## March 8, 2004

### Simple but not trivial

#### Posted by Urs Schreiber

While we are still discussing Ioannis Giannakis’ proposal for how to get worldsheet SCFTs for RR backgrounds and now that I have worked out how the deformations that I am considering reduce to the canonical deformations known in the literature when truncated at first order and how they relate to the vertex operators of the respective background, so that I am finally prepared to seriously begin to think about RR-backgrounds in NSR formalism - while all this is happening (at least in my life) a possibly interesting testing ground for any such ambitions is being discussed in a recent paper:

M. Billo, M. Frau, I. Pesando, $𝒩=1/2$ gauge theory and its instanton moduli space from open strings in R-R background.

The paper studies the 4d physics of Type II compactified on ${R}^{6}/\left({Z}_{2}×{Z}_{2}\right)$ with an R-R 5-form which is wrapped around a 3-cycle turned on, such that in 4d one sees a constant R-R 2-form ${C}_{\mu \nu }$ which (for some reason that escapes me) is known as a ‘graviphoton background’.

The point of the paper is to compute the amplitudes invoving the respective R-R vertex and find the respective effective field theory, which is a non-commutative one with respect to the fermionic degrees of freedom, because one gets something like

(1)$\left\{{\overline{\theta }}^{\stackrel{˙}{\alpha }},{\overline{\theta }}^{\stackrel{˙}{\beta }}\right\}={C}_{\mu \nu }\frac{1}{4}\left({\overline{\sigma }}^{\mu \nu }{\right)}^{\stackrel{˙}{\alpha }\stackrel{˙}{\beta }}$

for the fermion anticommutator.

Anyway, what caught my attention is that in the introduction it says:

It is a common believe that the RNS formalism is not suited to deal with R-R background; while this is true in general, it is not exactly so when the R-R field strength is constant. In fact, in this case one can represent the background by a R-R vertex operator at zero momentum which in principle can be inserted inside disc correlation function among open string vertices without affecting their dynamics.

So maybe this R-R background, being very simple and still non-trivial, would be a good testing ground for Ioannis Giannakis’ conjecture that construction of SCFTs for R-R backgrounds is possible.

Essentially, what one would need to do to check this is (according to the theory of superconformal canonical deformations) to take the R-R vertex (2.14) of that paper, integrate it at constant worldsheet time over the string and add the result to the (anti-)holomorphic stress tensor:

(2)$T\left(\sigma \right)\to {T}^{\prime }\left(\sigma \right):=T\left(\sigma \right)+\int d\sigma \phantom{\rule{thinmathspace}{0ex}}{C}_{\mu \nu }\left({\overline{\sigma }}^{\mu \nu }{\right)}_{\stackrel{˙}{\alpha }\stackrel{˙}{\beta }}\left({S}^{\stackrel{˙}{\alpha }}{S}^{\left(+\right)}{e}^{-\frac{1}{2}\varphi }{\stackrel{˜}{S}}^{\stackrel{˙}{\alpha }}{\stackrel{˜}{S}}^{\left(+\right)}{e}^{-\frac{1}{2}\stackrel{˜}{\varphi }}\right)\left(\sigma \right)$

and do something analogous to the supercurrent. (Here $S$ are spin fields.)

Then study the SCFT generated by ${T}^{\prime }$ and ${T}_{F}^{\prime }$.

Furthermore, I would like to check what one gets when adding also the second-order correction and possibly the full exact SCFT deformation associated with this background.

If for some reason this does not work, can maybe the superconformal canonical deformations be adapted to Berkovit’s covariant superstring formalism?

Posted at March 8, 2004 8:41 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/323

## 1 Comment & 0 Trackbacks

### Re: Simple but not trivial

Apparently this problem is studied in

P. Grassi & L. Tamassi Vertex operators for closed superstrings

cf. equation (2.12) and item b) in the outlook on p. 32.

Posted by: Urs Schreiber on May 22, 2004 12:44 PM | Permalink | Reply to this

Post a New Comment