quantumparticle

Spectral Standard Model and String Compactifications

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The Connes-Lott-Chamseddine-Barrett model is the observation that the standard model of particle physics — as a classical action functional, but including its coupling to gravity and subsuming a fair bit of fine detail —  may succinctly be encoded in terms of operator algebraic data called a “spectral triple“. This involves some non-commutative algebra,  and as such is often referred to as the “noncommutative standard model“. The key point however is the “spectral” aspect: in this model all spacetime geometry and particle content is encoded in the spectrum of a Dirac-like operator, hence in terms of what a spinning quantum particle sees that propagates through such a background.

Sometimes this perspective is perceived as, and sometimes it is also presented as, being drastically different from traditional perspectives in high energy particle physics. But this is not the case. Apart from some difference in terminology, the encoding of particle physics models in spectral triples is just the point-particle analog of the encoding of perturbative string theory backgrounds in 2-dimensional super-conformal field theories (2d SCFTs). Hence 2d SCFTs are really “2-dimensional spectral triples” (i.e. stringy spectral triples). Of course the concept of encoding spacetime geometry by 2d SCFTs precedes the concept of spectral triples, but in the theory of spectral triples it is made more systematic.

This relation has long been understood, conceptually, by some, and has meanwhile also partly been turned into theorems by others. But by and large the relation is underappreciated. Here I will informally explain the conceptual relationship and provide pointers (at the end) to references that provide technical details. There does remain much room to explore this relation in more depth.

 


 

Effective spacetime geometry seen by a quantum particle

The term “spectral triple” sounds technical, but the idea is really simple: it just means something like “abstract quantum mechanics”, or maybe “axiomatic quantum mechanics”. It’s a triple, because it means to collect the data that defines a quantum mechanical system, which is

  1. a Hilbert space ##\mathcal{H}## of quantum states of  some particle;
  2. a Hamiltonian operator ##H## encoding the time evolution of this particle;
  3. a sub-algebra ##A## of linear operators representing the smooth position observables of the particle.

The basic example to keep in mind is that of a particle propagating on some ##d##-dimensional spacetime ##X## subject to the force of gravity. In this case the Hilbert space is that of suitably square integrable complex functions on ##X## (i.e. ##\mathcal{H} = L^2(X)##), the Hamilton operator is the Laplace-operator of the given (pseudo-)Riemannian metric (##H = \Delta##), and the sub-algebra is that of smooth functions on spacetime (##A = C^\infty(X)##).

(For Euclidean spacetime signature and ##X## compact this is straightforward and would model a non-relativistic particle. For Lorentzian signature and ##X## non-compact one needs to be a tad more careful and invoke for instance Krein-Hilbert spaces, this describes then relativistic particles.)

The key idea is now the following:

If we forget the spacetime geometry ##X## which we used to construct the quantum mechanics of the particle propagating inside it, is it still possible to reconstruct the spacetime background by just inspecting the quantum dynamics of the particle?

Say we knew all the states of definite energy of the quantum particle — hence the spectrum of its Hamiltonian operator ##H##– could we somehow read off from this what spacetime the particle sees, and which forces it experiences? In other words:

Is there an “emergent effective target spacetime” encoded in the spectrum of a quantum particle?

In mathematics this issue is famously known as hearing the shape of a drum : If we think of the Hamilton operator ##H## as the wave operator ##\Delta## on some elastic body  ##X##, such as the surface of a drum, then its spectrum encodes the standing waves on this elastic body, hence the sound that the drum makes. The question then reads: from knowing just this sound spectrum of the drum, is it possible to deduce the shape of the drum? That is, anyway, the best imagery that Riemannian geometers came up with. Had they known just a little quantum physics, they might have been able to phrase this as:

Is it possible to deduce background spacetime physics by just looking at single quantum particle propagators, single particle spectra?

