deformation of SCFTs

December 15, 2003

Posted by Urs Schreiber

This environment is very inspiring, I’ll start with a couple of questions right away: :-)

Is there anything known about deformations for SCFTs?

From papers like

Förste, Roggenkamp,
Current-current deformation of conformal field theories, and WZW models

I see that there is some theory about deformations of CFTs based on perturbations of correlation functions.

On the other hand, an apparently unrelatred approach to (infinitesimal) deformations has been studied for a while by I. Giannakis. In his most recent

I. Giannakis,
Strings in nontrivial gravitino and Ramond-Ramond backgrounds

he looks at deformations of the BRST charge of the superstring which leave it nilpotent. He claims that he can incorporate RR backgrounds this way, and indeed, he seems to get the correct (linearized) background equations. This looks kind of mysterious to me, though, because the stress-energy superfield cannot be reobtained from a BRST charge deformed by spin fields, as the author emphasises himself. So what is going on here? Is the BRST charge more fundamental then the superconformal generators?

If Giannakis’ approach is viable, wouldn’t it have relevance for covariant quantization of strings in AdS5 and similar backgrounds? However, I see no citations of that paper.

Posted at December 15, 2003 6:41 PM UTC

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25 Comments & 1 Trackback

Re: deformation of SCFTs

Hi Urs,

Many of the AdS/CFT papers discuss deformations of CFTs, often by turning on fields in the bulk. The N=1* theory (POlchinski and Strassler) is only one such example.

On this paper by Giannakis, I don’t see how it could work. It is known that in the presence of RR fields, the string action no longer breaks up into independent left-moving and right moving sectors. So it would seem that no holomorphic modification of string theory could possibly reproduce the effect of RR fields.

Posted by: Arvind on December 15, 2003 9:20 PM | Permalink | Reply to this

Re: deformation of SCFTs

Hi Arvind,

thanks for your answer!

How certain is it that we cannot have an SCFT in RR backgrounds? Can you point me to a reference where the possibility is ruled out?

I am asking for two reasons:

1) I am aware (only) of the following related papers:

S. Antusch: Coupling the Superstring to a D-Brane Ramond-Ramond Background

which proposes a new way to couple the RR background, as well as

Berenstein, Leigh
Quantization of Superstrings in Ramond-Ramond Backgrounds

which gives a “perturbative study” of superstrings in RR backgrounds.

2) I have played around with deformations of ‘classical’ SCFTS (where by classical I mean using functionals and Poisson brackets without paying attention to operator normal ordering) and unless I am confused I think I have found in addition to the SCFT describing strings in NSNS 2-form backgrounds a familily of classical superconformal algebras which is also parametrized by a background 2-form, but is inequivalent to the algebra associated with Kalb-Ramond backgrounds.

I am trying to understand what background this superconformal algebra might describe. Since it is not the NSNS 2-form background one guess could be that it is the RR 2-form.

As a possible check, I am considering the following:

On the algebraic level at which I am working dualities manifest themselves as algebra isomorphisms as in

Lizzi, Szabo Duality Symmetries and Noncommutative Geometry of String Spacetime,

which is mostly concerned with T-dualiy. I have tried to find further algebra isomorphisms in a similar spirit and indeed, unless I am confused, I think I have found such an isomorphism which maps the SCFT for NSNS 2-form background to that other SCFT in question, while at the same time sending the dilaton to its negative and keeping the spacetime metric fixed. It therefore seems like this isomorphism corresponds to S-duality. This would imply that this other 2-form background SCFT is indeed that of the RR 2-form background.

I am trying to understand if this is possible or if it violates some known results, in which case I must have made a mistake somewhere.

Thanks for any comments!

Best,
Urs

Posted by: Urs Schreiber on December 17, 2003 8:28 PM | Permalink | Reply to this

Re: deformation of SCFTs

$MathML-enabled post (click for more details).$

I feel like making myself acquainted with the MathML capabilities of this blog, so let me give a couple of equations that illustrate what I wrote in my previous reply:

In canonical language the supercurrents for a background having a constant metric $G_{\mu\nu}$ and a constant NSNS 2-form field $B_{\mu\nu}$ are

(1)

D_\pm = \Gamma^a_\pm E_a{}^\mu\left( \partial_\mu \mp iT(G_{\mu\nu} \pm B_{\mu\nu})X^{\prime \nu} \right)\,.

Here $\Gamma^a_\pm$ are the fermions with canonical anticommutator

(2)

\{ \Gamma^a_\pm(\sigma), \Gamma^b_\pm(\sigma^\prime) \} = \pm 2 \delta(\sigma,\sigma^\prime) G^{ab}

and $\partial_\mu$ is the functional derivative with canonical commutator

(3)

[\partial_\mu(\sigma), X^{\prime\nu}(\sigma^\prime)] = -\delta_\mu^\nu \delta^\prime(\sigma,\sigma^\prime) \,.

It is well known that we can implement T-duality by looking at the algebra isomorphism $\mathbf T$ which acts as

(4)

{\mathbf T}[\Gamma^a_\pm] = \pm \Gamma^a_\pm

(5)

{\mathbf T}[-i\partial_\mu] = X^{\prime\mu}

(6)

{\mathbf T}[X^{\prime\mu}] = -i\partial_\mu

which manifestly preserves the canonical brackets.

