Before the flood

March 5, 2004

Posted by Urs Schreiber

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There is an old paper

A. Chamseddine, An Effective Superstring Spectral Action

which I have mentioned before and which keeps haunting me.

This paper is from an era where some people (Chamseddine, Fröhlich and others) have tried to identify the spectral aspect of Connes’ noncommutative geometry - namely that emphasizing the role of the Dirac operator - in string theory. According to Alejandro Rivero this was before the flood released by BFSS Matrix Models, strings in $B$ -field backgrounds, etc., where the algebraic aspect of noncommutative geometry - namely the noncommutativity! - is emphasized instead.

I am not sure why the original spectral string activity didn’t survive the great flood. But from reading section 3 of the above paper by Chamseddine I get the impression that maybe a certain idea was missing. That’s what I want to talk about here. That, and how superstring theory in terms of spectral triples relates to DDF operators, classical invariants of string, deformations of SCFTs, discrete differential geometry and all that.

Let me briefly indicate what this section 3 is concerned with:

In section 2 the author had discussed how the superconformal constraints of the Type II string for gravitational and Kalb-Ramond background gives rise to Dirac operators on loop space. He then outlines the role that he imagines these Dirac operators should play. He writes:

Most of the considerations of the last section could be looked at from the non-linear sigma model study and one may ask for the relevance of noncommutative geometry. The point of view we like to advance is that once a spectral triple $(𝒜, ℋ, D)$ is specified it is possible to define a noncommutative space and use the tools of noncommutative geometry. […]
For the example studied in the last section we have $𝒜 = C^{∞} (Ω (M))$ , the algebra of continuous functions on the loop space over the manifold $M$ . Elements of the algebra are functionals of the form $f [X^{μ} (σ)]$ where $σ$ parameterizes the circle. […]
There is also an advantage in treating this model with the noncommutative geometric tools as this would allow us to consider, in the future, more complicated examples which could only be treated by noncommutative geometric methods. […]
To illustrate, consider the operator $D = Q_{+} + Q_{-}$ [the sum of the supercharges of the string] […]. restricting to states which are reparameterization invariant […] it is possible to build the universal space of differential forms. A one-form is given by $π (ρ) = \sum_{i} f^{i} [D, g^{i}] = \sum_{i} \int d σ {[f^{i} [X] (ψ_{+}^{μ} + ψ_{-}^{μ}) \frac{δ g^{i}}{δ X^{μ}}]}_{P = 0} .$

The idea here is clear: Take the supercharges of the string as Dirac operators, identify the algebra on which these act, construct the corresponding spectral triple and – do something sensible with this machinery that boosts our understanding of string theory!

I want to suggest an approach similar but different to what is sketched at the end of this section 3. I want to show how we can construct an abstract differential calculus with exterior differential $d$ and forms $ω$ such that the equations $(d \pm d^{†}) ω = 0$ are equivalent to the full set of worldsheet constraints of Type II strings. I don’t claim that this is a particularly difficult problem and in fact the solution is rather trivial, but I think that something interesting can be learned from it.

(For the start, I will consider the classical superstring only, i.e. work at the level of Poisson brackets.)

The crucial differences of the approach that I am going to discuss here to that of Chamseddine will be the choice of algebra as well as the definition of differential forms.

Namely I will follow the approach to spectral geometry which is sketched here and which starts by defining an abstract exterior calculus over a given algebra $𝒜$ .

For this algebra I do not choose the algebra of reparameterization invariant functions, but the algebra $𝒜$ generated by integrals (along the string at fixed worlsheet time) of fields of reparameterization weight 1/2.

Now let $d$ be the 0-mode of the $K$ -deformed exterior derivate over loop space, as described here and consider the differential calculus $Ω (𝒜, d)$ generated from $𝒜$ and $d$ .

The trick is that this way all exact forms of this differential calculus are automatically reparameterization invariant objects, because they are integrals over unit weight fields. Using this reparameterization invariance it is now easy to see that the equations $[d, ω] = 0$ $[d^{†}, ω] = 0$ for $ω \in Ω (𝒜, d)$ (where the bracket is the graded Poisson bracket) indeed imply the full superstring constraints $[G_{m}, ω] = 0$ $[{\bar{G}}_{m}, ω] = 0 .$ That they are even equivalent to them follows from noting that every solution to the latter set of equations is indeed $d$ -exact! Namely this is the result of the construction of the classical supersymmetric DDF operators which shows that all physical states of the superstring are obtained from acting with $d$ repeatedly. Indeed, by looking at the construction principle of the DDF invariants of the classical superstring one sees that they have precisely the form of an exact abstract differential form $ω = [d, a_{1}] [d, a_{2}] \dots [d, a_{p}] .$

Phew, I am running out of time. I have an appointment at the cinema tonight which I am about to miss. I’ll have to continue this here tomorrow. Let me just say that my evil plan is to use the above construction to modify the algebra $𝒜$ and get deformed worldsheet theories this way. I believe that this was originally Chamseddine’s et al.’s plan. Find the spectral triple of superstrings and then deform. Before the flood, at least.

Ok, gotta run.

Posted at March 5, 2004 10:02 PM UTC

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Re: Before the flood

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Hi Urs,

As usual, you are boggling my mind with how fast you progress with all this stuff.

I won’t pretend to understand everything here, but one thing seems like it is too coincidental to be an accident.

