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added more references to 2-spectral triple (as far as I can see Jürg Fröhlich with his students was the first to try to formalize this to some extent)
Over on G+ Stam Nicolis kindly points out a more recent contribution to the “2d SCFT = 2NCG” idea, I have added the following to the references:
Analogous detailed discussion based not on the vertex operator algebra description of local CFT but on the AQFT description via conformal nets is in
- {#CHKL09} 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.
I expanded the text in the Idea-section at 2-spectral triple and added this reference here, which is currently the earliest that I am aware of which observes that the 0-mode sector of a 2d SCFT yields a spectral triple:
I have expanded at 2-spectral triple: more references and more comments and pointers to the references from the main text.
This was prompted by an exposition on PhysicsForums Insights that I wrote: Spectral standard model and String compactifications.
What does ’super’ add to VOA in terms of the role it plays in a 2-spectral triple?
Perhaps another way of asking about things here is how the connection goes: the monster vertex operator algebra is a super VOA, the monster group/Leech lattice shows up relating to the bosonic string, but that’s a version of the string lacking worldsheet supersymmetry.
Regarding the first question:
Connes’ first definition of spectral triple had the Laplace operator instead of the Dirac operator. That turned out to work, too, but only to some extent, and all proofs were tedious (such as that a Riemannian manifold may be reconstructed from its spectral triple).
Then later he found that the theory flows much better if one uses the Dirac operator, hence the “square root” of the Laplace operator.
But in terms of worldline quantum mechanics, passing from the Laplace operator to the Dirac operator means introducing worldline supersymmetry. The Dirac operator is the worldline supercharge.
Now passing from one to two worldvolume dimensions, the analogous step is that from VOAs to super VOAs.
Regarding the second question:
The super VOA here is that of the heterotic string, which has (1,0) supersymmetry (“only”). The holomorphic part looks like a 10d chiral superstring, but the antiholomorphic part looks like a 26d chiral bosonic string, compactified on the Leech lattice.
This heterosis of (non-)supersymmetries is what gives the heterotic string its name and which, one way or another, is responsible for its magic properties.
Here I think what happens is that the non-trivial part of the super VOA automorphisms all comes from the Leech lattice hidden in its anti-holomorphic sector, and so the computation may be reduced to automorphisms of the Leech lattice.
I think. Still have to absorb in real detail all things moonshiny. Experts should please correct me.
Incidentally, this combination of chiral sigma-models in the heterotic string is another instance of “non-geometric” string vacua: The two chiral halfs of a 2d sigma model jointly determine a geometric background, but discarding one and replacing it with the corresponding chiral half of another sigma model gives in total something that is not the two chiral halfa of a single sigma-model anymore, hence which is “non-geometric”.
Thanks! So of course we should include this wisdom on the relevant pages.
So, re #7, that first table at 2-spectral triple should really say ’super VOA’?
So of course we should include this wisdom on the relevant pages
made a quick note here
re #7, that first table at 2-spectral triple should really say ’super VOA’?
But then also the “operator algebra” in the slot right above should be “super operator algebra”. Alternatively the term headlining this column could say “bosonic underlying algebra”.
Not all fine print of definitions can be captured by these survey tables, though! I hope the reader understands that these tables cannot be a substitute for the actual definitions.
So if M-theory is behind the 5 string theories, including the heterotic ones, does it “know” about what the heterotic string has to do with the Leech lattice, moonshine, etc? Does it know about bosonic M-theory?
Did I see the latter has 2-branes and 21-branes? Anything cohomotopic going on there?
does it “know” about what the heterotic string has to do with the Leech lattice, moonshine, etc?
One way it “knows” about this is via KK-compactification on K3-surfaces by the cohomology lattice of K3-surfaces. These embed into suitable Niemeier lattices.
Am on my phone now, but when I am back, we’ll start an entry on HET-IIA. This is all about K3.
among mathematical accounts this one here seems useful: https://doi.org/10.1007/s40687-018-0150-4
to the discussion on how the Connes-Lott models strikingly match properties of superstring vacua, I have added the full pertaining quote from Connes 06, p. 8:
When one looks at the table (7.2) of Appendix 7 giving the KO-dimension of the finite space $[$ i.e. the noncommutative KK-compactification-fiber $]$ one then finds that its KO-dimension is now equal to 6 modulo 8 (!). As a result we see that the KO-dimension of the product space $M \times F$ $[$ i.e. of 4d spacetime $M$ with the noncommutative KK-compactification-fiber $F$$]$ is in fact equal to $10 \sim 2$ modulo 8. Of course the above 10 is very reminiscent of string theory, in which the finite space $F$ might bea good candidate for an “effective” compactification at least for low energies. But 10 is also 2 modulo 8 which might be related to the observations of Lauscher-Reuter 06 about gravity.
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