number. A spin network with loose edges is handled by making a duplicate copy of the spin network and attaching corresponding loose edges so the conglomerate has no loose ends and thus evaluates to a number. It sounds like the inner product in string theory does something similar. It glues the worldsheets of two string states together into some kind of a closed comglomerate that evaluates to a number. Is this anywhere in the vicinity of the truth? :)

So this implies to use an inner product of the form $〈1\mid \cdots U\mid 1〉$.

This seems like it could maybe be related to a $U$-deformation of the original inner product

(1)$〈\alpha \mid \beta 〉\to 〈\alpha \mid \beta {〉}_{U}=〈\alpha \mid U\beta 〉.$

That would explain the appearance of new physics, but it is not obvious to me that the result is a true inner product.

Eric

Posted by: Eric on June 16, 2004 3:03 PM | Permalink | Reply to this

Re: Minimal Principle

Hi Eric -

yes, that’s pretty much the idea. The inner product I am referring to is known as the BPZ inner product and it describes the ‘correlator’ of one string state coming from $t=-\infty$ with one coming from $t+\infty$. As you say, using a conformal transformation this can be mapped to the string states coming from the south and the north pole of a sphere. The sphere itself is the worldsheet of the two strings and the poles correspond to $t=±\infty$.

Concerning your last remark, let me clarify a bit:

What we are really interested in is the correlator which, given some string state, computes the amplitude that a given ‘Feynman graph’ between them is realized. By what I wrote above, this is closely related to the inner product for closed strings.

Consider three open string states ${\Phi }_{1}$, ${\Phi }_{2}$, ${\Phi }_{3}$. We want to know the (tree level) correlator, which in SFT language can be written as $\int {\Phi }_{1}\star {\Phi }_{2}\star {\Phi }_{3}$. This can be evaluated by inserting the state that results from the triple star product into an inner product of the closed string which has an appropriate boundary state $U\mid \alpha 〉$ inserted that emulates the boundary condition for the open strings:

(1)$\int {\Phi }_{1}\star {\Phi }_{2}\star {\Phi }_{3}=〈1\mid \left({\Phi }_{1}\star {\Phi }_{2}\star {\Phi }_{3}\right)\mid U\mid \alpha 〉\phantom{\rule{thinmathspace}{0ex}}.$

(It is really not easy to explain all the details behind the scenes in a couple of sentences, but this is pretty much the idea.)

Concerning the relation to spin networks: There might be a conceptual similarity, but I think that’s only because the concept is sufficiently general. You can even find it in 1-particle QM, where the scalar product can be related to the amplitude that an ‘in’ state goes to some given ‘out’ state.

Posted by: Urs Schreiber on June 16, 2004 3:47 PM | Permalink | PGP Sig | Reply to this

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