When acting with $\mathbf T$ on the supercurrent one finds

(7)

{\mathbf T}[D_\pm] = \tilde D_\pm

where $\tilde D_\pm$ is the supercurrent associated with the T-dual metric

(8)

\tilde G^{\mu\nu} = T^2 \left( (G \pm B)G^{-1}(G\pm B) \right)_{\mu\nu}

and the T-dual B-field

(9)

\tilde B_{\mu\nu} = \pm\left( (G\pm B)^{-1} - \tilde G \right)_{\mu\nu} \,.

Now, there is a slightly different algebra homomorphism $\mathbf S$ given by

(10)

{\mathbf S}[\Gamma^a_\pm] = \pm \Gamma^a_\pm

(11)

{\mathbf S}[-i\partial_\mu] = TG_{\mu\nu}X^{\prime\nu}

(12)

{\mathbf T}[X^{\prime\mu}] = -\frac{i}{T}G^{\mu\nu}\partial_\nu

which is obviously also an isomorphism but which apparently does not give T-duality.

Instead, one finds that

(13)

{\mathbf S}\left[ \Gamma^a_\pm E_a{}^\mu \left( (G_\mu{}^\nu \pm C_\mu{}^\nu) \partial_\nu \mp iT (G_{\mu\nu}\pm B_{\mu\nu}) X^{\prime\nu} \right) \right]

gives the same expression but with $B_{\mu\nu} \leftrightarrow C_{\mu\nu}$ interchanged. If one had included the dilaton in this discussion one would see that also $\Phi \rightarrow -\Phi$ under the action of $\mathbf S$ .

Of course, when one goes through these calculations, one finds that there is an ambiguity concerning the definition of the vielbein field depending on how much of $C_{\mu\nu}$ is included in $\tilde E_a{}^\mu$ . However, it can be shown that the above expression generalizes to nonconstant background fields, still giving classically the supercurrent of an SCFT, and that then terms depending on $C_{\mu\nu}$ appear which cannot be absorbed into a redefinition of $G_{\mu\nu}$ , nor can they be absorbed into a redefinition of $B_{\mu\nu}$ . Thus, by turning on $C_{\mu\nu}$ as indicated above one finds SCFTs that are inequivalent to those described by pure $G_{\mu\nu}$ and $B_{\mu\nu}$ backgrounds and which are however parameterized by a background 2-form.

Or so I think.

Posted by: Urs Schreiber on December 18, 2003 6:42 PM | Permalink | Reply to this

Re: deformation of SCFTs

Hi Urs,

I didn’t say you don’t have a SCFT, but rather that you don’t have separate SCFTs for the left and right movers. The Berenstein-Leigh paper states this, though I didn’t see an argument. You could also try the Vafa-Witten-Berkovits paper.

I don’t really understand the claim of your mapping. Strings do not couple to RR field, only the field strength. Is C a field or a field strength?

If C is a field, as I suspect it must be, you are most likely looking at a map from a F-string to a D-string. Canonical example would be the map from the heterotic string to the type I D-string, which I believe has been worked out. The map is still interesting, but one would not interpret the new SCFT as a theory of a string worldsheet.

Posted by: Arvind on December 18, 2003 7:16 PM | Permalink | Reply to this

Re: deformation of SCFTs

Yes, I thought that C should be a field (not a field strength) and if the S-duality interpretation is correct the result should describe a D-string, as you say.

Ah, ok, I see, I wasn’t precise about this point before. So this is not a big deal, it does not give us F-strings coupled to flux.

Hm, however one thing is still disturbing me: Any SCFT of the correct central charge defines F-strings in some background (by definition). Now, that C-field SCFT which I mentioned seems to be equivalent to a B-field SCFT only for constant (in a given holonomic basis) C_mn, B_mn . When interpreted as an F-string vacuum this SCFT should therefore describe something else than NSNS 2-form backgrounds.

But I am rambling. I’d have to provide further formulas to make this point precise.

I’ll read that Berkovits-Vafa-Witten paper now. Many thanks for the reference!

Posted by: Urs Schreiber on December 18, 2003 8:15 PM | Permalink | Reply to this

Re: deformation of SCFTs

Hi again,

could you be so kind and give the reference to that Polchinski&Strassler paper? I am not quite sure which one you meant.

Thanks a lot,
Urs

Posted by: Urs Schreiber on December 22, 2003 8:14 PM | Permalink | Reply to this

Re: deformation of SCFTs

Hi Urs,

I totally missed this post of yours. The paper in question is hep-th/0003136.

Arvind.

Posted by: Arvind on February 23, 2004 6:38 PM | Permalink | Reply to this

Re: deformation of SCFTs

Ioannis Giannakis has kindly taken the time to reply to some issues discussed here. With his permission I’ll make his comments available here at the Coffee Table

[begin forwarded email]

Hi Urs,

I finally started reading your paper. I will be able to make some
intelligent comments on Monday. But today I took a look at your
discussions on the web. Let me try to clarify few things

1) Issue of holomorphicity

In general when you turn on a background field, the resulting
stress tensor (action, supercurrent) appear to contain both
holomorphic and antiholomorphic mixed parts. It is not correct
to assume that holomorphicity is destroyed.
The reason is that holomorphicity is a consequence of equations
of motion. So when you discuss strings in flat spacetime
the equations of motion are {\partial}{\overline\partial}X=0

When we deform the theory, the equation of motion deforms.
But in the deformed theory we use operators from the undeformed
theory (we use the X and \partial X of the string theory in a flat
background rather than in curved backgrounds). We can of course
redefine our operators (use a basis from the deformed background
that obey the deformed equation of motion)
and then all our operators will appear to be either holomorphic or
antiholomorphic. This is what is happening in the case of RR backgrounds
too.

