New York, New York

April 23, 2004

Posted by Urs Schreiber

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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 $\exp (i λ)$ induces of course a modification of the constraint

(1)

(p - A)^{2} \to (p - A + d λ)^{2}

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 $p \to p + d λ$ 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 $ϕ$ the BRST operator $Q$ for flat Minkowski background and a star product $⋆$ , where the (classical) equations of motion for $ϕ$ are

(2)

Q ϕ + c ϕ ⋆ ϕ = 0,

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 $ϕ$ I considered adding an infinitesimel ‘test field’ $ψ$ to $ϕ$ and checking what equations of motion $ψ$ has to satisfy in order that $ϕ + ψ$ is still a solution of string field theory. To first order in $ψ$ one finds

(3)

(Q + c {ϕ ⋆, \cdot}) ∣ ψ ⟩ = 0 .

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

(4)

Q^{'} : = Q + c {ϕ ⋆, \cdot}

as the deformed BRST operator which corresponds to the background described by the background string field $ϕ$ .

It is due to the fact that $a ⋆ b$ and $b ⋆ a$ in string field theory have no obvious relation that I find it hard to see whether $Q^{'}$ 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 $ϕ$ 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.

Campus of Rockefeller University, New York

The campus of Rockefeller University.

Posted at April 23, 2004 5:25 PM UTC

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Re: New York, New York

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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 $AB$ is a 2-path (as in our notes) and finally

(1)

T (AB)

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

(2)

A ⋆ B = T (AB) .

This also explains why

(3)

A ⋆ B \neq B ⋆ A

because we know that

(4)

AB \neq BA .

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

(5)

e^{a \partial / \partial x} f (x) = f (x + a) .

)

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

Posted by: Eric on April 25, 2004 6:39 AM | Permalink | Reply to this

Re: New York, New York

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Hi Eric -

First, I was wondering if you could point out a good reference for these BRST operators.

A standard textbook on BRST formalism is

Marc Henneaux & Claudio Teitelboim: Quantization of Gauge Systems Princeton University Press (1994).

But probably you would like to see some quicker introduction. Surely there are some available, but I’d have to search for them right now.

But the basic idea is this (see Polchinski p. 133 for the first part):

Pick a theory with (bosonic) constraints $G_{I}$ with $I$ some index. Assume that the constraints generate a Lie algebra with structurte constant $f_{I}^{K}_{J}$

(1)

[G_{I}, G_{J}] = i f_{I}^{K}_{J} G_{K} .

Think of the $G_{I}$ as (invariant) vector field on the gauge group of the theory. The physical constraints

(2)

{\hat{G}}_{I} ∣ ψ ⟩ = 0

say that physical states have to be constants on the gauge group, i.e. they have to be gauge invariant. If we regard physical states as 0-forms on the gauge group this can succinctly be expressed as

(3)

d ∣ ψ ⟩ = 0

where $d$ is the exterior derivative on the gauge group.

In order to write down $d$ explicitly introduce form creators $c^{I}$ and form annihilators $b_{I}$ and using them write down the exterior derivative in the usual way (e.g. (A.39) of hep-th/0311064 )

(4)

d = c^{I} \nabla_{I}

where

(5)

\nabla_{I} = G_{I} + ω_{I}^{J}_{K} c^{K} b_{I}

is the covariant derivative on differentials forms, with $ω$ the Levi-Civita connection in the orthonormal basis associated with the orthonormal vector fields $G_{I}$ . Since on Lie groups $ω$ is proportional to the structure constants (e.g. equation D.8 of the above paper) one finds in this case that the exterior derivative on the gauge group reads explicitly

(6)

d = c^{I} (G_{I} - \frac{i}{2} g_{I}^{K}_{J} c^{I} c^{J} b_{K}) .

In the context of quantum field theory people call the differential form creators and annihilators $c^{I}$ and $b^{I}$ ghost fields (for historicaly reasons, since these are fields that appear in the path integral formalism alongside the real physical fields but have ‘strange’ properties from such a point of view) and call

(7)

Q_{B} : = d

the BRST operator.