It turns out that with just a tuple consisting of a Hilbert space and a Laplace operator on it, then it is not in general possible to do this reconstruction! But Alain Connes observed what algebraic ingredient is missing that does make the reconstruction possible: it is the algebra that in physics we think of as the algebra of smooth position observables of the particle, completing the tuple (Hilbert space, Hamilton operator) to a triple, hence called a spectral triple.

While it is possible to reconstruct a Riemannian manifold from the spectrum of its Laplace operator this way, the construction is a little involved. It turns out that it becomes much more elegant if the Laplace operator is replace by a Dirac operator, hence if the quantum particle used to detect the spacetime is taken to be a spinning particle.

There is still a triple of data then: a ##\mathbb{Z}/2\mathbb{Z}##-graded Hilbert space ##\mathcal{H}## (hence a super-Hilbert space), an odd-graded operator  ##D## acting on it whose square acts like a Hamiltonian operator, ## H = D^2##, and an algebra ##A## of operators acting as the algebra of smooth position operators.

The classical example now is obtained from a Riemannian manifold with Spin structure, by taking the Hilbert space to be that of square-integrable sections of a spinor-bundle (##\mathcal{H} = L^2(\mathcal{S})##), taking the odd-graded operator to be the corresponding Dirac operator (##D = \nabla\!\!\!\!\!\,/\;##), and specifying the algebra to be that of smooth functions (##A = C^\infty(X)##), acting by multiplication on spinors.

Incidentally, such a spectral triple axiomatized with a Dirac operator ##D## instead of just a Hamilton operator ##H## axiomatizes not just plain quantum mechanics, but supersymmetric quantum mechanics : the anticommutator-relation ##\{D,D\} = 2 D^2 = 2H##, that is the supersymmetry algebra in dimension one (on the worldline of a particle). Indeed, every spinning particle in the ordinary standard model (such as an electron) exhibits worldline supersymmetry this way (see here). For the moment this is just a curiosity, but this fact reveals a deeper meaning as we pass attention from spectral triple to “2-spectral triples” below.

It is remarkable that in terms of this spectral data, essentially all geometric constructions on Riemannian spin manifolds find an operator algebraic counterpart. This may be thought of as a differential geometric and metric refinement of the reconstruction of (compact) topological spaces in terms of their ##C^*##-algebras of continuous complex-valued functions (“Gelfand duality“).

The force of this reformulation however comes from the fact that it provides generalization: there are more ##C^*##-algebras than arise as continuous functions on topological spaces (namely there are also non-commutative ##C^*##-algebras) and there are more spectral triples than arise from Riemannian spin manifolds in the above way: the Dirac-like operator in the spectral triple just needs to satisfy some basic properties of actual Dirac operators, it need not actually come from a Riemannian spin manifold. Similarly the sub-algebra of operators thought of as the algebra of smooth position operators need not arise as the actual algebra of smooth functions on a smooth manifold. In particular, it could be a non-commutative algebra.

The point is that once all the actual Riemannian spin geometry of smooth maniolds is equivalently re-encoded in terms of operator algebra, then we may take these very same algebraic formulas, apply them to more exotic “non-geometric” situations, and still interpret the result from a geometric perspective, thus obtaining a generalized concept of geometry.

For instance if ##X## is an actual Riemannian manifold with spin structure, then on a given chart of the manifold its Dirac operator has the form

$$D = \gamma^a e_a{}^{\mu} \left( \partial_\mu + \tfrac{1}{4} \omega_{\mu a b} \gamma^a \gamma^b \right)\,, $$

(where ##\gamma^a## are the “Gamma-matrices” for flat space, ##e^a{}_\mu## is the vielbein that encodes the metric, and ##\omega_\mu^{a b}## is the Levi-Civita connection in its incarnation as a spin connection, for more on the latter see the review at 11d  Gravity from just the Torsion Constraint).

It follows that for ##\phi## a smooth function on spacetime, regarded as a smooth position operator, then the commutator

$$ [D,\phi]  = \gamma^\mu \nabla_\mu \phi = c(d \phi)$$

is essentially the de Rham differential ##d \phi## of the function (or rather the image of that under the “symbol map” ##c## which replaces 1-forms by Clifford algebra elements).