2) Issue of Superconformal Invariance

You can define a nilpotent BRST operator that describes to first
order (actually I can do it for all orders if I turn on a gravitational
field too) a RR background. We all know and believe (dangerous word) that
BRST and Superconformal Invariance are equivalent. This is definitely
true for NS-NS backgrounds (including NS-NS backgrounds in non-canonical
pictures). The reason is that you can extract the stress tensor and
the supercurrent from the BRST operator by commuting it with b and \beta
(ghosts). In the RR backgrounds this fails

As I have explained to you previously two things might happen

a) SC invariance is broken and the equivalance between SC and BRST
invarianve fails. If this happens strings in RR backgrounds are
perfectly consistent since BRST invariance guarantees decoupling
of negative norm states

b) SC is realized in a nonlocal manner. This means that the expressions
for the supercurrent (there is no problem with the stress tensor)
are nonlocal but the commutator with the stress tensor still produces
local results-delta functions or derivatives of delta functions.
The nonlocality I am talking about is on the world sheet not in
spacetime. This is not as outrageous as it sounds

The supercurrent has an OPE with the RR vertex operator in flat
spacetime that includes
branch cuts. This is usually not a problem since we deal with
physical operators (operators that survive GSO projection) and the
supercurrent is not one of them. But if you ask the question
What is the commutator of the supercurrent with the RR vertex
operator the answer is: It is non local
But at the deformed theory we use the set of operators (including
spin fields from the undeformed theory) so these nonlocalities
might not be surprising.

I personally put my money on option b) although I cannot discount
option a)

Re: deformation of SCFTs

Hi,

On rereading the paper, I realized that the construction is only supposed to work upto first order in the RR deformation. Is that correct?

I would like to understand one point. Your construction is supposed to give a SCFT for any deformation by a field strength which is constant. Now we know that that is not the case: we need a specific relation between different field strengths for supersymmetry to be preserved (e.g. for $AdS_3 \times S^3$, the field strengths on the two factors have to be the same.) It must be the case that this shows up only when we consider consistency to higher order in deformations. Is this the case?

Arvind.

Posted by: Arvind on February 21, 2004 12:42 AM | Permalink | Reply to this

Re: deformation of SCFTs

$MathML-enabled post (click for more details).$

Ioannis Giannakis wrote:

When we deform the theory, the equation of motion deforms. But in the deformed theory we use operators from the undeformed theory […]. We can of course redefine our operators […] and then all our operators will appear to be either holomorphic or antiholomorphic.

Yes, I completely understand what you mean. I have thought about this point quite a bit in the past, because I would like to understand how to identify the new holomorphic and anti-holomorphic objects in the deformed SCFTs that I considered.

The problem can nicely be studied in cases where it is well understood what happens, like in SWZW models. If we start with superstrings on a flat background and then gradually turn on gravitational and Kalb-Ramond fields so as to (locally in this framework) obtain an SWZW background, we know that the new (anti-)holomorphic objects are the Kac-Moody (affine Lie) algebra currents. But writing these out in terms of the original $U(1)$ supercurrents shows that they are highly nontrivial combinations of these original currents of both of the original chiralities. (The explicit fomula is for instance given in (2.63) of hep-th0311064 ).

It is not obvious to me how one would systematically find the new holomorphic objects in terms of the old ones for a deformed theory. For instance what happens to the Kac-Moody currents when we have gravitational and Kalb-Ramond backgrounds but not on a group manifold, i.e. not an SWZW model. According to equation (3.10) of the above mentioned paper the analogs of the chiral Kac-Moody currents can still be written down, but their commutator is an object that mixes both chiralities.

I expect that in general identifying the chiral objects can be highly nontrivial, for the following reason: If we know the chiral operators of conformal weight 0 and 1 we can write down the DDF operators for the theory and hence explicitly and exactly solve it. This won’t be possible in general. Right?

You can define a nilpotent BRST operator that describes to first order (actually I can do it for all orders if I turn on a gravitational field too) a RR background.

You say you can do deformations to all orders? That I would find immensely interesting. Can you give an example or some further hints? As you will have seen, in hep-th/0401175 I argue that exact deformations to higher orders will look different than what you call canonical deformations (e.g. in hep-th/9902194) to first order, because they will involve similarity transformations not of the chiral supercurrents $G$ and $\tilde G$ but of their ‘polar’ combinations $G \pm i \tilde G$ . Even though your approach in hep-th/0205219 is a little different, deformations of the BRST operator still translate to canonical deformations of the superconformal currents as the components of the BRST operator.

I personally put my money on option b) although I cannot discount option a)

Though I don’t have the required experience with non-local worldsheet objects, b) sounds quite plausible to me.