In the context of (bosonic) string theory the $G_{I}$ are the Virasoro constraints $L_{m}$ , their (classical) algebra is $[L_{m}, L_{n}] = (m - n) L_{m + n}$ from which you can read off the structure constants. All the infinitely many constraints which govern the string can then be succinctly be rewritten in one line:

(8)

Q_{B} ∣ ψ ⟩ = 0 .

There are more advantages to the BRST formalism than just notational convenenience. A couple of quantum effects which are somewhat mysterious in standard formalism become very clear in BRST formalism, such as the ground state energy offset $a$ and the critical diemsnion. Note how the latter comes about:

Formally we have always $d^{2} = 0$ . But for the string the Lie group that we are dealing with is infinite diemsnional! It turns out that this infinite dimensionality invalidates the naive general conclusion $d^{2} = 0$ . It turns out that due subtle effects when computing the square of the above $Q_{B}$ operator it squares to zero only under the condition that the central charge $c$ which characterizes the anomaly in the Virasoro algebra

(9)

[L_{m}, L_{n}] = (m - n) L_{m + n} + \frac{c}{12} (m^{3} - m) δ_{m + n, 0}

and which is for a flat Minkowski background equal to the number of spacetime dimensions

(10)

c = D

is precisely equal to

(11)

c = D = 26 .

That’s simply because the algebra of the ghosts that enter $Q_{B}$ picks up an anomaly, too, which happens to be equal to $- 26$ ! But nilpotency of $Q_{B} = d$ requires that all these anomalies cancel out, so this tells us that thebososnic string is consistent only in 26 dimensions.

I assume that you should love that: The fact that $d^{2} = 0$ for the string tells us the number of spacetime dimensiuons that we must live in! :-)

Another thing is that I once got very excited about the star product

I recall that we talked about that a long time ago already. I think that I pointed out that the star product that you are referring to is not the same as that used in string field theory. In the latter case the star product is something that takes to string states and spits out the state obtained by merging these two strings.

If I recall correctly you then pointed me to some literature which apparently showed or at least indicated that the string field star product is indeed related to the Moyal star product (was it some post by Aaron Bergman, I forget). Anyway, I have never understood how that works! :-)

I think I see pretty well what you are getting at in your comment concerning parallel tranport, fan products and so on, but currently I don’t see how it relates to the string field star product. Let’s try to clarify that, first.

(if you keep at me with an open mind I’m sure we could make this precise

I do! :-) Let’s try. What exactly is the relation between the Moyal star and the string field star? Anyone?

Posted by: Urs Schreiber on April 26, 2004 11:00 AM | Permalink | PGP Sig | Reply to this

Re: New York, New York

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Good morning! :)

Thanks for your post :)

I assume that you should love that: The fact that $d^{2} = 0$ for the string tells us the number of spacetime dimensiuons that we must live in! :-)

Prior to our adventures with diamonds and looking at “geometric paths” as those for which $\partial^{2} = 0$ , I would have been less receptive to this idea. Now, I am definitely interested in trying to understand it better :)

I recall that we talked about that a long time ago already. I think that I pointed out that the star product that you are referring to is not the same as that used in string field theory. In the latter case the star product is something that takes to string states and spits out the state obtained by merging these two strings.

That last URL I gave takes you directly to the neat observations that unfolded as the thread progressed. If you look here:

Beginning of thread on NCG/SUSY/*-Product

You can see the story unfold. I’ll paste the initial post because I think it sums up the direction I started out in:

Hello,

I recently stumbled onto something that could potentially open up mountains of research/reading and before starting the journey I was hoping someone might say a few words summarizing what this *-product is that pops up in noncommutative geometry. It looks suspiciously like something I have cooked up on my own. If that is the case, then I have reinvented another wheel :)

Does the *-product have anything at all to do with composition of strings or paths. A quick lanl search in addition to a quick spr search turned up tons of hits regarding NCG and string theory. I saw Aaron mention something about string algebra and Prof Baez mention it in regard to products on phase space.

