The String Coffee Table

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 $\mathbf{d} \propto \bar G_0 + i G_0$ and $\mathbf{d}^\dagger \propto \bar G_0 -i G_0$. (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 $\mathcal{A}$ the object $[\mathbf{d},\cdot]$ becomes nilpotent and invariance follows already from supercommutation with only $G_0$ and $\bar G_0$, i.e. with only $\mathbf{d}$ and $\mathbf{d}^\dagger$.

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 $\mathcal{A} = C^\infty(M)$ which is generated by instanton/D(-1)-brane positions, namely (smeared) delta-functions on $M$ (spacetime). Let $\mathbf{d}$ be the ordinary exterior derivative acting on this algebra and let $\mathcal{H} = (\Omega(\mathcal{A},\mathbf{d}),\langle\cdot|\cdot \rangle)$ be the inner product space of differential forms associated with this algebra. The spectral triple $(\mathcal{A},\mathbf{d},\mathcal{H})$ together with the equations of motion $[\mathbf{d},\omega] = 0 = [\mathbf{d}^\dagger,\omega]$ ($\omega : \mathcal{H}\to \mathcal{H}$ 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 $[\mathbf{d}+\mathbf{d}^\dagger, \omega] = 0$ and $[(\mathbf{d}+\mathbf{d}^\dagger)^2,\omega] = 0$).

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 ($D1$-branes) from the algabra of super D0-branes.

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

For instance

is such a weight (1/2) algebra generator which describes a collection of superparticles with coordinates sitting at the fixed coordinates $x_0^\mu(\sigma)$. Here $\gamma^\mu(\sigma) = {e}^{\dagger \mu}(\sigma) + e^\mu(\sigma)$ is a ‘gamma-matrix’ observable for the $\sigma$th superparticle.

Now let $\mathbf{d}^{(1)} \propto i G_0 + \bar G_0$ 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 $\sigma$. But the elements of $\mathcal{A}^{(1)}$ are constructed in such a way that they are invariant under these rigid reparameterizations, so $\mathbf{d}^{(1)}$ is nilpotent on $\mathcal{A}^{(1)}$ 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

where $a = [\mathbf{d}^{(1)},\omega_1] \cdots [\mathbf{d}^{(1)},\omega_p]$ is a $p$ form with respect to the new spectral triple, where all the $\omega_p$ 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 $C^\infty(M)$ 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 $\mathbf{d}$ encodes the connectivity of these points. More precisely, the 1-form

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 $\rho$ the gluing 1-form which is just the sum of all edges of the graph:

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

In this sense $\mathbf{d}$ 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’.

However, it seems the algebra $\mathcal{A}$ 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 $d^\dagger$ following our prescription. If we took these operators and defined

and

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 $\overline{G}_0$ and construct $d$ and $d^\dagger$ 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

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 $a_0 [D,a_1]\cdots [D,a_p]$ 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 $\mathbf{d}_K$ and $\mathbf{d}^\dagger_K$ and we can/should use the standard differential forms $a_0 [\mathbf{d}_K,a_1]\cdots [\mathbf{d}_K,a_p]$.

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

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

Hence it is more of a matter of notation whether I use $\mathbf{d}_K$ or $D$ to define these differential forms. The advantage of $\mathbf{d}_K$ 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.

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 $\mathcal{M}$ that intersect at a point. Then their union $L_1\cup L_2$ is also a loop in $\mathcal{M}$. However, it seems to me that $\psi_*(L_1)$, $\psi_*(L_2)$ and $\psi_*(L_1\cup L_2)$ would be three distinct points in $\mathcal{L}(\mathcal{M})$. Keeping track of this kind of thing and making it consistent seems nontrivial to me.

Eric

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 $\omega$ 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!