One question I’d have is the following: In your approach of hep-th/0205219 the BRST operator is deformed in a way that it remains nilpotent. Do you expect that any deformation whatsoever which sends the nilpotent BRST operator to another nilpotent operator can be interpreted as yielding the BRST operator of another SCFT/superstring background?

Posted by: Urs Schreiber on February 21, 2004 3:09 PM | Permalink | Reply to this

Re: deformation of SCFTs

$MathML-enabled post (click for more details).$

Ioannis Giannakis replies:

[begin forwarded mail]

Arvind Rajaraman wrote:

On rereading the paper, I realized that the construction is only supposed to work upto first order in the RR deformation. Is that correct?

Indeed in the paper the construction works only to first order in $\delta Q$ .

The reason is that I have dropped terms $\{\delta Q, \delta Q\}$ from the deformation equation - the requirement that the deformed BRST operator $Q + \delta Q$ is nilpotent. Actually if you calculate this term you will realise that the deformed operator is not nilpotent.

In order to make it nilpotent you need to enlarge your deformation to include non-trivial gravitational backgrounds.

I would like to understand one point. Your construction is supposed to give a SCFT for any deformation by a field strength which is constant. Now we know that that is not the case: we need a specific relation between different field strengths for supersymmetry to be preserved (e.g. for $AdS_3 \times S^3$ , the field strengths on the two factors have to be the same.) It must be the case that this shows up only when we consider consistency to higher order in deformations. Is this the case?

The deformation to first order is consistent-gives nilpotency of the BRST operator-for any field strength of the RR field (not necessarily constant) as long as it satisfies the equation of motion. Your question is different. You ask about a deformation that preserves spacetime susy. This is a stronger condition. If you deform by turning on a RR background only starting from flat spacetime then for any value of $F$ (field strength) spacetime susy is spontaneously broken. If you want to preserve susy then you need to turn on gravitational backgrounds (like $AdS_3 \times S^3$ ).

The condition then that SUSY is unbroken - which is extra - imposes constraints on the field strengths of RR fields. So this has nothing to do with higher orders but rather from the requirement that SUSY is unbroken.

Now to the questions raised by Urs.

One question I’d have is the following: In your approach of hep-th/0205219 the BRST operator is deformed in a way that it remains nilpotent. Do you expect that any deformation whatsoever which sends the nilpotent BRST operator to another nilpotent operator can be interpreted as yielding the BRST operator of another SCFT/superstring background?

Well the set of solutions to the deformation equations (either nilpotency or supervirasoro) can be expressed in terms of superprimary fields or combinations of spin fields that can be interpreted as turning on spacetime fields-describe emission of particle states of the spectrum of the string. As a result I would expect that all deformations yield BRST operators that describe string backgrounds.

Concerning the deformation to all orders in RR, you can do it only if you turn on appropriate gravitational backgrounds too. You simply calculate $\{\delta Q, \delta Q\}$ which provides a term that can be cancelled upon picture changing by a nontrivial gravitational field.

Regards Ioannis

[end forwarded mail]

Posted by: Urs Schreiber on February 22, 2004 11:14 PM | Permalink | Reply to this

Re: deformation of SCFTs

$MathML-enabled post (click for more details).$

Hi Ioannis -

many thanks for your replies!

You wrote:

In order to make it nilpotent you need to enlarge your deformation to include non-trivial gravitational backgrounds.

Ok, since the background has to satisfy its equations of motion, turning on finite RR fields will excite other fields, like for instance gravitational ones. (Not necessarily and exclusively gravitational, though, I assume. Right?)

Currently I don’t understand yet how by just deforming the, say, left-moving, BRST current we could get beyond first-order backgrounds. Probably I am missing something. Please help.

My problem is related to what I wrote last time: We know from tractable examples, like SWZW models, that the chiral currents of non-trivial such models are mixed combinations of the chiral $U(1)$ currents of the trivial theory on flat Minkowski space. The left-moving and right-moving currents of the unperturbed theory rearrange and recombine to form new left-moving, say, currents of the perturbed model.

As a result I would expect that all deformations yield BRST operators that describe string backgrounds.

Let me see if I understand what you are saying:

We start with nilpotent BRST charges $Q$ and $\bar Q$ , $Q^2 = 0 = \bar Q^2$ .

As on page 2 of hep-th/0205219 a deformation $Q \to Q + \delta Q$ to first order preserves nilpotency to this order if

(1)

\{Q, \delta Q\} = 0 \,.

In principle it seems easy to construct finite (i.e. beyond 1st order) deformations that preserve nilpotency. Just let $A$ be any invertibe operator and use the similarity transformation

(2)

Q \to A^{-1}\,Q\,A \,.

For $A$ of the form

(3)

A = \exp({i\, :h:})

with $h$ any (hermitian, probably) operator we’d get to first order $\delta Q = i [Q,h]$ as in equation (7) of your paper.

Now are you saying that the full deformed operator

(4)

A^{-1}\,Q\,A = 1 + [Q,ih] + \frac{1}{2}[[Q,ih],ih] + \cdots

is still in general the BRST operator of some 2d SCFT?