This means that given a general spectral triple ##(\mathcal{H},D,A)##, which need not come from a smooth geometry, hence in which case we do not a priori have any concept of differentiating a smooth function, then we still have an operator-algebraic analog of the construction of the de Rham differential: for any element ##\phi \in A## in the given (possibly non-commutative) algebra, we may still form the commutator ##[D,\phi]## with the Dirac operator, and that operator will behave for all practical purposes just as the actual de Rham differential ##c(d\phi)## of a smooth function does.

This is just the simplest example of a long list of operator algebraic constructions that mimic constructions in Riemannian spin geometry and that allow to think of spectral triples equivalently as encoding generalized Riemannian spin geometries — “spectral geometries”.

 

Lecture notes that explain the content of the previous paragraphs in detail are in

  • Jürg Fröhlich, Oiver Grandjean, Andreas Recknagel,
    “Supersymmetric quantum theory and (non-commutative) differential geometry”,
    Commun.Math.Phys. 193 (1998) 527-594
    (arXiv:hep-th/9612205)

Together with section 7 of

  • Jürg Fröhlich, Oiver Grandjean, Andreas Recknagel,
    “Supersymmetric quantum theory, non-commutative geometry, and gravitation”,
    Lecture Notes Les Houches (1995)
    (arXiv:hep-th/9706132)

and with the content of

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

this discusses most of the ideas that I am reviewing here. Highly recommended reading. For more references see towards the end.

 

Quantum aspects of Spacetime

A more sophisticated example that will be relevant is the algebraic incarnation of the geometric concept of dimension. It turns out that there are spectral geometries that classically look like a point, but that still have some exotic internal structure left, as if a Riemannian manifold has collapsed to a point while still remembering some of its original structure in the process.

One such structure that it could remember is the behaviour of vector bundles over it. A way to characterize this in turn is via their K-theory class (or its dual, the K-homology class). The closest to a physical interpretation of what this mean is via D-branes in superstring theory: On a “stack” of ##N## coincident D-branes, the open strings that end on them induce quanta of ##SU(N)## gauge fields (Chan-Paton gauge fields),  coupled to fields which are sections of rank-##N## complex vector bundles.

But there are also anti-D-branes. A stack of ##n’## of them similarly carries a rank ##n’## vector bundle, but as the ##n## D-branes and ##n’## anti-D-branes are brought together, then they may locally annihilate, to ##n-n’## D-branes, but globally there may remain twists, due to the fact that the Chan-Paton vector bundles may have global twists. As a result, the Chan-Paton fields on D-branes are actually not so much vector bundles themselves, as rather the equivalence classes of pairs of “positive” and “negative” vector bundles (in math the latter are called “virtual” vector bundles), subject to the relation that locally they may cancel their rank. These equivalence classes form groups called the K-theory groups.  Similarly, in type I string theory there are real vector bundles instead of complex vector bundles, and one speaks of KO-theory.

Now, using a magnificent idea called KK-theory,  all these K-theory groups have a completely operator algebraic formulation (the “K”s here are not for “Kaluza-Klein”, but for “K-homology and K-cohomology”). This means that given any Riemannian spin manifold, we may compute its KO-groups purely algebraically in terms of its Dirac operator. This in turn means that given any spectral geometry encoded by a spectral triple, there is still a concept of KO-groups.

Very roughly, the KO-dimension of a spectral geometry is its dimension as seen by KO-homology classes over this geometry, hence as seen by D-branes. This yields a number which is only defined modulo addition of multiples of 8 (due to a phenomenon called Bott periodicity). So it is a coarser measure of dimension than that of classical dimension. But on the other hand it is a more sensitive measure: there are spectral geometries that classically look like a point, but which still have non-vanishing KO-dimension. We may roughly think of these as spaces that classically look like a point, but which to a D-brane probe reveal a remaining spacetime quantum fluctuation.