What exactly is a string algebra? Is there some operation in string theory that composes strings/paths? A path algebra is a very obvious construction and I reinvented it on my own and I even called it “path algebra”. Is it at all related? Could path algebra be a kind of discrete string algebra? A path algebra is simply a directed graph with product of paths in the graph defined by concatenation when such a concatenation makes sense. When it doesn’t, i.e. the end of the first path doesn’t correspond to the beginning of the next path, then the product is zero.

It would be really fascinating to me if the work I have been doing on path algebras is remotely related to string theory. I already know that my stuff is related to NCG, and I have heard rumors that NCG and string theory were related, so it would be quite cool if string theory came into the picture. I’m fairly certain I would be the first electrical engineer to write his thesis on string theory :)

Anyway, I’d really appreciate some words of wisdom from people who know about this stuff.

Thank you very much,
Eric

The point is that already even before I knew about the string field theory *-product, I could see (“feel” is probably a better word) a relation between the Moyal *-product, our product on discrete spaces, and a possible to link to a product that concatenates strings. I wish I could make it more than a “gut feeling”, but right now that is all it is :) I’ll do what I can to make the idea as precise as I can and maybe with your more refined vision, you can maybe pick out something worth keeping. So be forewarned. I will spew a bunch of garbage before we find the connection, but just like the old gold miners. You need to sift though tons of mud to find a small gold nugget :)

The first thing we (I?) should do is to dig up those old references (again) that Aaron pointed out and try to understand what other people think of it. I’ll be coming from a different perspective thinking in terms of discrete geometry. I see a neat picture that can unfold if we massage it right.

Let’s go! :)

Eric

Posted by: Eric on April 26, 2004 3:18 PM | Permalink | Reply to this

Re: New York, New York

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Hello,

With very little effort I am already prepared to provide my first wild speculation :)

Let the sifting begin! :)

Open String Star as a Continuous Moyal Product
Michael R. Douglas, Hong Liu, Gregory Moore, Barton Zwiebach

We establish that the open string star product in the zero momentum sector can be described as a continuous tensor product of mutually commuting two dimensional Moyal star products. Let the continuous variable $κ \in [~ 0, ∞)$ parametrize the eigenvalues of the Neumann matrices; then the noncommutativity parameter is given by $θ (κ) = 2 \tanh (π κ / 4)$ . For each $κ$ , the Moyal coordinates are a linear combination of even position modes, and the Fourier transform of a linear combination of odd position modes. The commuting coordinate at $κ = 0$ is identified as the momentum carried by half the string. We discuss the relation to Bars’ work, and attempt to write the string field action as a noncommutative field theory.

Ok. I think that the Moyal *-product is definetely related to our product via that map that I gave earlier (at least heuristically)

(1)

A ⋆ B = T (AB) .

The above abstract indicates a relation between the open string *-product and a continuous tensor product of Moyal *-products.

This is interesting because we have already conjectured that the discrete loop space is a tensor product of discrete spaces (as developed in our notes)

Hence, this is the desired course of action I think:

1.) Make the relation between the Moyal *-product and our product on a discrete space precise.

2.) Make the conjecture that discrete loop space is a tensor product of copies of a single discrete space precise. (It follows that the continuum limit would involve a continuum of tensor products).

3.) Assuming all this goes through, make the relation between the open string *-product and our product on discrete loop space precise using arguments similar to those found in the above paper.

Sounds easy enough, right? Yeah right! :)

What do you think so far?

Eric

Posted by: Eric on April 26, 2004 3:52 PM | Permalink | Reply to this

Re: New York, New York

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Hi Eric -

I have now read through all (or at least most) of the stuff that you mentioned here.