You write that

(1)

[d, ω] = 0, [d^{†}, ω] = 0

implies

(2)

[G_{m}, ω] = 0, [{\bar{G}}_{m}, ω] = 0 .

This reminds me a LOT of Equation (2.44) in our notes.

You even use the same letter :)

Is there anything deep and meaningful here or is it just a coincidence after all? I always had a gut feeling that this discrete differential geometry was somehow related to string theory. I’m glad to see you starting to make things explicit.

Eric

PS: I hope you had fun at the cinema. I’m trying to mentally prepare for a 30km run tomorrow *gulp!* :)

PPS: I think this abstract exterior calculus dates back to Koszul. If you haven’t seen it, another one of my all time favorite papers is

Geroch, R. Einstein Algebras, Communications in Mathematical Physics, 26, 271, 1972

Posted by: Eric on March 7, 2004 5:23 AM | Permalink | Reply to this

Re: Before the flood

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Hi Eric -

in what I wrote, $G_{m}$ , as conventional in the string literature, is supposed to be a mode of the worldsheet supercurrent. It is not a gluing 1-form or graph operator which we denoted by $G$ in our notes - but of course it is related…

The exterior derivative that I was talking about is constructed from the 0-modes $G_{0}$ and ${\bar{G}}_{0}$ as $d \propto {\bar{G}}_{0} + i G_{0}$ and $d^{†} \propto {\bar{G}}_{0} - i G_{0}$ . (See section 3.1 of this for the details.) The point is that a classical invariant observable of superstring has to supercommute with all the $G_{m}$ and ${\bar{G}}_{m}$ , $\forall m \in Z$ but that with an appropriate choice of algebra $𝒜$ the object $[d, \cdot]$ becomes nilpotent and invariance follows already from supercommutation with only $G_{0}$ and ${\bar{G}}_{0}$ , i.e. with only $d$ and $d^{†}$ .

The point is that this way the Type II superstring is equivalently described by an $N = 2$ spectral triple (and all what we did in our notes for these spectral triples is applicable in principle).

Above I ran out of time with describing the formulation that I have in mind. Here is another attempt:

An interesting aspect of this construction is that it kind of looks like higher-order genelization of the NCG description of the superparticle. So let me describe things from this point of view:

Start with the algebra $𝒜 = C^{∞} (M)$ which is generated by instanton/D(-1)-brane positions, namely (smeared) delta-functions on $M$ (spacetime). Let $d$ be the ordinary exterior derivative acting on this algebra and let $ℋ = (Ω (𝒜, d), ⟨ \cdot ∣ \cdot ⟩)$ be the inner product space of differential forms associated with this algebra. The spectral triple $(𝒜, d, ℋ)$ together with the equations of motion $[d, ω] = 0 = [d^{†}, ω]$ ( $ω : ℋ \to ℋ$ is an ‘observable’) describe the R-R-sector of the NSR form of the superparticle, namely nothing but quanta of form fields. (The NS-R and NS-NS sectors of the superparticle are equivalently obtained by using the equations of motion $[d + d^{†}, ω] = 0$ and $[(d + d^{†})^{2}, ω] = 0$ ).

Note that by starting with the an algebra associated with $D (- 1)$ branes (spacetime events) we obtain an algebra of D0 branes (point particles) this way.

I claim that by roughly repeating this process we obtain strings ( $D 1$ -branes) from the algabra of super D0-branes.

To this end, consider a collection of the above superparticles indexed by a parameter $σ$ . Let $𝒜^{(1)}$ be the algebra generated by formal superpositions of D0-brane observables of the form $\sum_{σ} ω (σ)$ , where $ω (σ)$ is an observable of the superparticle with index $σ$ and we impose a certain condition on the relation between the $ω (σ)$ at different $σ$ (namely that the integrand is of rep weight 1/2) as well as appropriate boundary conditions.

For instance

(1)

ω^{(1)} = \int_{0}^{2 π} d σ γ^{μ} (σ) δ (x^{μ} (σ) - x_{0}^{μ} (σ))

is such a weight (1/2) algebra generator which describes a collection of superparticles with coordinates sitting at the fixed coordinates $x_{0}^{μ} (σ)$ . Here $γ^{μ} (σ) = e^{† μ} (σ) + e^{μ} (σ)$ is a ‘gamma-matrix’ observable for the $σ$ th superparticle.

Now let $d^{(1)} \propto i G_{0} + {\bar{G}}_{0}$ be the $K$ -deformed exterior derivative on loop space, i.e. a linear combination of left- and rightmoving supercharges of the NSR string. This object squares to the generator of rigid rotations along $σ$ . But the elements of $𝒜^{(1)}$ are constructed in such a way that they are invariant under these rigid reparameterizations, so $d^{(1)}$ is nilpotent on $𝒜^{(1)}$ and hence an honest exterior derivative. We can again construct the associated spectral triple and I claim that it now contains all physical observables of the superstring. In particular, the equations of motion are now

(2)

[d^{(1)}, a] = 0

(3)

[d^{† (1)}, a] = 0

where $a = [d^{(1)}, ω_{1}] \dots [d^{(1)}, ω_{p}]$ is a $p$ form with respect to the new spectral triple, where all the $ω_{p}$ are integrals over weight 1/2 fields as described above.

