Spectral Standard Model and String Compactifications - Comments

[Urs Schreiber]

Urs Schreiber submitted a new PF Insights post

Spectral Standard Model and String Compactifications

Continue reading the Original PF Insights Post.

 

[john baez]

Typos:

emberrassingly -> embarrassingly

(modul0 8) -> (modulo 8)

conicidence -> coincidence

Those that do are called sigma-model -> Those that do are called sigma-models

I've got a bunch of things to say, but here are three preliminary comments, in order of increasing seriousness:

1) Has anyone ever claimed that KK-theory is named after Kaluza-Klein? A naive reader of your post might think so.

2) Are there noncommutative geometries that can seen as the target spaces for the Gepner models? Have people studied them using ideas from noncommutative geometry?

3) I'm really happy to see there's a systematic way to take the "point particle limit" of a vertex operator algebra. Is the result a spectral triple? I somehow imagine that VOA experts would not have phrased it this way, since they are more algebraic and less C*-algebraic, but you're sort of claiming it can be, at least modulo a bunch of technical details. It would be nice to straighten this out.

 

[Urs Schreiber]

Typos:

Thanks for spotting typos. Have fixed them now, and have clarified that "KK-theory", where it appear above, refers to "K-homology and K-cohomology".

 

[JorisL]

1) Has anyone ever claimed that KK-theory is named after Kaluza-Klein? A naive reader of your post might think so.

Thanks for spotting typos. Have fixed them now, and have clarified that "KK-theory", where it appear above, refers to "K-homology and K-cohomology".

I did not know that, thanks for clarifying this. Other than that I'm still digesting the insight.

 

[Greg Bernhardt]

I may save more interesting comments for when Greg Bernhardt or someone enables comments "in the Forum"

oops, added, thanks!

 

[Urs Schreiber]

Are there noncommutative geometries that can seen as the target spaces for the Gepner models? Have people studied them using ideas from noncommutative geometry?

By the story of the flop transition, the Gepner models describe some degenration limit of Calabi-Yau 3-folds. As it says on p.127 of Fröhlich-Grandjean-Recknagel 97 :

"Superconformal field theories of considerable interest in string theory would be the Gepner models, whose target spaces are expected to correspond to (non-commutative deformations of) Calabi-Yau spaces. They remain to be understood more precisely."

I'm really happy to see there's a systematic way to take the "point particle limit" of a vertex operator algebra. Is the result a spectral triple?

Yes. The review Roggenkamp-Wendland 08 is probably the best place to start. Then best to look at the original article in this direction, which is Fröhlich-Gawedzki 93

(I have now edited the article a little such as to highlight this point better.)

 

[john baez]

That review article is nice. I might have some technical questions/comments, but I'm watching the US presidential debate so right now I can only muster a very simple-minded general question. Roggenkamp and Wendland seem to present a systematic construction that takes us from 2d SCFTs (or technically VOAs) to noncommutative geometries (or technically spectral triples, though I don't actually see them claiming to have proved their triples obey the spectral triple axioms). So, it's interesting to think about what information is being thrown out here.

To understand this I need to describe the two items in a way that makes them sound similar. So I'll say a 2d SCFT is a way of presenting a "space in which strings can move", while a noncommutative geometry is a way of presenting a "space in which particles can move". Since I love categorification this makes me want to say the former is a "2-space" while the latter is a "1-space". But the word "noncommutative" shows up in the second item, so I feel like saying the former is a "noncommutative 2-geometry", while the latter is a "noncommutative 1-geometry".

First of all, I wonder if this is the right general idea. Of course it's sort of vague.

Second of all, it's a bit amazing how the adjective "noncommutative" sneaks in here. In the case of spectral triples we put it there on purpose. But for 2d SCFTs we didn't. I realize this is the point of your article. But it still seems a bit shocking to me.

Can we understand this a bit better? Is there some natural concept of "1d quantum field theory" such that we can show any such thing can be seen as a theory of a particle propagating in a noncommutative geometry?

(I am suppressing the adjective "super", which should appear all over the place.)

 

[Urs Schreiber]

Roggenkamp and Wendland seem to present a systematic construction that takes us from 2d SCFTs (or technically VOAs) to noncommutative geometries (or technically spectral triples, though I don't actually see them claiming to have proved their triples obey the spectral triple axioms).

Ah, that's because this is just the exposition of their technical article. For the proof that they get spectral triples (in fact a slight variant which they call "spectral pre-triples") see section 1.2 of

• Daniel Roggenkamp, Katrin Wendland,
  "Limits and Degenerations of Unitary Conformal Field Theories",
  Commun.Math.Phys. 251 (2004) 589-643
  (arXiv:hep-th/0308143)

It starts on the bottom of p. 8, where it says "Let us extract a spectral pre-triple from a CFT..."