The Douglas et. al paper hep-th/0202087 on how the string field star (without the ghost sector) is the same as a continuous copy of Moyal stars is pretty interesting. It takes a while to get an intuition for this, though, in particular due to involved relations such as (3.2). But the dipole picture mentioned right after (2.13) (and (2.13) itself) helps.

So, ok, we know how the Moyal star and the string field star are related, formally. You need to help me, though, with how this relates to the noncommutativity in general NCG and maybe in particular to the path/fan algebras that you have in mind.

In particular, you need to help me see how you think that the Moyal product $A ⋆ B$ is related to what you wrote as $T (AB)$ .

- - From your discussion with Thomas Larsson at spr I understand how (oriented) areas spanned by two vectors play a role in the Moyal product. It seems to me that John Baez’s comment on how this gets a nice physical interpretation for charged particles in 2d with a transverse EM field should be relevant for making your intuitive connection to parallel transport precise.

BTW, in that thread Thomas Larsson mentioned that the ‘enclosed area’ construction fails in $d > 2$ and you wondered what he might mean by that.

Thomas Larsson certainly meant that in $d > 2$ there is no matrix $Θ$ such that $\overset{⇀}{m} \cdot Θ \overset{⇀}{n}$ is proportional to the area spanned by $\overset{⇀}{m}$ and $\overset{⇀}{n}$ . But you are perfectly right that this is irrelevant for one obvious generalization of the Moyal star in one natural sense: You would just define in arbitrary $d$

(1)

e^{i \overset{⇀}{m} \cdot \overset{⇀}{x}} ⋆ e^{i \overset{⇀}{n} \cdot \overset{⇀}{x}} = e^{i A (m, n)} e^{i (\overset{⇀}{m} + \overset{⇀}{n}) \cdot \overset{⇀}{x}}

where $A (m, n)$ is defined to be the area spanned by the two vectors, not something obtained from matrix multiplications.

But: Do we still have associativity in this generalized case for three non-coplanar vectors?

Concerning the question how the relation between Moyal star and string field star help to find some NCG generalization of this stuff I am not sure yet, maybe I haven’t thought about it sufficiently. I will first need to understand your $A ⋆ B \sim T (AB)$ ! :-)

Posted by: Urs Schreiber on April 28, 2004 5:58 PM | Permalink | PGP Sig | Reply to this

Re: New York, New York

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Hi Urs :)

So, ok, we know how the Moyal star and the string field star are related, formally.

Excellent :)

You need to help me, though, with how this relates to the noncommutativity in general NCG and maybe in particular to the path/fan algebras that you have in mind.

In particular, you need to help me see how you think that the Moyal product $A ⋆ B$ is related to what you wrote as $T (AB)$ .

I wish I knew :) It is still little more than a gut feeling, but I can try to make my ideas materialize a little more.

Maybe first, I’ll make some observations that are (obscurely) related. Consider three vectors $A_{1}, A_{2}, A_{3} \in R^{n}$ denoting points $[1]$ , $[2]$ and $[3]$ and let

(1)

A_{ij} = A_{j} - A_{i} .

The vectors $A_{12}$ , $A_{23}$ , and $A_{13}$ may be used to represent the sides $[12]$ , $[23]$ and $[13]$ of the triangle $[123]$ . Now, parallel transport $A_{23}$ back from node $[2]$ to node $[1]$ and denote the result $A_{13'}$ , i.e.

(2)

A_{13'} = H_{2 \to 1} A_{23} .

This “new” vector defines a new triangle with sides $[12]$ , $[23']$ , and $[13']$ .

I know this sounds completely mysterious and unfortunately I don’t have time to explain it right now, but the first triangle is analagous to a “path”, the second triangle is analogous to a “fan” and the process of parallel transporting an edge of a “path” to obtain a “fan” is the map $T$ from my earlier post :)

Note that the area of the two triangles is equivalent. This is a trivial (yet don’t over trivialize it because I think it will be relevent) observation :)

Keep in mind, we’ll need to sift through a lot of garbage before I’ll be able to say something sensible :)

Gotta run!