These $a$ are products of DDF observables of superstring. There is a subtlety with the 0-mode of the string’s coordinate fields which forces the total center-of-mass momentum of $a$ to vanish. (This does not mean that the string’s com momentum has to vanish, just that these observables are not sensitive to the com-momentum.) Hence the $a$ are really the superstring generalizations of the Pohlmeyer invariants (which can be rewritten in terms of DDF invartiants and vice versa).

Ok, this reformulates the superstring constraints in a peculiar NCG-like algebraic way. Now let me address the issue of the gluing 1-form.

Let met get back to the superparticle and let’s replace $C^{∞} (M)$ with a discretized approximation for the moment, i.e. some algebra generated by discrete delta-functions which described a disrete set of points. Then it is known that $d$ encodes the connectivity of these points. More precisely, the 1-form

(4)

δ (x - x_{i}) [d, δ (x - x_{j})]

in nonvanishing if and only if there is an edge running from vertex $x_{j}$ to vertex $x_{i}$ , or, in other words, if a (super)particle can move from $x_{j}$ to $x_{i}$ in one ‘step’. One denotes by $ρ$ the gluing 1-form which is just the sum of all edges of the graph:

(5)

ρ = \sum_{i, j} δ_{i} [d, δ_{j}] .

It turns out that on graphs without opposite and without intermediate edges we have

(6)

[d, ω] = [ρ, ω] .

In this sense $d$ knows all about the graph connectivity.

Now suppose that we build the superstring in the above sense on the theory of the superparticle on a discrete space. Then the integrals over weight 1/2 fields play the role of vertices and the connectivity of these vertices is encoded in the worldsheet supercharges, which, indeed, are then related to higher order gluing-1 forms as described above. A higher-order edge in this sense should be a way for a string to evolve in one ‘step’.

Posted by: Urs Schreiber on March 7, 2004 2:30 PM | Permalink | Reply to this

Re: Before the flood

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Hi Urs,

After rereading this entry, I thought I’d ask how the efforts along this direction are going? You’ve probably explained it elsewhere, but we’re really at the fringes of my ability to follow so I could well have missed it :)

If I’m not mistaken, this framework seems perfectly suited for our discrete differential geometry. However, it seems the algebra $𝒜$ is probably not commutative (although that of closed loops may be commutative). Is that right? A node in the graph would correspond to a loop, which could be explicitly thought of as a polymer in a product of graphs. If we worked this out, my gut tells me that we’ll finally be able to make the relation to the Moyal *-product manifest.

I also think that if you worked in the discrete world, the formulation might be much simpler.

Eric

Posted by: Eric on April 3, 2004 6:52 PM | Permalink | Reply to this

Re: Before the flood

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However, it seems the algebra $𝒜$ is probably not commutative (although that of closed loops may be commutative). Is that right?

There are a couple of ways to define a product of discrete strings. After I wrote that, I now think the product I had in mind is not very natural. There is a more natural product that is commutative where the discrete strings behave like Dirac delta functions (just as the algebra in our notes) on a discrete loop space. This is probably the product you would use.

It was your idea to consider products of a complex as a discrete kind of loop space and I think I’m starting to see how it will all fit together. If only I could clone myself to spend some time on it :)

I wish I could better understand your work with the DDF stuff. Although I don’t understand it very well at all, it feels right.

On a discrete loop space, we could easily construct $d$ and $d^{†}$ following our prescription. If we took these operators and defined

(1)

G_{0} \propto d - d^{†}

and

(2)

{\bar{G}}_{0} \propto d + d^{†}

would these correspond to the “0-modes” you are talking about? This seems to be kind of the reverse of what you are talking about here where you start with $G_{0}$ and ${\bar{G}}_{0}$ and construct $d$ and $d^{†}$ from that.

On another note…

I’m making really slow progress learning string theory, but I see that in the conformal gauge, the classical equation of motion is just the 1d wave equation. This is curious from a discrete point of view because the discretization of the 1d wave equation is exact, i.e. there is no error introduced by discretizing the 1d wave equation. Solutions propagate without error. It seems to me like this fact may turn out to be very interesting for a discrete string theory.

Eric

Posted by: Eric on April 4, 2004 3:42 AM | Permalink | Reply to this

Re: Before the flood

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Ok. It is 3:52AM and I’ve got a horrible cold and have completely lost my voice. That should be fair enough warning for the wild speculation I’m about to make :)

Recently, I’ve heard (somewhere) that string theory predicts an “infinite tower” of particles rather than the presently observed three (there may be more observed with higher energy accelerators, right?). Am I totally off base in saying that I think the reason for this infinite tower is due to taking a Fourier transform on a continuum line segment, i.e. the string is taken as a continuum of points?

The wild thought that just occurred to me (based upon thinking in terms of a discrete loop space) is what if a string really consisted of only a finite set of nodes? For example, if the string consisted of 3 nodes, then there would only be 3 Fourier modes available. Could this possibly correspond to a tower of three particles? One for each Fourier excitation/mode?

I am just about convinced now that if we take a directed graph $𝒢$ and construct an abstract differential calculus on it (precisely as in our notes), then we could construct a discrete loop space $ℒ (𝒢)$ over $𝒢$ by taking direct products of $𝒢$ with itself. For example, a loop space where each string consisted of three nodes, giving a “tower” of three excitations would simply be

(1)

ℒ (𝒢) = 𝒢 \times 𝒢 \times 𝒢 .