 

Spectral Kaluza-Klein models

The availability of such generalized spacetimes, which classically look like points but which to quantum particles probing them reveal inner structure, allows to give one a deeper meaning to an old idea: that of Kaluza-Klein compactification of spacetime. The old observation here is that on a spacetime ##X## that is the Cartesian product ##X = \mathbb{R}^{3,1} \times F## of 3+1-dimensional Minkowski spacetime with a tiny compact space ##F##, then the Einstein equations of pure gravity on the total, higher dimensional, spacetime ##\mathbb{R}^{3,1} \times F##, effectively look like Einstein-Yang-Mills theory in 3+1 dimensions: the components of the field of gravity along the tiny compact dimensions effectively look like gauge fields in 3+1 dimensions.

Hence Kaluza-Klein compactification is a way to unify gravity and gauge forces, by realizing the latter as components of the former, but in higher dimensions. This is an attractive idea.

As a theory of higher dimensional gravitational field theory, this idea however has a serious flaw: Since everything gravitates, the shape and size of the fiber space ##F## is dynamical, and in gravitational field theory there is no reason why it should stay the way it is, even if at some point it happens to have a specific size and shape. This problem with the Kaluza-Klein scenario is known as the moduli stabilization problem.

This is what eventually led people to abandon the idea of Kaluza-Klein-compactification in the early 20th century. Recently Roger Penrose amplified this problem again, by arguing that his singularity theorem is one way to systematically see that Kaluza-Klein-compactifications must generically be unstable against collapsing into singularities in gravitational field theory (Penrose 03, section 10.3).

On the other hand, if gravity is thought of as being embedded in string theory, then there are arguments of various sorts that imply that here the problem of moduli stabilization has a solution; by stringy effects kicking in that serve to stabilize the compactified dimensions. See at moduli stabilization for pointers. There is much room left to substantiate these arguments and make them more rigorous, but it is this way via embedding into string theory that Kaluza-Klein compactification is again regarded as a viable scenario for high energy physics these days.

Hence Kaluza-Klein compactification deserves a second look, with modern mathematical tools. Such as those of spectral geometry.

Because, from the perspective of the effective 4-dimensional physics, ideally one wants to consider the actual limit where the spatial extension of the fiber space ##F## literally shrinks to zero, leaving behind nothing but the imprint of its shape on the spectrum of effective gauge fields seen in 4d.

In classical geometry this is absurd — but it is just the kind of effect possible in spectral geometry that we discussed in the previous section. In spectral geometry there is a way to take Kaluza-Klein theory to its logical conclusion: we may consider spacetimes of the product form ##X = \mathbb{R}^{3,1} \times F## such that the compact fiber space ##F## is classically a point, but with non-vanishing KO-dimension. It should then be true that the spectral Riemannian geometry of ##\mathbb{R}^{3,1} \times F## is exactly equivalent to gravity coupled to gauge fields on 4d-spacetime. The precise nature of the gauge fields would be controled by the precise internal structure of ##F## as seen by spinning quantum particles propagating inside it.

This allows a new kind of Kaluza-Klein model building:

Is there a spectral geometry of the form ##\mathbb{R}^{3,1} \times F##, hence a corresponding spectral triple , such that the resulting KK-compactification yields the standard model of particle physics coupled to gravity in 4d.

The answer is: yes. The Connes-Lott-Chamseddine-Barrett model is just such a spectral triple. Moreover, the choices made for ##F## are comparatively simple, and yet, with these choices the model reproduces the standard model of particle physics not just in broad outline, but reflecting a fair bit of its fine detail.

Side remark.