To understand this I need to describe the two items in a way that makes them sound similar. So I'll say a 2d SCFT is a way of presenting a "space in which strings can move", while a noncommutative geometry is a way of presenting a "space in which particles can move".

Yes, that's the way to think about it. Or maybe better, as amplified in section 7 of Fröhlich-Grandjean-Recknagel 97, we should think about spectral triples as being like spacetime manifolds (i.e. (pseudo-)Riemannian manifolds) while 2d SCFTs are like path spaces or loop spaces of these (the configuration spaces of strings inside these spacetime manifolds).

First of all, I wonder if this is the right general idea. Of course it's sort of vague.

Yes, I think this is the right general idea.

Second of all, it's a bit amazing how the adjective "noncommutative" sneaks in here. In the case of spectral triples we put it there on purpose. But for 2d SCFTs we didn't. I realize this is the point of your article. But it still seems a bit shocking to me.
Can we understand this a bit better?

Conceptually I think of this as a generalization of the famous observation of

• Nathan Seiberg, Edward Witten,
  "String theory and non-commutative geometry",
  JHEP 9909:032,1999,
  arXiv:hep-th/9908142

which observed that the finite extension of the open string may induce non-commutative geometry in its effective point-particle geometry.

One sees this maybe more formally in the construction of Roggenkamp-Wendland 03, which produces the algebra data in the spectral triple from the vertex operator algebra of the 2d SCFT. The latter is a non-commutative algebra due to the fact that it is really the algebra of quantum observables on the 2-dimensional string worldsheet. As that worldsheet is shrunk to a worldline, some of the non-commutativity of the OPE reflecting its 2-dimensional extension may survive.

That's at least the rough picture. I suppose a more detailed analysis should be possible. In section 1.3 of Roggenkamp-Wendland 03 they discuss how to extract commutative spectral (pre-)triples from a CFT, calling them "commutative sub-geometries".

Is there some natural concept of "1d quantum field theory" such that we can show any such thing can be seen as a theory of a particle propagating in a noncommutative geometry?

Here I am not sure what you are after. Because we have already been saying that the answer to this question is "Yes, that's a spectral triple!"

 

[john baez]

Here I am not sure what you are after. Because we have already been saying that the answer to this question is "Yes, that's a spectral triple!"

What I mean was this. I can easily see how "conformal field theory" is a reasonable answer to "what's a 2d quantum field theory?". But I would never have guessed that "spectral triple" is the answer to "what's a 1d quantum field theory?" I would instead have imagined the answer is "a Hilbert space $\mathcal{H}$ with a self-adjoint operator $H$, the Hamiltonian", since that's enough to give a functor from a 1d cobordism category (with 1d Riemannian cobordisms as morphisms) to Hilb (at least if $\exp(-tH)$ is trace class for all $t > 0$). I wouldn't have guessed that we also want an algebra of operators on that Hilbert space. If you'd told me we're interested in 1d field theories that are really describing particles moving around on a space, I would have added a commutative algebra of operators on the Hilbert space, namely an algebra of position operators. And if you'd said we should also allow "noncommutative geometries" as our spaces, I'd allow the algebra to be noncommutative. But this amounts to dragging me toward the desired answer "spectral triple".

[...] the finite extension of the open string may induce non-commutative geometry in its effective point-particle geometry. One sees this maybe more formally in the construction of Roggenkamp-Wendland 03, which produces the algebra data in the spectral triple from the vertex operator algebra of the 2d SCFT. The latter is a non-commutative algebra due to the fact that it is really the algebra of quantum observables on the 2-dimensional string worldsheet. As that worldsheet is shrunk to a worldline, some of the non-commutativity of the OPE reflecting its 2-dimensional extension may survive [...]

Okay, this sounds like a more "principled" was to see how the noncommutative algebra gets into our description of a "1d quantum field theory". I'd like to understand it better. But interestingly it arises "from above" - from a study of 2d QFTs, not directly from the study of 1d QFTs.

It reminds of another story, which goes

commutative algebra : set :: noncommutative algebra : groupoid​

the groupoid algebra of a groupoid being noncommutative, but reducing to a commutative algebra when the groupoid is discrete and thus merely a set of objects. Here we see noncommutative geometry as a way of thinking about 2-geometry.