0-forms on $ℒ (𝒢)$ would be spanned by elements of the form

(2)

e^{i} \otimes e^{j} \otimes e^{k},

where the $e^{i}$ are the bases for 0-forms on $𝒢$ . The coboundary acts on these elements in the obvious manner via

(3)

d (e^{i} \otimes e^{j} \otimes e^{k}) = ({de}^{i}) \otimes e^{j} \otimes e^{k} + e^{i} \otimes ({de}^{j}) \otimes e^{k} + e^{i} \otimes e^{j} \otimes ({de}^{k}) .

You might then say that the ground state of $ℒ (𝒢)$ is spanned by elements of the form

(4)

e^{i} \otimes e^{i} \otimes e^{i},

which are formally indistinguishable from $e^{i}$ . Then the ground state fields would be indistinguishable from ordinary forms on $𝒢$ . That seems to resonate well with some things you have told me in the past.

I am way past being delirious, so I’ll stop there for now :)

Best regards,
Eric

Posted by: Eric on April 4, 2004 9:40 AM | Permalink | Reply to this

Re: Before the flood

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Ack! 5:45AM and I still can’t put this stuff down! :)

I just wanted to highly recommend a paper. It is by far the most beautiful paper I have read in a long time. Interestingly, it is coauthored by the author of the notes you are using in the string seminar.

Noncommutative Geometry and String Duality
Fedele Lizzi, Richard J. Szabo

A review of the applications of noncommutative geometry to a systematic formulation of duality symmetries in string theory is presented. The spectral triples associated with a lattice vertex operator algebra and the corresponding Dirac-Ramond operators are constructed and shown to naturally incorporate target space and discrete worldsheet dualities as isometries of the noncommutative space. The target space duality and diffeomorphism symmetries are shown to act as gauge transformations of the geometry. The connections with the noncommutative torus and Matrix Theory compactifications are also discussed.

I am quickly becoming of a fan of Szabo :)

In the paper, he defines the algebra $𝒜$ very similarly to the way you are suggesting, but he defines forms in the unusual (to me) way that Connes does, i.e.

(1)

w = a_{0} [D, a_{1}] . . . [D, a_{p}]

using the Dirac operator as opposed to

(2)

w = a_{0} [d, a_{1}] . . . [d, a_{p}] .

I prefer the latter way because the degrees makes more sense. Hey! Then again, these could be the same because

(3)

[d^{†}, a_{i}] = 0

for all $a_{i} \in 𝒜$ . It looks like what Lizzi and Szabo do could be precisely what you have suggested after all.

Hmm…

A great paper!

Good night!
Eric

Posted by: Eric on April 4, 2004 11:03 AM | Permalink | Reply to this

Re: Before the flood

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Hi Eric -

I enjoyed reading these comments. I would like to answer in detail, but right now I am short of time and will need to content myself with a quick answer.

First, I am aware of the work by Lizzi and Szabo. What I did in hep-th/0401175 is supposed to be a generalization/elaboration of part of their approach (which goes back to Froehlich, Gawedzky and others).

You are completely right about your observation about the definition of differential forms in NCG. Usually, when only one Dirac operator is present one defines them as $a_{0} [D, a_{1}] \dots [D, a_{p}]$ modulo a technical detail. This is also certainly what one should to in the context of spectral description of open strings. But for closed strings we have two Dirac operators which can be superposed to yield $d_{K}$ and $d_{K}^{†}$ and we can/should use the standard differential forms $a_{0} [d_{K}, a_{1}] \dots [d_{K}, a_{p}]$ .

The point is, this is, up to a small subtlety, pretty much the same, as you already guessed. With $d_{K} = \propto D + i \tilde{D}$ we have for a chiral element $w$ which supercommutes with $\tilde{D}$ the relation

(1)

[d_{K}, w] \propto [D, w] .

This is more or less what you wrote yourself in your last comment.

The only subtlety is that one has to deal correctly with the coordinate 0-mode which has nontrivial commutator with both $D$ and $\tilde{D}$ . But one can do that along the lines discussed for instance on p.238 of Polchinski’s book. (I can say more about that later.)

So if one takes care of this one can note that the DDF invariants are exact differential forms in the above sense: Let $w_{i}$ be rightmoving and ${\tilde{w}}_{i}$ leftmoving elements of the algebra, then a DDF invariant is of the form

(2)

[d_{K}, w_{1}] \dots [d_{K}, w_{p}] [d_{K}, {\tilde{w}}_{1}] \dots [d_{K}, {\tilde{w}}_{q}] \propto [D, w_{1}] \dots [D, w_{p}] [\tilde{D}, {\tilde{w}}_{1}] \dots [\tilde{D}, {\tilde{w}}_{q}] .

Hence it is more of a matter of notation whether I use $d_{K}$ or $D$ to define these differential forms. The advantage of $d_{K}$ is that this way it is easier to make contact with what is known (by us :-) about discrete/noncommutative differential geometry.

Finally regarding that ‘tower of massive states’. Could it be that you tried in one of your comments to associate this with the three generations of leptons in the standard model? This is not quite how things are expected to work. All the massive string excitations are assumed to correspond to currently unobservable partciles. All observed particles are (assumed to be) massless excitations of string which acquire mass by the usual Higgs mechanism.

