### New York, New York

#### Posted by Urs Schreiber

I have visited Ioannis Giannakis at Rockefeller University, New York, last week, and by now I have recovered from my jet lag and caught up with the work that has piled up here at home enough so that I find the time to write a brief note to the Coffee Table.

Ioannis Giannakis has worked on infinitesimal *superconformal deformations* and I became aware of his work while I happened to write something on finite deformations of superconformal algebras myself. In New York we had some interesting discussion in particular with regard to generalizations of the formalism to open strings and to deformations that describe D-brane backgrounds.

The theory of superconformal deformations was originally motivated from considerations concerning the effect of symmetry transformations of the background fields on the worldsheet theory. It so happened that while I was still in New York a heated debate concerning the nature of such generalized background gauge symmetries and their relation to the worldsheet theory took place on sci.physics.strings.

People interested in these questions should have a look at some of the literature, like

Jonathan Bagger & Ioannis Giannakis: Spacetime Supersymmetry in a nontrivial NS-NS superstring background (2001)

and

Mark Evans & Ioannis Giannakis: T-duality in arbitrary string backgrounds (1995) ,

but the basic idea is nicely exemplified in the theory of a single charged point-particle in a gauge field $A$ with Hamiltonian constraint $H=(p-A{)}^{2}$. A conjugation of the constraint algebra and the physical states with $\mathrm{exp}(i\lambda )$ induces of course a modification of the constraint

which corresponds to a symmetry tranformation in the action of the *background field* $A$. In string theory, with its large background gauge symmetry (corresponding to all the null states in the string’s spectrum) one can find direct generalizations of this simple mechanism. (Due to an additional subtlety related to normal ordering, these are however fully under control only for infinitesimal shifts or for finite shifts in the classical theory.)

More importantly, as in the particle theory, where the trivial gauge shift $p\to p+d\lambda $ tells us that we should really introduce gauge connections $A$ that are *not pure* gauge, one can try to guess deformations of the worldsheet constraints that correspond to physically distinct backgrounds. This is the content of the theory of (super)conformal deformations. My idea was that there is a systematic way to find finite superconformal deformations by generalizing the technique used by Witten in the study of the relation of supersymmetry to Morse theory. The open question is how to deal consistently with the notion of normal ordering as one deforms away from the original background.

In order to understand this question better I tried to make a connection with string field theory:

Consider cubic bosonic open string field theory with the string field $\varphi $ the BRST operator $Q$ for flat Minkowski background and a star product $\star $, where the (classical) equations of motion for $\varphi $ are

for some constant $c$.

In an attempt to understand if this tells me anything about the propagation of single strings in the background described by $\varphi $ I considered adding an infinitesimel ‘test field’ $\psi $ to $\varphi $ and checking what equations of motion $\psi $ has to satisfy in order that $\varphi +\psi $ is still a solution of string field theory. To first order in $\psi $ one finds

If we think of the ‘test field’ $\psi $ as that representing a single string, then it seems that one has to think of

as the *deformed* BRST operator which corresponds to the background described by the background string field $\varphi $.

It is due to the fact that $a\star b$ and $b\star a$ in string field theory have no obvious relation that I find it hard to see whether ${Q}^{{\textstyle \prime}}$ is still a nilpotent operator, as I would suspect it should be.

But assuming it is and that its interpretation as the BRST operator corresponding to the background described by $\varphi $ is correct, then it would seem we learn something about the normal ordering issue referred to above: Namely as all of the above string field expressions are computed using the normal ordering of the *free* theory it would seem that the same should be done when computing the superconformal deformations. But that’s not clear to me, yet.

The campus of Rockefeller University.

Posted at April 23, 2004 5:25 PM UTC
## Re: New York, New York

Hi Urs,

I hope you don’t mind a couple of random thoughts…

First, I was wondering if you could point out a good reference for these BRST operators. You seem to relate them to exterior derivatives. In your post you suggest something that looks like a modification of the exterior derivative. By now, you probably know what to expect from me when I hear this kind of thing :) The exterior derivative should not be thought of as a “derivative” operator so to speak. Rather, the exterior derivative is the transpose of the boundary map. If you want to deform an exterior derivative, you should think about what this does to the boundary map. Like has happened before, I suspect that you can instead move any such deformations to the Hodge star. In this case, it looks like it might get mapped to a deformation of the star product? That is probably nonsense. Sorry :)

Another thing is that I once got very excited about the star product and wrote up some neat observations on s.p.r.

NCG/SUSY/*-Product

It turns out that the star product has an interpretation in terms of parallel transport. Now that you have a good handle on the discrete theory you might appreciate why I got excited. In the discrete world we have directed edges with a simple algebra (basically concatenation)

e_ij e_kl = delta_jk e_ijl

You can picture this as a “path” on a graph. There is another algebra that I talk about in my thesis that doesn’t appear in our notes. This is given by

e_ij e_kl = delta_ik e_ijl.

These elements can be thought of as “fans”. So we have “fans” versus “paths”. I call them fans because they remind me of those chinese fans that fold out from two sticks :)

In the first algebra (paths), segments get concatenated end to end forming a path. In the second algebra (fans) segments all attach to a given base point forming a fan.

It is hard to do this without ascii art and I don’t trust this font, but if you start with a 2-path, then imagine that you want to transform this 2-path to a 2-fan. To do so, you want to transport the base of the second edge back so that it coincides with the base of the first edge (the bases all coincide for a fan). Imagine this as some kind of abstract parallel transport. Let me denote this process by the map T: Paths -> Fans. Now if $A$ and $B$ are two 1-paths (as in our notes), then $\mathrm{AB}$ is a 2-path (as in our notes) and finally

is a 2-fan. I know that I am sucking at this explanation, but it is my conjecture that the star product can be interpretted this way (if you keep at me with an open mind I’m sure we could make this precise). In other words, I am pretty sure that

This also explains why

because we know that

If I can manage to convince you that this idea is not completely bogus, then I have a hunch that it might help you gain a better intuition for the meaning of the star product. I hope :)

I know I am not describing my idea here very clearly (as usual), but I think there is something worth thinking about. Especially if it helps gain understanding regarding the star product.

Best wishes,

Eric

PS: If you stair at the usual definition of the star product you will see exponentiatied derivative operators. Exponentiated derivatives operators are screaming out saying they way to transport something (think of Fourier analysis

)

PPS: The star product seems like a way to make a continuum theory act like a discrete theory.