### deformation of SCFTs

#### Posted by urs

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
TrackBack URL for this Entry: http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/266

### 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.

### 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

or

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)

More on Monday

Regards

Ioannis

**[end forwarded email]**

Read the post

Research Blogging
**Weblog:** The n-Category Café

**Excerpt:** On research blogging.

**Tracked:** December 20, 2006 5:27 PM

## 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.