Posted by: Urs Schreiber on April 4, 2004 2:43 PM | Permalink | PGP Sig | Reply to this

Re: Before the flood

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Good morning! :)

I would like to answer in detail, but right now I am short of time and will need to content myself with a quick answer.

No problem. I know you are extremely busy with conferences and travel, etc. I thought I’d continue putting my thoughts down here and you can take a look whenever/if ever you find the time.

Finally regarding that tower of massive states. Could it be that you tried in one of your comments to associate this with the three generations of leptons in the standard model?

Yeah. I guess they are different, eh? :) That would have been too easy of a fix I suppose :) The possibility of having a discrete string with only a finite number of possible excitations might still be interesting though. It seems to me that the question whether a string really is finitary or a continuum is possibly an interesting one. Like I said, it would almost be impossible to tell the difference due to fact that the 1d (I see some papers refer to it as 2d, maybe I should call it (1+1)d to be completely unambiguous) wave equation is exactly solvable on a discrete space without introducing any errors and the fact that the only difference in the spectrum would be that a discrete string would have a maximum energy mode, where the continuum string would have this infinite tower. Experimentally, it would be hard to rule out a discrete string I would imagine since it boils down to the question of whether an infinite energy string is allowed.

Wait a second…

If a string became energetic enough, is it possible for it to form a black hole? Is there some other mechanism that would limit the possible energy of the string? I vaguely remember reading about this (probably in a dialogue between you and Lubos). If so, this would possibly provide an argument for a discrete string. Unless I’m mistaken, the only real difference between a continuum string and a discrete string would be that the discrete string has a maximum energy.

Bear in mind, I am working on less than 3 hours of troubled sleep and am probably (most definitely) not thinking clearly :)

Best wishes,
Eric

Posted by: Eric on April 4, 2004 3:38 PM | Permalink | Reply to this

Re: Before the flood

Hi Eric -

you are perfectly right that a very energetic string should become a black hole. Did you see the discussion over at Peter Woit’s blog? There I give a couple of links regarding this question.

A heuristic but detailed description for how this works is summarized in here. Note that indeed a sort of discretization appears already in this heuristic picture.

Best,
Urs

Posted by: Urs Schreiber on April 4, 2004 9:55 PM | Permalink | PGP Sig | Reply to this

Re: Before the flood

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Hi Urs,

I don’t know if it is relevant, but the dialog between you and Lubos (and Robert) I was thinking about was on s.p.r. at

D1-brane action

Apparently there is a maximum E-field. It seem like this is something different though.

Eric

Posted by: Eric on April 5, 2004 2:00 AM | Permalink | Reply to this

Integrals: Loops space vs target space

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Hi Urs,

When you say things like

We can again construct the associated spectral triple and I claim that it now contains all physical observables of the superstring. In particular, the equations of motion are now

(1) $[d^{(1)}, a] = 0$

(2) $[d^{† (1)}, a] = 0$

where $a = [d^{(1)}, ω_{1}] . . . [d^{(1)}, ω_{p}]$ is a $p$ -form with respect to the new spectral triple, where all the $ω_{p}$ are integrals over weight 1/2 fields as described above.

of course I will try to relate this to what we know about spectral (discrete) differential geometry. There, the $ω_{p}$ would actually be 0-forms, which made me start thinking about integration on loop space as opposed to target space.

I’m probably missing some of the more subtle issues, but it is my understanding that a point in a loop space $ℒ (𝒯)$ corresponds to a loop in target space $𝒯$ .

Given any manifold $ℳ$ and a 0-form $f \in Ω^{0} (ℳ)$ , then the integral of $f$ over a 0-dimensional domain, i.e. a point, $p \in ℳ$ is simply evaluation at the point

(3)

\int_{p} f = f (p) .

Now it seems to me, although it sounds a little weird, that a 0-form on loop space $ℒ (ℳ)$ would somehow correspond to a 1-form on $ℳ$ .

Given a point $p \in ℒ (ℳ)$ , let $ϕ_{*} (p)$ denote the corresponding loop in $ℳ$ . Then, it is tempting to write down something like

(4)

\int_{p} ϕ^{*} (α) = \int_{ϕ_{*} (p)} α,

where $α$ is a 1-form on $ℳ$ and $ϕ^{*} (α)$ is a 0-form on $ℒ (ℳ)$ . However, this is not really like your typical push forward and pull back, which preserves degree/dimension.

Actually, come to think of it, is it really natural to consider the map $ϕ : ℒ (ℳ) \to ℳ$ because this isn’t really a nice map because it maps one point in $ℒ (ℳ)$ to many points in $ℳ$ . Perhaps I should swap the direction.

Given a map $ψ : ℳ \to ℒ (ℳ)$ that maps loops in $ℳ$ to points in $ℒ (ℳ)$ , then we can rewrite the above as

(5)

\int_{L} ψ^{*} (f) = \int_{ψ_{*} (L)} f,

where now $f$ is a 0-form in $ℒ (ℳ)$ and $ψ^{*} (f)$ is a 1-form in $ℒ (ℳ)$ .

I understand that this is not the way push forward and pull back usually work, but if this could be made rigorous (which I’m pretty sure it could be with some work), then instead of saying that your $ω_{p}$ are “integrals over weight 1/2 fields” on target space, could you just say that the $ω_{p}$ are 0-forms on loop space? This would, it seems to me, make the relation to spectral (and discrete) geometry more manifest I think.