Eric

PS: I think we should be thinking in terms of parallel transport on spheres. Instead of triangles, I believe we are dealing with spherical triangles. The Moyal *-product relates to parallel transport on a space with constant curvature so that parallel transport around a loop is proportional to the area enclosed by the loop. This is only true, i.e. holonomy proportional to area, when the curvature is constant.

Posted by: Eric on April 28, 2004 7:12 PM | Permalink | Reply to this

Re: New York, New York

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Hello! :)

I am at a library with IE and no ability to check my TeX, so forgive me if this doesn’t look right :)

On a discrete space $G$ , construct the product space $G \times G$ . Functions on this space will look like

(1)

f = \sum_{ij} f (i, j) e^{i} \otimes e^{j} .

with evaluation given by

(2)

(e^{i} \otimes e^{j}) (k, l) = δ_{k}^{i} δ_{l}^{j} .

Let’s define a product of functions on $G \times G$ by first defining the product of basis elements via

(3)

(e^{i} \otimes e^{j}) ⋆ (e^{k} \otimes e^{l}) = {delta}_{jk} e^{i} \otimes e^{l} .

Extending by linearity we have

(4)

f ⋆ g = \sum_{i, j, k} f (i, j) g (j, k) e^{i} \otimes e^{k}

so that

(5)

(f ⋆ g) (i, k) = \sum_{j} f (i, j) g (j, k) .

Does this look familiar? :)

Sorry, it is not exactly related to the product of 1-forms as I thought, but rather the product of 0-forms on a product space. This is about the most obvious way to define a product on a product space and it gives you a discrete version of the Moyal *-product.

Another example where the discrete theory brings illumination to an otherwise cloudy picture :)

I still need to relate this to the stuff in the spr thread.

Cheers!
Eric

PS: It probably wouldn’t hurt to learn from the masters

Moyal Deformation, Seiberg-Witten-Map, and Integrable Models
Aristophanes Dimakis, Folkert Muller-Hoissen

Posted by: Eric on April 29, 2004 2:44 AM | Permalink | Reply to this

Re: New York, New York

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Hello,

This is kind of neat and it gives a pretty cool interpretation of a function on phase space even in the continuum. If you have a function $f$ on phase space, then the value of the function evaluated at a point, i.e.

(1)

f (x_{0}, p_{0})

is like an entry of a matrix having a continuum of entries. In other words, you can think of $f$ as a matrix with a continuum of entries and then the Moyal *-product is just matrix multiplication, which is of course associative :)

The reason I thought it would involve parallel transport is that it is like you are multiplying functions evaluated at different points. The exponentiated differential operators really just offset the functions prior to multiplying them.

It would still be nice to make the connection to the spr thread. In particular to that example of particles confined to a plane in the presence of a B field.

But for now, good night!

Eric

Posted by: Eric on April 29, 2004 5:06 AM | Permalink | Reply to this

Dipoles

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Does this look familiar? :)

Yup! :-)

Ok, I get the idea. You are probably saying that somehow hep-th/0202087 shows that the string field star product can be interpreted as an infinite number of concatenations of continuously many dipoles $(\oplus \to \otimes)$ which somehow make up the open string

(1)

(\oplus \to \otimes) (\oplus \to \otimes) (\oplus \to \otimes) \dots (\oplus \to \otimes) .

You are imagining that by reducing the number of dipoles in the string from $ℵ_{1}$ to some natural number we get a discretization and can still define a string field product by concatenating the dipoles with the discrete Moyal product that you mentioned above. Right?

Yes, maybe that’s a good way to think about it. I’ll have a look at the D&MH paper that you mentioned as well as at

D. Bigatti & L. Susskind: Magnetic fields, branes and noncommutative geometry (2000).

That should help me to get a better feeling for what is going on here.

