### Before the flood

#### Posted by Urs Schreiber

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 $(\mathcal{A},\mathscr{H},D)$ 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 $\mathcal{A}={C}^{\mathrm{\infty}}(\Omega (M))$, the algebra of continuous functions on the loop space over the manifold $M$. Elements of the algebra are functionals of the form $f[{X}^{\mu}(\sigma )]$ where $\sigma $ 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 $$\pi (\rho )=\sum _{i}{f}^{i}[D,{g}^{i}]=\sum _{i}\int d\sigma {\left[{f}^{i}\left[X\right]({\psi}_{+}^{\mu}+{\psi}_{-}^{\mu})\frac{\delta {g}^{i}}{\delta {X}^{\mu}}\right]}_{P=0}\phantom{\rule{thinmathspace}{0ex}}.$$

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 $\omega $ such that the equations
$$(d\pm {d}^{\u2020})\omega =0$$
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 $\mathcal{A}$.

For this algebra I do *not* choose the algebra of reparameterization invariant functions, but the algebra $\mathcal{A}$ 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 $\Omega (\mathcal{A},d)$ generated from $\mathcal{A}$ 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
$$[d,\omega ]=0$$
$$[{d}^{\u2020},\omega ]=0$$
for $\omega \in \Omega (\mathcal{A},d)$ (where the bracket is the graded Poisson bracket) indeed *imply* the full superstring constraints
$$[{G}_{m},\omega ]=0$$
$$[{\overline{G}}_{m},\omega ]=0\phantom{\rule{thinmathspace}{0ex}}.$$
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
$$\omega =[d,{a}_{1}][d,{a}_{2}]\cdots [d,{a}_{p}]\phantom{\rule{thinmathspace}{0ex}}.$$

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 $\mathcal{A}$ 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
## Re: Before the flood

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

implies

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 calculusdates back to Koszul. If you haven’t seen it, another one of my all time favorite papers isGeroch, R. Einstein Algebras, Communications in Mathematical Physics, 26, 271, 1972