Hmm… just before hitting “Post” another thought occurred to me. Maybe this is precisely what you ARE saying, but using different words :)

What I described above is a way to obtain a 0-form on loop space given a 1-form on target space via this generalized notion of pull back.

Is it just me or do stringy things that seem convoluted on target space turn out to be trivial if you consider geometry of loop space?

If anything I said here makes sense, I propose describing the $ω_{p}$ as simply 0-forms on loop space to make the connection to spectral (discrete) geometry more obvious.

Best regards,
Eric

Posted by: Eric on April 5, 2004 4:27 PM | Permalink | Reply to this

Re: Integrals: Loops space vs target space

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Oops!

This statement

What I described above is a way to obtain a 0-form on loop space given a 1-form on target space via this generalized notion of pull back.

should be replaced with what I meant to say

What I described above is a way to obtain a 1-form on target space given a 0-form on loop space via this generalized notion of pull back.

By the way, I’m already starting to have second thoughts about this whole idea. Consider two loops $L_{1}$ and $L_{2}$ in $ℳ$ that intersect at a point. Then their union $L_{1} \cup L_{2}$ is also a loop in $ℳ$ . However, it seems to me that $ψ_{*} (L_{1})$ , $ψ_{*} (L_{2})$ and $ψ_{*} (L_{1} \cup L_{2})$ would be three distinct points in $ℒ (ℳ)$ . Keeping track of this kind of thing and making it consistent seems nontrivial to me.

Eric

Posted by: Eric on April 5, 2004 4:43 PM | Permalink | Reply to this

Re: Integrals: Loops space vs target space

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Hi Eric -

nice observations! :-)

Yes, there should be some relation between $p$ forms on Loop space and $p + 1$ -forms ‘on’ target space. I think we should be careful here since this would need to be made precise, but morally it is true. If we want to map a single loop to a number then we will essentially always find us integrating something along that loop.

I always wanted to look up the stuff again that John Baez said about higher order connections that allow you to parallel transport loops around. He has once on spr mentioned giving a talk on the strings in Kalb-Ramond backgrounds. There we have a 2-form on target space and after integrating it along the string it becomes a 1-form on loop space.

Yes, you are right that my $ω$ are something like 0-forms on loop space. But actually there is a subtlety, since they are (for the superstring at least) not strictly number-valued but take values in something constructed from operators of the single superparticle, roughly.

Of course that’s related to the fact that their algebra is non-commutative! So they are 0-forms which take values in a peculiar non-commutative algebra.

I am a little short of time, that’s why I say sketchy thinks like that. But I hope from what I said before this does make sense.

One should probably sit down and try to find nice rigorous formulations for all these vague ideas that are currently populating my poor brain! :-) Great that you are (still) interested in this stuff!

Posted by: Urs Schreiber on April 5, 2004 6:00 PM | Permalink | PGP Sig | Reply to this

Re: Integrals: Loops space vs target space

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Hi Urs! :)

This is very interesting to me. In a way, I am glad I’ve got a cold so I can spend some time away from work thinking about this stuff :)

Yes, there should be some relation between $p$ -forms on Loop space and $p + 1$ -forms ‘on’ target space. I think we should be careful here since this would need to be made precise, but morally it is true.

I see from Chamseddines’s paper that he states

Some attempts were made to understand string theory based on loop space geometry [2], but this had limitations because the needed geometrical tools are not available.

Maybe it won’t be so easy to make this precise after all :)

There we have a 2-form on target space and after integrating it along the string it becomes a 1-form on loop space.

Neat! :) This is one degree up from the analogous statement: starting with a 1-form on target space and after integrating it along the string it becomes a 0-form on loop space.

I’m a little concerned about the “direction” here though. It would almost seem more natural to start with a $p$ -form on loop space and obtain a $(p + 1)$ -form on target space somehow (since the map from $ℒ (ℳ) \to ℳ$ is multivalued as opposed to $ℳ \to ℒ (ℳ)$ ).

Another thought that came to me when reading what you just said…

I wonder if 2-form electromagnetism on target space may somehow be related to 1-form electromagnetism on loop space?

Similarly, could 1-form EM on target space be related to 0-form EM on loop space?

If there is any truth to this, then I wonder if Baez’ stuff on 2-form Yang-Mills can be thought of as ordinary Yang-Mills on loop space? :)

The mind boggles :)

I definitely think it is worth the effort trying to understand geometry on loop space better. In fact, I’m beginning to think that things would seem more natural if you formulated them on loop space in the first place rather than struggling through messy formulations on target space. I think your work on DDF operators and their relation to spectral geometry makes a strong argument for this idea.

Eric

Posted by: Eric on April 5, 2004 7:10 PM | Permalink | Reply to this

Re: Integrals: Loops space vs target space

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I am now skimming over the paper:

Yang-Mills Theory on Loop Space
S. G. Rajeev

which goes over some of the issues of defining calculus on loop space. One of the most striking things that pops out at me is the statement that the integrands are actually tensors on direct products of the target space. Does this sound familiar? This sounds precisely like what we are talking about with discrete loop space being the direct product of copies of a discrete (target) space with itself :)

I’m definitely learning a lot these days :)

Cheers,
Eric

Posted by: Eric on April 5, 2004 7:42 PM | Permalink | Reply to this

Re: Integrals: Loops space vs target space

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One thing is nice about this paper: It tells us that, indeed, all functions on loop space come from integrating stuff around the loops (top of p. 7). Not too surprising, maybe, but nice that the mathematicians agree! ;-)

I still don’t know what to think of the exterior derivative defined in equation (24). This is apparently not related to the naive exterior derivative $d = {dX}^{(μ, σ)} \frac{δ}{δ X^{μ} (σ)}$ . But the latter is the one which is needed for what I have in mind. Of course it is not well defined without adding that term $i X_{(μ, σ)} {dx}^{(μ, σ)} \leftarrow$ , but still.