It should however not be forgotten that, as in every model in physics, there are choices involved in the Connes-Lott-Chamseddine-Barrett model. In fact details of the model have been changed over the years, as more of its consequences were worked out. A certain excitement about the fact that details come out so well has led some proponents to claim that the prediction of the Higgs boson mass made by one version of the model, subject to some assumptions, are actually generic predictions of the the general theory of spectral geometry. When these predictions were falsifed by experiment right away, the model was accordingly adjusted. Unfortunately, also the claim was made that now this adjusted prediction is generic (Connes-Chamseddine 12). But this is unlikely to be true either: There is the general theory of spectral geometry, and then there are models built inside it, and just as it was embarrassingly wrong of some string theorists in the 80s to hope that the theory would have an essentially unique model, so I think it is likely wrong of some proponents of spectral Kaluza-Klein-compactification models to hope that the theory has an essentially unique model. See at (How do physical theories make predictions anyway?, for more details). The reason that I am saying this is not to pick on the proponents of the spectral standard model. On the contrary: I am saying this to warn sceptics not to fall for the convenient but false conclusion that investing energy into understanding the spectral Kaluza-Klein-model building is a waste of time “because it has already been falsified by experiment”. No, the theory has not been falsified by experiment, just a particular model of the theory has been falsified. The situation is just the usual one as everywhere in phenomenology: in the context of a given theory, people fit the parameters of their model to those that are experimentally, known. The remaining properties of the model are its predictions. As soon as one of these contradicts experiments, one adjusts the parameters again to also fit the new observation. And so on. The theory itself is only falsified once the whole parameter space for all its models is ruled out by experiment. Some people these days seem to find this state of affairs unfortunate, but it’s just the basic principle of the exact sciences.

End of side remark.

That said, one prediction of the spectral Kaluza-Klein Connes-Lott-Chamseddine-Barrett model needs to be highlighted: Barrett and Connes independently found that, apparently generically, it predicts that the KO-dimension of the compact fiber space needs to be … six dimensions (modulo 8).

This number of course coincides with the prediction of another Kaluza-Klein theory: perturbative superstring theory.

 

Stringy Kaluza-Klein models

While this may or may not be a coincidence, what is known for sure is that the encoding of effective target space geometry in perturbative string theory is of just the same form as in Connes-style spectral geometry, the only difference being that effective geometry in string theory is an enrichment of Connes-style spectral geometry:

Where Connes-style spectral geometry encodes effective spacetime in terms of what a quantum spinning particle sees that propagates through it, so perturbative superstring theory encodes effective spacetime in terms of what a quantum spinning string  sees that propagates through it. In the limit where the string’s oscillations are in their quantum ground state, this reproduces the particle perspective (this is the point-particle limit of the string), but generally the higher oscillations of the string make it see more details in the effective quantum spacetime.

Algebraically, this situation is modeled in fairly straightforward analogy, up to technical details of course. The technical term for the abstract axioms of the worldsheet quantum mechanics of the string (i.e. the stringy generalization of the spectral triples above) is “2-dimensional super-conformal field theory“, or just “2d SCFT”, for short.

Where the spectral triple of a spinning quantum particle has a single Dirac operator, so the 2d SCFT for the spinning string has the Dirac-Ramond operator, which describes the dynamics of the center-of-mass point of the string, together with an infinite tower of “higher modes of Dirac operators”, encoding the dynamics of the oscillation modes of the string. (Technically, the jargon is that this tower of higher Dirac operators is the Fourier model of the “supercharge vertex operator” in the 2d SCFT.)

Otherwise, the situation is directly analogous. In particular, the way that a 2d SCFT encodes a target space geometry proceeds by the same kind of mechanism as for spectral triples (only that the procedure had originally not been formalized as neatly the way Connes then did for spectral triples).

This is amplified and explained nicely in section 7 of

  • Jürg Fröhlich, Oiver Grandjean, Andreas Recknagel,
    “Supersymmetric quantum theory, non-commutative geometry, and gravitation”,
    Lecture Notes Les Houches (1995) (arXiv:hep-th/9706132).

Better yet, there is an algebraic procedure that takes the “degeneration limit” or “point particle limit” of a 2d SCFT, and, with due technical care, the result is … a spectral triple.