Next, it will be necessary (at least for me) to heuristicaslly understand what happens in the transformation in section 3 of Douglas et al’s hep-th/0202087:

What is the $κ$ -th ‘dipole’ on the string, heuristically?

Comparison of (4.11),(4.13) with (2.12) suggests that it is an object consisting of two endpoints $x_{κ L}$ and $x_{κ R}$ given by

(2)

x_{κ L} = x_{κ} + \tanh (π κ / 4) q_{κ}

and

(3)

x_{κ L} = x_{κ} - \tanh (π κ / 4) q_{κ}

with $x_{κ}$ and $q_{κ}$ given by (4.11) and (4.13).

In order to interpret what these $x_{κ L / R}$ mean I’d need to understand the nature of the coefficients $v_{n} (κ)$ that enter their definition in terms of the Fourier modes $x_{n}$ , whose meaning I do understand.

Unfortunately, all that I really know about the $v_{n} (κ)$ is their implicit definition in (3.4).

Before I can conjure up a discrete picture of what is happening in the string field star product I’d need to understand the $v_{n} (κ)$ ! :-)

Posted by: Urs Schreiber on April 29, 2004 12:11 PM | Permalink | PGP Sig | Reply to this

Re: Dipoles

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Good morning! :)

You are probably saying that somehow hep-th/0202087 shows that the string field star product can be interpreted as an infinite number of concatenations of continuously many dipoles $(\oplus \to \otimes)$ which somehow make up the open string

I don’t know if concatenation is the word I would use to describe it.

You are imagining that by reducing the number of dipoles in the string from $ℵ_{1}$ to some natural number we get a discretization and can still define a string field product by concatenating the dipoles with the discrete Moyal product that you mentioned above. Right?

Yes, basically :)

The thing is that I “think” that the discrete loop space (instead of “discrete loop space” maybe “polymer space” is a better word because the discrete loops would look a lot like polymers :)) would be $n$ copies of a single discrete (point) space. This, so far, is just a conjecture, but since it was a conjecture made by you, I have more faith in it than I would in one of my own conjectures :)

If this conjecture is true, then given a discrete space $G$ , let $G^{n}$ denote $n$ copies of $G$ , i.e.

(1)

G^{n} = \underset{n copies}{\underset{⏟}{G \times . . . \times G}}

should describe some kind of discrete loop space.

The discrete phase space of discrete loop space would then be

(2)

G^{n} \times G^{n} .

My suggestion would then be to go ahead and complete the description of discrete loop space and define an abstract calculus $Ω (𝒜, d)$ on this discrete loop space. Once we have this, then we define the discrete open string *-product on the discrete phase space $G^{n} \times G^{n}$ of the discrete loop space $G^{n}$ in analogy to the way the discrete Moyal *-product is defined on the discrete phase space $G \times G$ of a single discrete (point) space $G$ .

In other words, now that we seem to have a discrete Moyal *-product on a discrete phase space $G \times G$ , I conjecture that the discrete open string *-product is essentially just the discrete Moyal *-product on the discrete phase space $G^{n} \times G^{n}$ of a discrete loop space $G^{n}$ .

[note: if the abundance of the word “discrete” seems cumbersome, I’d blame it on historical accident. If the discrete theory had been developed first, we would have constructed physics completely from the discrete perspective. Then to compare it to the continuum theory, you’d have to keep saying “continuum (blah)”, which would then seem cumbersome and you’d say, “Why do we even need the continuum?!” :)]

If this conjecture turns out to be true, then that would give a nice interpretation of the continuum theory as well :) It would mean that the usual continuum open string *-product is just the Moyal *-product in the phase space of loop space rather than the phase space of particles. Once you say it, it seems kind of obvious and natural :)

That would probably make a nice paper if it were true :) Because I have no confidence in myself, it is probably too good to be true. Maybe you can find the flaw before I get too excited :)

Ciao,
Eric

Posted by: Eric on April 29, 2004 2:23 PM | Permalink | Reply to this

Re: Dipoles

My problem is that you seem to be thinking of doing the Moyal product for every single ‘bit’ of string. But that’s probably not the correct way to do it, because from hep-th/0202087 we learn that instead we have to apply the Moyal star to each of the $x_\kappa$ of equation (4.11). This are modes of the string, but not the $\delta$-‘modes’ as in your proposal.