I certainly understand that it is still nilpotent, by construction. But is it also still a product of (super)conformal ghosts with (super)conformal generators of some 2d SCFT? Probably not in general, I would guess. Can we preserve nilpotency and the superconformal algebra? Is this trivial?

Posted by: Urs Schreiber on February 23, 2004 1:14 PM | Permalink | Reply to this

Re: deformation of SCFTs

$MathML-enabled post (click for more details).$

Ioannis Giannakis replies:

[begin forwarded email]

Ok, since the background has to satisfy its equations of motion, turning on finite RR fields will excite other fields, like for instance gravitational ones. (Not necessarily and exclusively gravitational, though, I assume. Right?)

I think that it has to be either gravitational or two form gauge field. You simply have to calculate ${\delta Q, \delta Q}$ with $\delta Q$ the expression in my paper. The result is of order $F^2$ where $F$ is the RR field strenght. This $F^2$ is multiplied by the graviton-two form gauge vertex operator. In order to cancel this term you need to add to $\delta Q$ an appropriate term.

Currently I don’t understand yet how by just deforming the, say, left-moving, BRST current we could get beyond first-order backgrounds. Probably I am missing something. Please help.

Although probably it is not clear I discuss deformations of both $Q$ and ${\overline Q}$

By finite deformations I mean $Q^\prime = Q+{\delta Q}$ that satisfies

(1)

\{Q, \delta Q\}+\{\delta Q\}+\{\delta Q, \delta Q\}=0 \,.

For infinitesimal ones you can drop the last term.

Now this condition has to give you equations of motion (linearized or nonlinear) for the spacetime fields.

I have few questions concerning your paper. I will send them in a separate email

The deformations that you are refering (similarity transformations) are physically non interesting (they induce spacetime symmetries) from the point of view of deformations but extremely important from the point of view of symmetries.

Ioannis

[end forwarded email]

Posted by: Urs Schreiber on February 23, 2004 7:34 PM | Permalink | Reply to this

Re: deformation of SCFTs

$MathML-enabled post (click for more details).$

Ioannis Giannakis replies:

[begin forwarded mail]

Hi Urs,

Some comments concerning your paper.

The first question to address is “How does a SCFT deform as we deform the classical solution of the string equations of motion?”

In general this deformation will not correspond to a symmetry transformation, it’s just a physically different nearby solution (if we discuss infinitesimal deformations). For example in General Relativity two nearby solutions are

1) flat empty Minkowski space and

2) a weak gravitational wave propagating in flat Minkowski space.

In general 1) and 2) are not related by any symmetry transformation (similarity transformation), they are physically distinct. Similarity transformations cannot change the physics.

Similarity transformations generate a subset of the deformations that correspond to symmetry transformations of the spacetime fields.

Of course I have to be careful and maybe I have misunderstood your work. I mean a similarity transformation of the full operator algebra. Of course by construction in the case of deformations the two SuperVirasoro algebras are isomorphic but they will usually correspond to inequivalent embeddings in the full operator algebra.

In your case I noticed that you do not get as a result of deforming, equations of motion for your spacetime fields which might suggest that you are turning on gauge degrees of freedom.

But maybe I do not understand your construction.

Ioannis

P.S.

Also you mention in your paper that one of the unpleasant characteristics of the approach that I developed with Evans is that you turn on spacetime fields in a particular gauge. This problem was solved in

Evans and Giannakis, Phys.Rev.D44:2467-2479,1991

and

Bagger and Giannakis (Appendix) hys.Rev.D65:046002,2002 e-Print Archive: hep-th/0107260

The resolution is that the most general solution to the deformation equations is not $\delta T=(1, 1)$ primary field but $\delta T=(1, 1)+(2, 0)+(0, 2)$ in other words the total dimension of the deformation has to be $2$ .

Then you find gauge covariant equations of motion.

[end forwarded mail]

Posted by: Urs Schreiber on February 24, 2004 2:04 PM | Permalink | Reply to this

Re: deformation of SCFTs

$MathML-enabled post (click for more details).$

Hi -

yes, I do understand that a similarity transformation of the entire algebra corresponds to a symmetry or duality of the target space and not to a different background. This has been discussed in great detail by yourself and others and aspects of this are discussed in section 4 of my hep-th/0401175.

Please note, though, that what I am considering in section 3 of that paper are not similarity transformations of the entire algebra, so they do describe nontrivial new backgrounds which are not related by symmetry transformations.

This can probably most clearly be seen in equation (3.36). This is the result of a deformation which reproduces precisely the result obtained by analyzing the nonlinear sigma-model (3.38) for superstrings in general gravitational and Kalb-Ramond backgrounds.

But I see why you might think that all I am considering are similarity transformations of the entire algebra. This is probably due to the fact the deformations that I am considering look like

(1)

T_F + i \bar T_F \to e^{-W} (T_F + i \bar T_F) e^{W}

(2)

T_F - i \bar T_F \to e^{W^\dagger} (T_F - i \bar T_F) e^{-W^\dagger} \,.

But note that this are two different similarity transformations in general and that hence we don’t have a single similarity transformation for the entire algebra, so that this is not a trivial deformation.