This is discussed in sections 7.4 and 7.5 of the above article. As these authors exclaim on their p . 123:

2d Superconformal field theories and their “target space geometry” fit perfectly into the framework of spectral triples!

Below at the end I provide a list of further references that work out this relation in more detail. Presently the most detailed analysis of how 2d CFTs degenerate to spectral triples is that of Roggenkamp and Wendland, due to

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

In section 1.1 there spectral triples are recalled, and section 1.2 there discusses the procedure that extracts a spectral (pre-)triple from a 2d CFT.

Genuine quantum spacetimes

Just as there are spectral triples that do not come from a classical Riemannian spin geometry (such as the fiber spaces ##F## in the Connes-Lott-Chamseddine model), so there are 2d SCFTs that do not describe strings propagating on classical spacetimes. Those that do are called sigma-models , those that do not are sometimes called “non-geometric” models.

A standard example of non-geometric models in 2d CFT language are the Gepner models. And these nicely illustrate what’s going on here. Namely a famous result of string theory is that the flop transition is actually a smooth path in the space of 2d SCFT. Here one is looking at a 1-parameter family of Calabi-Yau geometries. At each parameter value except at one, these are actual manifolds to which a sigma-model 2d SCFT is associated, decribing the propation of the spinning string on these geometries. But at one value of the parameter these geometries collapse and become singular, they no longer exist as classical geometries at that parameter point. Hence it might seems as if the continuous families of Calabi-Yau spaces on the left and on the right of this critical parameter value are disconnected. However, it turns out that as spectral geometries, namely as 2d SCFTs, they are connected! At that one parameter value where the classical geometry breakes down, there is the Gepner model 2d SCFT, and this serves to continuously connect the two families of geometric sigma models. The quantum geometry is actually well defined for all parameter values, also where the classical geometry collapses to a point.

So the Gepner model in 2d CFTs is analogous to the singular fiber space ##F## in the spectral geometry of the Connes-Lott-Chamseddine model.

 

References with more details

Here is a list of references that provide more technical details on the relationship between spectral triples and 2d SCFTs, beyond the lecture notes by Fröhlich-Grandjean-Recknagel already cited above in the text.

That the 0-mode sector of a 2d SCFT – hence the quantum point particle limit of a quantum superstring dynamics – yields a spectral triple was maybe first highlighted in

  • Jürg Fröhlich, Krzysztof Gawędzki,
    “Conformal Field Theory and Geometry of Strings”,
    extended lecture notes for lecture given at the Mathematical Quantum Theory Conference, Vancouver, Canada, August 4-8 (1993)
    (arXiv:hep-th/9310187)

by way of a series of concrete examples, such as the WZW model.

That hence the superstring quantum theory should be regarded as a kind of higher spectral triple was maybe first suggested in

  • Ali Chamseddine,
    “An Effective Superstring Spectral Action”,
    Phys.Rev. D56 (1997) 3555-3567
    (arXiv:hep-th/9705153)

together with arguments that the associated spectral action indeed reproduces the action functional of the string’s target space effective supergravity theory. (See at Why supersymmetry? Because of Deligne’s theorem for more on this supergravity aspect.)

Later it was shown more formally in

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

following

  • Maxim Kontsevich, Yan Soibelman,
    section 2 of “Homological mirror symmetry and torus fibrations”,
    Proceedings of KIAS Annual International Conference on Symplectic Geometry and Mirror Symmetry
    (arXiv:math/0011041, spire)

and reviewed in

  • Daniel Roggenkamp, Katrin Wendland,
    “Decoding the geometry of conformal field theories”,
    Proceedings of the 7th International Workshop “Lie Theory and Its Applications in Physics”, Varna (2008)
    (arXiv:0803.0657)

that there is a precise algebraically formalization of taking the “point particle limit” of a quantum string, by sending its vertex operator algebra to a spectral triple obtained by suitably retaining only worldsheet 0-modes.