See what I mean?

Posted by: Urs Schreiber on April 29, 2004 2:31 PM | Permalink | Reply to this

Re: Dipoles

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

I see what you mean, but as I’ve mentioned before, I think a “bit” of a discrete string can be thought of as a mode of a continuum string. At least they are in correspondence (I think) :)

Eric

Posted by: Eric on April 29, 2004 2:36 PM | Permalink | Reply to this

Re: Dipoles

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

I should probably explain what I meant by that. If you have a finite number of nodes describing a discrete string, fields on the discrete string are analogous to a digital signal. If you have $m$ nodes then you can $m$ (kronecker) delta functions to describe the “signal”. A delta function corresponds to a mode in the dual (Fourier) space and vice versa. So you can completely describe the fields on a discrete string via $m$ delta functions or $m$ modes and there is a map between the two descriptions.

Eric

Posted by: Eric on April 29, 2004 2:45 PM | Permalink | Reply to this

Re: Dipoles

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

Yes, agreed. We can make a cutoff or something in mode space. But in order to understand what we are doing we need to know how the $κ$ -modes look like.

It should be a straightforward exercise. But currently I need to do something else, unfortunately. But we will solve this.

Posted by: Urs Schreiber on April 29, 2004 2:50 PM | Permalink | PGP Sig | Reply to this

Re: Dipoles

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

Ok, good. If we want to make any progress along this way of thinking then we need to figure out in which sense this is true. With respect to equation (4.11) of the Douglas et al. paper:

What are the modes

(1)

{mode}_{κ} (σ) = \sum_{N = 1}^{∞} v_{2 n} (κ) \sqrt{2 n} e^{in σ},

where $σ$ is the coordinate along the string? I have posted a similar question to s.p.s.

Posted by: Urs Schreiber on April 29, 2004 2:47 PM | Permalink | PGP Sig | Reply to this

Re: Dipoles

Maybe the answer to our question is given in section 4.2 of

C. Chu & P. Ho & F. Lin: Cubic String Field Theory in pp-wave Background and Background Independent Moyal Structure (2002) .

Posted by: Urs Schreiber on May 7, 2004 4:12 PM | Permalink | PGP Sig | Reply to this

Re: New York, New York

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

Concerning my question whether

(1)

Q + c {ϕ, \dots}

is nilpotent, I’d need to refresh my unfortunatly insufficent familiarity with OSFT, but after thinking about it it seems to me that schematically the question is whether

(2)

(Q + ϕ)^{2} = 0,

where $⋆$ -multiplication is implicit. And it seems that it is if $ϕ$ is of odd ghost number (and it is indeed of ghost number 1, of course, e.g. equation (2.12) of the review

K. Ohmori: A Review on Tachyon Condensation in Open String Field Theories (2001) )

then this does indeed vanish due to the equations of motion satisfied by $ϕ$ .

If we think of $Q$ as the exterior derivative $d$ on the gauge group and of $ϕ$ as a 1-form connection $ω$ on the group, then what I am saying is that the covariant exterior derivative

(3)

d + ω \land

is nilpotent if

(4)

d ω + ω \land ω = 0 .

Hm, this means that the ‘connection’ is flat.

The condition on the ‘test field’ $ψ$ , which we might write as another 1-form $γ$ to give a consistent (i.e. still flat) new connection $ω + γ$ would then be simply that $γ$ is ‘covariantly constant’ in the sense that

(5)

{d + ω, γ} = 0 .

Hm…

Posted by: Urs Schreiber on April 29, 2004 2:08 PM | Permalink | Reply to this

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