So far my attitude to loop space is this: I tend to pretend that I can work on it as on any finite dimensional manifold. But I know I cannot. If forced to admit this I appeal to the fact that what I am really doing is quantum field theory and that normal ordering presciption etc. take care of the infinities that would appear otherwise. So the loop space framework is rather a suggestive way of writing down otherwise well-known and well-defined objects.

I think if we want to make progress with the idea of generalizing the algebra of integrals over weight-1/2 fields (subset of the set of all 0-forms on loop space which take values in the peculiar algebra generated by $X, \frac{δ}{δ X}$ and the form creators/annihilators on loop space) we should see if we can find a very simple toy model analogue of this algebra.

I don’t really know what I am looking for here, so don’t ask! ;-)

But of course somehow I would like to have an algebra generated by elements which are ‘supported’ at some sort of ‘vertices’.

Maybe one should think of the formal objects $\int d σ (X^{'} \cdot X^{'})^{1 / 4} δ (X^{μ} (σ) - x^{μ})$ .

(The quarter power is there to give weight 1/2, since $X^{'}$ has weight 1 and $X$ has weight 0.) This would describe a ‘loop’ concetrated at the point $x^{μ}$ in target space.

No, wait. Maybe one should rather uses the weight-0 objects $R_{\pm}$ from my Pohlmeyer/DDF paper and seek generalizations of $\int d σ (X^{'} \cdot X^{'})^{1 / 4} δ (X^{μ} (σ) - f^{μ} (R (σ)))$ .

Hm, I don’t know. This is really currently my main stumbling block for making further progress with this idea: I don’t know what a good analogue of a discrete point would be in this framework.

If you have an idea, let me know! :-)

Posted by: Urs on April 5, 2004 9:40 PM | Permalink | Reply to this

Re: Integrals: Loops space vs target space

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I still don’t know what to think of the exterior derivative defined in equation (24). This is apparently not related to the naive exterior derivative $d = {dX}^{(μ, σ)} \frac{δ}{δ X^{μ} (σ)}$ . But the latter is the one which is needed for what I have in mind. Of course it is not well defined without adding that term $i X_{(μ, σ)} {dx}^{(μ, σ)} \leftarrow$ , but still.

My opinion about the exterior derivative has always been that you should relate it to the boundary map. The exterior derivative is that thing which makes Stokes’ theorem valid :)

I’d approach the problem by trying to define intergation on loop space, the boundary map, and then (and only then) define the exterior derivative on loop space. This will tell you what the correct form for what $d$ should be.

If you have an idea, let me know! :-)

Of course I’m biased, but I would suggest trying to make things work out on discrete loop space. This would probably be much easier and would give a lot of insight into the continuum version (if someone actually cared about the continuum theory :))

That paper reminds me a lot of those “multitangent fields” in Gambini and Pullin’s “Loops, Knots, Gauge Theories and Quantum Gravity.”

Eric

Posted by: Eric on April 5, 2004 10:51 PM | Permalink | Reply to this

How to proceed

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Hi Eric -

maybe we should proceed as follows:

(Do you have the Pohlmeyer/DDF paper in front of you?)

Let’s concentrate on the fermionic DDF invariants, first. These come from commuting the supercharge (i.e. the exterior derivative) with

(1)

\int d σ \frac{𝒫^{μ}}{k inner 𝒫} \exp (- inR (σ)) .

Let’s conjure the faint idea of a discretized target space in our minds as a guiding principle, but not really consider any such discretization yet. We can insert a function of 0 reparameterization weight into the integrand of the above expression without ruining its crucial properties. In order to somehow localize this observable lets insert

(2)

δ (X^{μ} (σ) - X_{0}^{μ}),

where $X_{0}^{μ}$ is fixed and constant. (Sorry for the indices…)

The observable

(3)

f_{X_{0}}^{μ} : = \int d σ \frac{𝒫^{μ}}{k inner 𝒫} δ (X^{μ} (σ) - X_{0}^{μ}) \exp (- inR (σ))

vanishes on all of phase space except at those configuration where at least one point of the string sits at $X_{0}^{μ}$ in target space. Integrating $f_{X_{0}}^{μ}$ over all of target space with respect to $X_{0}$ gives us back the above pre-DDF observable.

Now I propose to consider the $f_{X_{0}}$ as analogs of the functions supported at vertices for the case of the point particle. They are very different, but obviously morally at least similar.

The point is, in analogy to the particle case where the exterior differential of the delta-functions give us the edges along which the particle can move, we can now compute the (Poisson/quantum)-commutator of the loop-space exterior derivative with $f_{X_{0}}^{μ}$ and try to read of the infinitesimal ‘shifts’.

From commuting the delta-function with $d_{K}$ one gets something completely analogous as known from the point particle and a similar interpretation might make sense. Then there is a further term from commuting $d_{K}$ with $𝒫^{μ}$ . This produces a derivative which can be moved over to the delta-function again, thus yielding another ‘infinitesimal shift’.