 

[Urs Schreiber]

The way that spectral triples encode 1d QFT was sorted out by Kontsevich and Soibelman. Unfortunately they tend not to produce polished exposition, but I had once reported on some aspects of their work here They explain that the algebra in the spectral triple is what allows to extend from a representation of 1d cobordisms, to a representation of 1d cobordisms with interaction vertices, i.e. to graphs (Feynman graphs!). This is just what one expects to see from the point particle limit of a string worldsheet theory.

There is a section with more details on this in

• Yan Soibelman, "Collapsing CFTs, spaces with non-negative Ricci curvature and nc-geometry", in H. Sati, U. Schreiber (eds.), "Mathematical Foundations of Quantum Field and Perturbative String Theory", Proceedings of Symposia in Pure Mathematics, AMS (2001)

See section 5 "Graphs and singular quantum Riemannian 1-geometry".

 

[john baez]

Okay, that makes sense: if you take a finite-dimensional Hilbert space $\mathcal{H}$ and equip with a nice commutative algebra structure, namely a commutative dagger-Frobenius structure, this does two things:

• it provides an isomorphism between $\mathcal{H}$ and 'an algebra of functions on a configuration space', meaning the commutative dagger-Frobenius algebra of functions on a finite set.
• it allows us to get an operator $\mathcal{H}^{\otimes m} \to \mathcal{H}^{\otimes n}$ from any graph with $m$ input edges and $n$ output edges, giving a representation of a certain category whose morphisms are graphs of this sort.

Here the lengths of the edges of the graph don't matter, so we can crush all the internal edges to zero length, but if we also equip $\mathcal{H}$ with a Hamiltonian $H$ then we can put a 'propagator' $\exp(-i t H)$ or $\exp(-t H)$ on each edge where $t$ is the length of that edge, and now we're getting a representation of a category of 'Feynman graphs' where the edges have lengths.

If we visualize the graph's edges as ribbons rather than arcs then we should use a noncommutative dagger-Frobenius structure. So, noncommutativity seems to show up from thinking of point particles as tiny open strings (as you said).

Now this stuff makes sense to me. Interestingly, from this viewpoint the singular 1-dimensional cobordisms, namely those containing interaction vertices, arise 'from above' - they come from perfectly smooth 2-dimensional cobordisms.

So the analogy we're talking about here is a bit lopsided because some exotic features of our concept of 1d QFT are arising from thinking of them as a limiting case of a 2d QFT, while we're - apparently - not equipping our concept of 2d QFT with the analogous exotic features that might arise from thinking of them as a limiting case of a 3d QFT. We'd get those exotic features if we considered some singular 2d cobordisms.

(In the 1d case, what I'm calling the 'exotic features' are first of all the very presence of an algebra of observables in our spectral triple, making $\mathcal{H}$ into a space of 'functions on configuration space', but second of all the fact that this can be noncommutative.)

(On the other hand, the 2d case has its own 'exotic feature': we're using cobordisms equipped with a conformal structure, a specially potent trick in 2 dimensions. This is the magic of string theory.)

 

[john baez]

By the way, the lopsided nature of the analogy is perfectly fine if we take the attitude that string theory - or more precisely SCFTs - is primary and then we're investigating the particle limit. And that's what you were doing. It only becomes an issue if we imagine ourselves trying to treat n-dimensional QFTs in a systematic way on an equal footing. And that's what I was doing, to help myself figure out what's really going on here.

 

[Urs Schreiber]

Now this stuff makes sense to me. Interestingly, from this viewpoint the singular 1-dimensional cobordisms, namely those containing interaction vertices, arise 'from above' - they come from perfectly smooth 2-dimensional cobordisms.

Incidentally, as you know, this is the standard informal explanation of why string scattering amplitudes are UV-finite at each loop order (which is supposed to be a theorem for the bosonic string, and almost a theorem for the superstring): because the singular interaction vertices in the point particle's Feynman diagrams have been smoothened by the string worldsheet.

On the other hand, the 2d case has its own 'exotic feature': we're using cobordisms equipped with a conformal structure, a specially potent trick in 2 dimensions.

It's good to remember that, as you know, fundamentally, the action functional on the string is not a conformal field theory, but a diffeomorphism invariant theory, namely 2d supergravity coupled to the string's "embedding"-fields, regarded as 2d matter fields. One quantizes this 2d diffeomorphism invariant by noticing that one may fix a diffeomorphism gauge that only leaves the Möbius group as a remnant of diffeomorphism invariance. In fixing this gauge, Fadeev-Popov ghosts need to be introduced, and so after this gauge fixing the original 2d diffeomorphism invariant theory looks like two super-conformally invariant theories, one being that of the diffeomorphism ghosts, the other being the actual string worldsheet 2d SCFT that we are discussing here. The diffeomorphism ghost system happens to have a conformal anomaly of central charge -15, and so, since the original 2d diffeomorphism invariant theory is supposed to be well defined, the remaining 2d SCFT has to have central charge 15, to make the total anomaly cancel. And since each effective target space dimension contributes $1\tfrac{1}{2}$ to the central charge, this is what fixes the target space dimension to be 4+6, coinciding mod 8 with the KO-dimension of the standard model of particle physics.