Indeed, I claim that this is (almost) the most general deformation possible. That’s for the following reason, which is discussed in more detail in section 2.1.2 of hep-th/0311064:

As noticed long ago by Witten and others, the combination $T_F + i \bar T_F$ is something like the exterior derivative on loop space, the configuration space of the string. In particular its square, which schematically looks like

(3)

(T_F + \bar T_F)^2 \sim T - \bar T

is the generator of spatial reparameterizations of the string. This object is independent of the background in which the string moves! So no matter how we deform the algebra the above relation must be preserved. The deformations that I am considering do precisely this, they ensure that the square of $T_F \pm i \bar T_F$ is always the generator of (worldsheet-)spatial reparameterizations. In other words, they ensure that $(T_F \pm i \bar T_F)$ is nilpotent up to reparameterizations.

One should see how this works in a specific example. Please consider equation (3.36) hep-th/0401175. There the result of a deformation of the above form (two different similarity transformations) is given and it is shown that the result is precisely what one alternatively gets from the nonlinear sigma-model Lagrangian (3.38) for superstrings in general gravitational and Kalb-Ramond backgrounds. Hence this does reproduce well knwon results.

Please note that the only reason why I do not discuss the equations of motion of the background fields is because I do not attempt to do so. But since the deformation displayed in (3.36) is just the usual result the usual beta-functional computations apply and give us the usual background equations of motion.

Furthermore note that the method that I am discussing should really go over into the method that you are using when the deformations are considered to first order only. I am in the process of working that out in more detail but it can already explicitly be seen in equation (3.36). Summing up these two terms gives you the deformed chiral supercurrent $T^\prime_F$ or $\bar T^\prime_F$ and it is obviously of the form

(4)

T^\prime_F = T_F + \delta T_F

where $\delta T_F$ is a field of total weight $3/2$ so that it follows that

(5)

T^\prime = T_F + \delta T

with $\delta T$ a field of total weight $2$ .

That’s why I think the two approaches are compatible. One perspective focuses on the first order perturbation and CFT formalism while the other considers the full perturbation.

Please let me know if you agree with the above statements or if you think there is something unclear or wrong.

I would like to come back to my question concerning deformations of the BRST operator which you consider. Unfortunately I did choose a bad example when asking about this last time, since, as you kindly pointed out, a similarity transformation of the BRST charge will just describe a symmetry or duality. Please let me try again to express my concern:

Say I am given a BRST chage $Q$ and an operator $\delta Q$ such that

(6)

(Q + \delta Q)^2 = Q^2 + \{Q,\delta Q\} + (\delta Q)^2 = \{Q,\delta Q\} + (\delta Q)^2 = 0 \,.

My question is: Are you saying that for all $\delta Q$ that satisfy the above equation the operator $Q^\prime = Q + \delta Q$ is another BRST operator of a (S)CFT?

For the reasons expressed above I would be surprised if there were not supplementary conditions on $\delta Q$ necessary to ensure that $Q + \delta Q$ can be interpreted as another BRST charge.

I apologize for having expressed this point badly, before. I am hoping that I am now making myself clearer.

Posted by: Urs Schreiber on February 24, 2004 3:04 PM | Permalink | Reply to this

Re: deformation of SCFTs

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Ioannis Giannakis replies:

[begin forwarded email]

Hi Urs,

More comments:

Furthermore note that the method that I am discussing should really go over into the method that you are using when the deformations are considered to first order only. I am in the process of working that

The method that I and collaborators are using is not confined to first order deformations. If you calculate terms $[\delta T, \delta T]$ when you demand that your deformation satisfies the SuperVirasoro algebra then you deal with finite deformations.

Please note that the only reason why I do not discuss the equations of motion of the background fields is because I do not attempt to do so. But since the deformation displayed in (3.36) is just the usual result the usual beta-functional computations apply and give us the usual background equations of motion.

I am puzzled by the fact that you do not get equations of motion. SC invariance should give you equations of motion. The point is that a collection of spacetime fields (metric, B field, RR etc) does not provide consistent string theories (strings propagating and interacting with those spacetime fields ) unless these fields obey Equations of motion. The beta function calculations mean exactly that. They constrain the coupling constants of the non-linear sigma model (spacetime fields) quantum mechanically so the theory is SC invariant.

But maybe you are working at the classical level only.

My question is: Are you saying that for all $\delta Q$ that satisfy the above equation the operator $Q^\prime = Q+\deltaQ$ is another BRST operator of a (S)CFT?

As long as we deal with NS-NS spacetime fields thats exactly what I am saying. For RR fields it is not clear that nilpotency implies SC invariance.

Regards Ioannis

[end forwarded email]

Posted by: Urs Schreiber on February 24, 2004 8:47 PM | Permalink | Reply to this

Re: deformation of SCFTs

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Yes, as I say in the abstract and the introduction of hep-th/0401175, I am working at the level of Poisson brackets in that paper. That is, I am ignoring higher order Wick contractions and normal ordering. I am not saying that they are not there, I am just not considering them.