Analogous discussion based not on the vertex operator algebra description of local CFT but on the AQFT description via conformal nets is in

  • Sebastiano Carpi, Robin Hillier, Yasuyuki Kawahigashi, Roberto Longo,
    “Spectral triples and the super-Virasoro algebra”,
    Commun.Math.Phys.295:71-97 (2010)
    (arXiv:0811.4128)

where 2d SCFTs are related essentially to local nets of spectral triples.

In

this was used as a means to systematically study the large volume limit of effective string spacetimes (and hence aspects of the landscape of string theory vacua) by studying the spectral geometries (i.e. the Connes-style noncommutative geometries) of the spectral triples arising from the string’s point particle limit this way.

 

Conclusion

In conclusion, the concept of spectral triples and of 2d SCFTs are directly alike: the first is the algebraic axiomatization of a worldline-supersymmtric quantum particle (such as any spinning quantum particle), whereas the second is the algebraic axiomatization of the worldsheet-supersymmetric quantum string. 2d SCFTs are “2-dimensional spectral triples” (stringy spectral triples).

2spectraltriple

Moreover, there is a systematic process that takes 2d SCFTs to their point particle limit, which are spectral triples. Conversely this means that for every spectral triple it makes sense to ask whether it lifts to a 2d SCFT, and if so, in how many ways.

This is an operator-algebraic formalization of what it means to find an embedding of any point-particle spectral geometry into perturbative string theory.

In view of this, the Connes-Lott-Chamseddine-Barrett model is naturally regarded like so: it provides the point-particle limit of a spectral geometry that does describe the standard model of particle physics in a fair bit of fine detail. An very interesting question to ask therefore is: which 2d SCFTs (if any) would lift the Connes-Lott-Chamseddine-Barrett spectral triple? These would be realistic string vacua.

 

 

 

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague. Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.
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  1. John Baez
    John Baez says:

    Sometimes we can click on something to discuss Insights posts "in the Forum", which allows for better editing tools.  But not this time – not yet, anyway.Typos:emberrassingly -> embarrassingly(modul0 8) -> (modulo 8)conicidence -> coincidenceThose that do are called sigma-model -> Those that do are called sigma-modelsI may save more interesting comments for when Greg Bernhardt or someone enables comments "in the Forum", but here are three I don't want to forget, 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.

  2. JorisL
    JorisL says:

    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.

  3. Urs Schreiber
    Urs Schreiber says:

    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

  4. john baez
    john baez says:

    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.)

  5. Urs Schreiber
    Urs Schreiber says:

    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

    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!"

  6. john baez
    john baez says:

    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 with a self-adjoint operator H, the Hamiltonian", since that's enough to give a functor from a 1d cobordism category (with open Riemannian 1-manifolds s 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 .  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.

  7. Urs Schreiber
    Urs Schreiber says:

    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 expects to see from the point particle limit of a string worldshee theory.

    There is a section with more details on this in

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

  8. john baez
    john baez says:

    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)## on each edge where ##t## is the length of that edge, and now we're getting a representation of a category of 'Feynman graphs'.

    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.)

  9. john baez
    john baez says:

    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.  It only becomes an issue if we imagine ourselves trying to treat n-dimensional QFTs in a systematic way on an equal footing.  I was imagining that to help myself figure out what's really going on here.

  10. john baez
    john baez says:

    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.)

  11. Urs Schreiber
    Urs Schreiber says:

    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 contribution ##1tfrac{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. )

    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 functions the membrane by operators — and people like to think of finite rank operators here and speak of matrices. 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.

  12. David Corfield
    David Corfield says:

    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.

  13. David Corfield
    David Corfield says:

    "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?

  14. Urs Schreiber
    Urs Schreiber says:

    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 by hyperlinked, then you click on the symbol that looks like a chain, then add the url into the dialogue box that appears.

  15. Urs Schreiber
    Urs Schreiber says:

    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.

    "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?

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

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