I haven’t worked this out in detail yet. But maybe we should try to understand this commutator and hence the generalized 1-forms

(4)

f_{X_{0}}^{μ} [d_{K} f_{Y_{0}}^{ν}] .

These might reasonably encode the movement of string(bits) from the point $Y_{0}$ to $X_{0}$ . When we understand these objects in the continuum we can then transfer the idea to the discretum.

Even if the above idea is flawed something like this is what we neeed to look for.

Posted by: Urs on April 6, 2004 5:51 PM | Permalink | Reply to this

Re: How to proceed

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Hi Urs! :)

I’m not fully recovered, but I’m back to work and I have a big presentation to give Friday so my time is going to be very limited until after then. Maybe you could state in a few sentences what it is you are trying to do?

I’m sure I am over simplifying things, but I thought the basic idea of formulating string theory in terms of spectral geometry was going to be fairly straight forward after I saw you write things like

(1)

α = [G, ω^{1}] . . . [G, ω^{p}]

because I would have taken this to mean that the $ω_{1}$ generated the space of 0-forms so that we’d write down immediately things like

(2)

ω^{ij} = ω^{i} [G, ω^{j}] ω^{j} .

For this to make sense, I guess we’d need to have

(3)

ω^{i} ω^{j} = δ_{ij} ω^{i} .

I don’t suppose this would be the case here would it? :)

This is where I thought the discrete world could come to the rescue. A point in the discrete loop space would be something like

(4)

ω^{I} = e^{i_{1}} \otimes e^{i_{2}} \otimes . . . \otimes e^{i_{n}}

with multiplication given by

(5)

ω^{I} ω^{J} = (e^{i_{1}} \otimes . . . \otimes e^{i_{n}}) (e^{j_{1}} \otimes . . . \otimes e^{j_{n}}) = δ_{i_{1} j_{1}} . . . δ_{i_{n} j_{n}} e^{i_{1}} \otimes . . . \otimes e^{i_{n}} = δ_{IJ} ω^{I} .

In other words, I would look for elements $ω^{i}$ that span the space of 0-forms on loop space (which might be exactly what you are doing :)) that satisfy

(6)

ω^{i} ω^{j} = δ_{ij} ω^{i} .

(no implied sums anywhere in this post).

Gotta run!

Eric

Posted by: Eric on April 6, 2004 7:30 PM | Permalink | Reply to this

Re: How to proceed

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You wrote:

Maybe you could state in a few sentences what it is you are trying to do?

Oh, sorry. :-)

I am trying to find the algebra relations which need to be preserved in order that we don’t loose the information that we are dealing with Nambu-Goto/Polyakov strings.

What you write is a plausible guess - but the problem is that this does not flow to the known continuum theory, I think. I.e. if we let the lattice spacing go to 0 for what you wrote in your previous comment we don’t seem to be left with anything resembling invariants of the NG/Polyakov string. But this is exactly what we need to achieve.

And I am trying (in my, very limited, spare time) to do that by studying suitable invariants in the continuum.

The $f_{X_{0}}^{μ}$ in my previous comment are correct continuum objects. I am trying to guess/derive the algebra relations which are analogous to

(1)

δ_{i} d δ_{j} \neq 0 \Leftrightarrow there is an edge from vertex i to vertex j

of these $f_{i}^{μ}$ . Apparently these are such that $f_{i}^{μ} d_{K} f_{j}^{ν}$ can be nonvanishing when there is an edge (in target space) from vertex $i$ to vertex $j$ . But the details are much more involved here.

Maybe this is a bling alley. But it is currently what comes to (my) mind. :-)

Posted by: Urs Schreiber on April 6, 2004 9:27 PM | Permalink | PGP Sig | Reply to this

Re: How to proceed

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Hi Urs,

Just a light-hearted comment in the 10 seconds I have to spare here :)

I think I have more faith in the discrete theory than you do. I am pretty sure that if we formulate a discrete loop space, define $d$ and $d^{†}$ on this space, then things will fall into place naturally. The continuum limit will highlight any problems with the continuum theory, not the discrete theory ;)

Of course we have to be a little careful what we mean by continuum limit. I would like to see something where the string maintains its size and does not shrink to 0 in the continuum limit. This would involve some subdivision.

What we should really be trying to do (I think) is define a discrete loop space that carries over to the continuum loop space in some limit. If the math works, the physics will follow (“If you build it, they will come.”) :)

Gotta run!

Eric

Posted by: Eric on April 6, 2004 9:50 PM | Permalink | Reply to this

Re: How to proceed

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I am trying to find the algebra relations which need to be preserved in order that we don’t loose the information that we are dealing with Nambu-Goto/Polyakov strings.

I wonder if you could say a few more words about what the goal is here? :)

I am beginning to wonder if we are making things too complicated. If all we want to do is study Nambu-Goto or Polyakov strings, we have enough machinery to do that already in the discrete world with what we have in our notes. What is stopping us from just writing down a discrete Nambu-Goto action or a discrete Polyakov action, turn the crank and see what falls out? Perhaps we don’t even need to think about loop space?

Just a quick thought before heading to bed.

Good night! :)

Eric

Posted by: Eric on April 7, 2004 4:40 AM | Permalink | Reply to this

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