There is a reason for why I am doing that, and I would be interested in your opinion about it:

Namely I believe that it is nontrivial to determine the correct form of the normal ordering for arbitrary backgrounds. I think that we cannot just use the normal ordering of the theory in flat Minkowski background, which sends the flat space annihilators $a^\mu_m = {const} \int d\sigma\, e^{-im\sigma}\left(-i\eta^{\mu\nu}\frac{\delta}{\delta X^\nu\left(\sigma\right)} \pm i T X^{\prime \mu}\left(\sigma\right)\right)$ to the right and creators $a_{-m}$ to the left. I think that this can be exemplified in the case of SWZW models: There the Kac-Moody current modes $j^a_m$ are nontrivial combinations of the $a^\mu_m$ and $\bar a^\mu_m$ . Normal ordering with respect to the $j^a_m$ is not the same as that with respect to the $a^\mu_m$ , I think. Don’t we have to be careful with how the notion of normal ordering changes when turning on background fields?

Currently I think that the usual normal ordering can only be maintained by working only to first order in the background fields. Isn’t that correct?

When I truncate the the superconformal constraints (3.36) in hep-th/0401175 at first order in the deviation from flat background and then proceed by CFT methods on flat space, following the approach in your papers, I do indeed get the correct equation of motion, namely

(1)

\nabla^\mu H_{\mu\nu\lambda} = 0 \,,

where $H = dB$ is the field strength of the $B$ field.

When I proceed in the same vein for the higher order terms I also find the other two equations of motion (to the given order in $\alpha^\prime$ )

(2)

R_{\mu\nu} - {const} H_{\mu\lambda\kappa}H_\nu{}^{\lambda\kappa} = 0

(3)

H_{\mu\nu\lambda}H^{\mu\nu\lambda} = 0 \,.

When I first did this calculation quite a while ago I was delighted. But then I realized a problem: The constant ${const}$ in the second equation should be $1/4$ but by the above methods I get a $1/2$ . I now believe the reason for this discrepancy is that the above equations are obtained by using the normal ordering associated with the flat space theory. But this is not the correct one as soon as higher order terms of the background fields are taken into account.

I have a set of private notes online where this calculation is sketched. Being just private notes, they are probably not very readable, but in case you are interested you can find the calculation on pp. 127 of these notes. The final result is given in equation (860) on p. 130.

Many thanks for your pointer to hep-th/0107260. That’s very interesting. I am puzzled about one point: The deformations given in equation (52) and (53) don’t seem to be the ones that one finds by writing down the Lagrangian for superstrings in these backgrounds. Why is that?

Posted by: Urs Schreiber on February 25, 2004 11:03 AM | Permalink | Reply to this

Re: deformation of SCFTs

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Ioannis Giannakis replies:

[begin forwarded email]

Hi Urs,

Few comments since I will be away for a week.

I think that it is clear now why you do not get EOM. It is extremely difficult to define normal ordering for arbitrary backgrounds.

After all in that case the two dimensional field theory is interacting.

The advantage of the method we are using is that we can work in the deformed picture with the normal ordering of the undeformed picture since we are using a basis of operators that was defined in the undeformed picture (flat spacetime). The disadvantage is that the operators we get in the deformed picture are complicated.

This is the reason also that equations (52) and (53) do not appear familiar. If you write them in terms of $\hat{\partial X}$ and not in terms of $\partial X$ (flat spacetime) they reproduce the terms that appear in the Lagrangian.

Ioannis

[end forwarded email]

Posted by: Urs Schreiber on February 26, 2004 9:26 AM | Permalink | Reply to this

Re: deformation of SCFTs

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It should be possible in my approach to get the equations of motion purely algebraically without any normal ordering following the idea of this calculation of the anomaly for flat space. I am currently trying to do that calculation.

Posted by: Urs Schreiber on February 26, 2004 3:01 PM | Permalink | Reply to this

Re: deformation of SCFTs

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Hi -

I may have made some progress in calculating the various background equations of motions conformal anomalies in the algebraic formalism that I am using. I have checked that I get the correct equations of motion for gravity by using just regulated algebraic relations, without considering any normal ordering or Fock space vacua. I expect that the method works for the Kalb-Ramond background just as well, but I still need to check that.

I also think I can show that the deformations that I am considering are to first order of the form of weight (1,1) vertices of the corresponding background fields (plus further terms) just as in your approach. I will write some LaTeX notes on that.

I would still very much enjoy to see in detail how the deformations that I am using and those that you are using are related. I am pretty convinced that they are and that something can be learned from understanding how.

I have looked at an old paper by Ovrut and Rama

B. Ovrut & S. Rama Deformations of superconformal field theories and target-space symmetries, 1992

which is very nice and which helped me better understand the nature of the ‘canonical deformations’.

For instance therein it is emphasized (p.553) that the ‘canonical deformation’ does not preserve the functional form of the Virasoro algebra, but only gives a consistent deformation when integrated over. I was wondering about this point before. From some papers on ‘canonical deformations’ one gets the impression that it is claimed that these preserve the functional Virasoro algebra.

Posted by: Urs Schreiber on March 3, 2004 8:19 PM | Permalink | Reply to this

Re: deformation of SCFTs

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Ioannis Giannakis replies:

[begin forwarded email]

Hi Urs,

Thanks for your message.

I may have made some progress in calculating the various background equations of motions conformal anomalies in the algebraic formalism that I am using. I have checked that I get the correct equations of motion for gravity by using just regulated algebraic relations, without considering any normal ordering or Fock space vacua.

This would be very nice and I will wait for your Latex notes.