I am recalling this just to highlight that worldsheet conformal invariance is not postulated, but comes out from starting with the evident diffeomorphism invariant system. On the other hand it is certainly true that part of what makes $p$-brane Feynman perturbation series give sensible results for $p=1$, but apparently not for $p > 1$, is that it can be related to conformal invariance this way.

There are some arguments for what happens for $p=2$: proceeding with the membrane the way one did for the particle and then the string runs into various technical problems, among them the fact that 3-manifolds don't have a nice classification as 2-manifolds do, and the fact that the membrane world-volume Hamiltonian has "flat directions" in its potential energy term, indicating that a naive quantum membrane will erratically spread itself throughout spacetime, instead of looking like tracing out a well-behaved worldline from far away.

(These two problems are the reason why Witten introduced the term "M-theory": as a "non-committal" abbreviation for "membrane theory", which he could use while remaining sceptical that the theory is a theory of membranes in analogy to how string theory is a theory of strings. See here. )

But people noticed that the Hamiltonian of the membrane involves a term that has the form of the square of a Poisson bracket between functions on its 2-dimensional spatial part. This led them to speculate that maybe the membrane's worldvolume theory wants to be regularized, by replacing the algbra functions on the membrane by a noncommutative algebra of operators -- and people like to think of finite rank operators here and speak of matrices -- and replacing Poisson brackets by commutators. If one works out the worldvolume theory of the M2-brane with this regularization... then it becomes the BFSS matrix model

This and more is reviewed in these nice articles:

• Hermann Nicolai, Robert Helling,
  "Supermembranes and M(atrix) Theory",
  Lectures given by H. Nicolai at the Trieste Spring School on Non-Perturbative Aspects of String Theory and Supersymmetric Gauge Theories,
  23 - 31 March 1998
  (arXiv:hep-th/9809103)
• Arundhati Dasgupta, Hermann Nicolai, Jan Plefka,
  "An Introduction to the Quantum Supermembrane",
  Grav.Cosmol.8:1,2002; Rev.Mex.Fis.49S1:1-10, 2003
  (arXiv:hep-th/0201182)

There was a time when some people actively explored these "membrane matrix models" as the thing in 3d that is the next thing in the sequence starting with spectral triples in 1d and 2d SCFTs in 2d. Various consistency checks were made (some of them recalled in the above articles). But then AdS/CFT arrived on the scene and all these interesting developments of the 90s died out, sociologically.

 

[David Corfield]

Is there formatting advice somewhere?

"In section 1.3 of Roggenkamp-Wendland 03 they discuss how to extract commutative spectral (pre-)triples from a CFT, calling them "commutative sub-geometries"."

I went looking to see if there is something Bohr topos-like (https://ncatlab.org/nlab/show/Bohr+topos) happening, but that section reminded me more of when two (dual) prequantum theories have the same quantization.

 

[David Corfield]

"Effective spacetime geometry seen by a quantum particle"

From this point of view, we would presumably expect then different kinds of particles to see different geometries. If so, can those geometries be made compatible?

"In the limit where the string’s oscillations are in their quantum ground state, this reproduces the particle perspective"

So this does allow for compatible particle-based geometries?

 

[David Corfield]

Oh, and more typos: of quantum some particle; supersymmtry algebra; dimenion; do not come describe

 

[Urs Schreiber]

Is there formatting advice somewhere?

You should make sure to reply via the "Forums" page here, not via the "Insights" page. (Though I wish somebody would fix this. Greg?) Then when replying, you find a toolbox bar above your edit box. Hovering the mouse over the symbols yields explanations of what the formatting tools will do for you. In particular, to make hyperlinks you first mark the text to be hyperlinked, then you click on the symbol that looks like a chain, then add the url into the dialogue box that appears.

 

[Urs Schreiber]

From this point of view, we would presumably expect then different kinds of particles to see different geometries.

Actually, in the spectral standard model the Hilbert space of the spectral triple has a tensor factor that contains the 32-dimensional rep of an entire generation of fermions (or rather three copies of them, to match the three generations). This means that the quantum particle modeled by the spectral triple is really all these particles at once, thought of as a composite system.