For instance therein it is emphasized (p.553) that the ‘canonical deformation’ does not preserve the functional form of the Virasoro algebra, but only gives a consistent deformation when integrated over. I was wondering about this point before. From some papers on ‘canonical deformations’ one gets the impression that it is claimed that these preserve the functional Virasoro algebra.

I am not sure I understand what is meant with “functional form of the Virasoro”. Canonical Deformations are (1,1) primary fields that satisfy the deformation equations-and thus preserve the Virasoro algebra, the full Virasoro algebra. Could you be more specific of what is the functional form of the Virasoro?

Regards Ioannis [end forwarded email]

Posted by: Urs Schreiber on March 4, 2004 3:57 PM | Permalink | Reply to this

Re: deformation of SCFTs

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I have prepared LaTeX notes which demonstrate how I imagine to calculate the EOM without using normal ordering. These can be found here. Please note that I am aware that this still needs some work.

Furthermore I have prepared notes which demonstrate how the deformations that I am using reproduce canonical deformations to first order. These can be found here.

Posted by: Urs Schreiber on March 4, 2004 4:04 PM | Permalink | Reply to this

Re: deformation of SCFTs

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Hi again -

I have just made an addition to the above mentioned LaTeX notes (p. 39) where I demonstrate in detail how the deformations that I am considering when being purely non-gauge (i.e. for hermitean deformation operators $\mathbf{W}$ ) are to first order of the form

(1)

\delta T \propto \{\bar T_F, [T_F, \mathbf{W}]\} \,.

This is nice, because it proves the observation that all contributions to $\delta T$ which are not of weight $(1,1)$ are due to background gauge transformations.

Furthermore, I think that this should help understand how RR-form background deformations are to be handled beyond leading order while preserving superconformal invariance. Namely, when I look at the RR-background deformation in equation (17) of your hep-th/0205219 I see that, up to a change of picture, this is of the above form with

(2)

\{ \bar T_F, [T_F \mathbf{W}]\} \propto \int d\sigma\, F^{\alpha\beta}(X) e^{-\phi/2} S_\alpha e^{-\bar \phi/2} \bar S_\beta \,,

where $\mathbf{W}$ is of weight (1/2,1/2) (when the background equations of motion hold) and hence in particular reparameterization invariant. ( $S_\alpha$ are worldsheet spin fields.)

So my conjecture is now that the worldsheet SCFT for RR backgrounds is generated by the superconformal generator $T^{(\mathrm{RR})}_F$ given by

(3)

T^{(\mathrm{RR})}_F = \frac{-i}{2} \left( e^{-W} \left( \bar T_F + i T_F \right) e^{W} - e^{W} \left( \bar T_F - i T_F \right) e^{-W} \right) \,,

(cf. p. 38 of my notes.), where $\bf W$ is as above and $T_F$ , $\bar T_F$ are the superconformal generators for the trivial Minkowski background.

To leading order this reduces precisely to the result you give in hep-th/0205219. To higher orders it is guaranteed to preserve the superconformal algebra due to section 3.2 of hep-th/0401175. (Of course there is still the issue with higher-order Wick-contractions.)

Posted by: Urs Schreiber on March 7, 2004 6:20 PM | Permalink | Reply to this

Re: deformation of SCFTs

Ioannis Giannakis writes:

[begin forwarded email]

Hi Urs,

You raise some very interesting issues. I would like to try to understand your approach better. I will get back to you soon.

Regards
Ioannis

Do you plan to visit the States? If you ever decide to come to the States I can arrange for you to stay in NY for 4-5 days.

[end forwarded email]

Posted by: Urs Schreiber on March 9, 2004 5:42 PM | Permalink | PGP Sig | Reply to this

Re: deformation of SCFTs

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Hi -

you wrote:

You raise some very interesting issues. I would like to try to understand your approach better. I will get back to you soon.

I realize that I should have worked out the relation between my approach and the canonical deformations in more detail earlier. I am aware that all this loop-space formalism in my paper is very non-standard and might have obfuscated the basic idea.

But the loop space formalism has the virtue that it illuminates the deeper meaning of my constructions, I think, namely that the deformations that I am considering are a direct generalization of the deformations that E. Witten originally considered for N=2 susy quantum mechanics in his Morse Theory paper, which are mathematically quite deep, in a sense.

Today a replacement of my hep-th/0401175 has appeared on the arXive, wherein I have added some paragraphs which should help clarify what I am doing:

For instance, you might want to have a look at the new section 3.2, which I have expanded by a discussion of how the deformation prescription that I am considering reads in more standard notation.

Also, what is now section number 3.4 and 3.5 contains a detailed analysis of how the deformations that I am considering reduce to canonical deformations when truncated at first order and how this implies that for every vertex operator + background gauge transformation there is a deformation operator $\mathbf{W}$ in my formalism. (Up to the subtlety discussed in section 3.5).

Do you plan to visit the States? If you ever decide to come to the States I can arrange for you to stay in NY for 4-5 days

Currently I don’t plan to visit the States, but I could arrange it. I will keep this offer at the back of my mind. Many thanks!

All the best, Urs

Posted by: Urs Schreiber on March 9, 2004 5:49 PM | Permalink | PGP Sig | Reply to this

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December 15, 2003