So this does allow for compatible particle-based geometries?

Hm, here I am not sure what you are asking. Could you expand?

 

[Urs Schreiber]

Oh, and more typos

Thanks! Fixed now.

 

[David Corfield]

Hm, here I am not sure what you are asking. Could you expand?

Now you've told me about the spectral triple modelling all particles at once as a composite system, that shifts the thought I was vaguely pointing to towards something like: Does the 2d SCFT of a spinning string contain within it all particles in a more structured way than as the spectral geometry of a composite system?

But perhaps that is merely to ask about string theory's capacity to unify particles as excitation modes.

 

[Urs Schreiber]

Now you've told me about the spectral triple modelling all particles at once as a composite system, that shifts the thought I was vaguely pointing to towards something like: Does the 2d SCFT of a spinning string contain within it all particles in a more structured way than as the spectral geometry of a composite system?

Indeed, that's the way to think about it: There is just one species of string (apart from the choice of numbers of worldsheet supersymmetry, anyway) and a rich spectrum of fundamental particle species arises by decomposing its spectrum. As we collapse the string's worldsheet 2d SCFT to a spectral triple, we see the massless ones of these particle species. Now they look like particles, but the spectral triple still remembers them as forming one single composite system, of sorts.

 

[arivero]

Actually, in the spectral standard model the Hilbert space of the spectral triple has a tensor factor that contains the 32-dimensional rep of an entire generation of fermions (or rather three copies of them, to match the three generations). This means that the quantum particle modeled by the spectral triple is really all these particles at once, thought of as a composite system.

Urs, given that the usual standard triples do not have explicit susy, I wonder if there is some work starting from a "two-spectral triple", taking some limit towards the standard spectral triple, and pointing out where the superpartners decouple and how do they appear then in such limit.

 

[Urs Schreiber]

Urs, given that the usual standard triples do not have explicit susy, I wonder if there is some work starting from a "two-spectral triple", taking some limit towards the standard spectral triple, and pointing out where the superpartners decouple and how do they appear then in such limit.

Both "2-spectral triples" (aka 2d SCFTs) as well as "1-spectral triples" have manifest worldvolume supersymmetry. The former describes strings with worldsheet supersymmetry, the latter describes particles with worldline supersymmetry. (Remember that every ordinary spnning particle has worldline supersymmetry, see here).

However, in order for a 2d SCFT to correspond to a target spacetime theory that has global ("low energy") spacetime super-symmetry, very special conditions need to be met. Generically there is no global spacetime supersymmetry preserved. The condition for global spacetime supersymmetry is existence of global Killing spinors (e.g. CY compactifications). These are generically absent, just as Killing vectors are absent from generic spacetimes. The special attention that compactifications with a global Killing spinor receive(d) originates in the (decaying) prejudice that reality exhibits supersymmetry at the (comparatively "low") weak scale, not in theoretical necessity that this needs to be so.

Hence it is no mystery that global spacetime supersymmetry is broken by these models. It would be more of a mystery if it were not. String theory (i.e. 2-spectral triples) does not imply "low" energy supersymmetry, it only implies high energy supersymmetry, generically broken right away, just as global Poincare symmetry is broken in our world.

 

[arivero]

Hence it is no mystery that global spacetime supersymmetry is broken by these models. It would be more of a mystery if it were not. String theory (i.e. 2-spectral triples) does not imply "low" energy supersymmetry, it only implies high energy supersymmetry, generically broken right away, just as global Poincare symmetry is broken in our world.

Well but still String Theory (and this 2-spectral triples, I assume) implies high energy "target-space" supersymmetry, and 1-spectral triples have not got any of it. So you are telling that the only way to connect them is via the usual mechanism of susy breaking and then that 1-spectral triples are inherently "low energy"?

EDIT: I am reading this as telling that the "low energy" susy breaking and the "compactificacion of dimensions" are really the same mechanism, or at least happen at the same time, leaving the 1-spectral triples as the 3+1 dimensional object we see. Still, susy breaking has scalars, Connes et al models haven't.

 

[Urs Schreiber]

Clearly, the particle limit of the string corresponds to its low energy limit. Conversely, any 2-spectral triple lifting a 1-spectral triple may be thought of as providing a UV-completion of the corresponding target effective field theory.

But the good think about the articles referenced above is that they give you a direct mathematical way to see this, without need to go through a long sequence of semi-precise folklore arguments. There is a precise way to take the point particle limit of 2d SCFT, hence to reduce string backgrounds to their point particle limit, and the result are spectral triples of the kind that appear also in the Connes-Lott model. That's the mathematical fact you should be contemplating.