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created spectral triple, but so far a bit bizarre:
I give an unorthodox category-theoretic VAGUE definition, which I have reason to think is the right one
and then I record an unusual reference on vonNeumann spectral triple (just because at MO somebody asked for this and I don't like to dig out a link just to throw it away after one reference use like a paper napkin )
I have to admit I haven't actually looked at that article on "vonNeumann spectra triples". But I guess some aspect of the definition must differ, otherwise the author wouldn't have called it that way.
added references on point particle limits of 2d CFTs to spectral triples, and also some other links
more references (now also the standard ones! :-)
If
A spectral triple is algebraic data that mimics the geometric data provided by a smooth Riemannian manifold with spin structure and generalizes it to noncommutative geometry.
Is it possible to have algebraic data that mimics the geometric data provided by a smooth Riemannian manifold with spin structure and string structure (and then fivebrane structure) and generalizes it to noncommutative geometry?
It is a $Spin^{\mathbb{C}}$ structure not usual $Spin$ structure if this makes a difference.
Is it possible to have algebraic data that mimics the geometric data provided by a smooth Riemannian manifold with spin structure and string structure
That’s supposed to be what a 2-spectral triple is, which is supposed to be the data given by a 2d superconformal QFT that behaves as if it is the worldvolume theory of the heterotic/type II superstring.
Pretty close to fully formalizing this is the work surveyed at (2,1)-dimensional Euclidean field theories and tmf. (The String structure condition that you are looking for appears right at the beginning in this entry.)
But there are many other known approaches that must be effectively encodings of the same kind of data, such as certain vertex operator algebras, speicifcally certain chiral de Rham complexes. Also Costello has more on this in terms of his factorization algebras. Hopefully eventually somebody writes it all out nicely.
But the general upshot is: yes, the algebraic data we are talking about it is that encoding the worldvolume theory of the fundamental super $p$-brane propagating on the corresponding type of geoemtry. There ought to be also a 6-spectral triple that corresponds to geometries with Fivebrane structures, yes, but I am not aware of much concrete progress in getting that.
It is a $Spin^{\mathbb{C}}$ structure not usual $Spin$ structure if this makes a difference.
I think these variants all corespond to slight variants in the axioms of the spectral triple and can all be accounted for. For instance see the sentence right beneath theorem 1.2 here which says that the “reality condition” on the spectral triple makes it pick Spin-geometry among $Spin^{c}$-geometries.
I have slightly expanded 2-spectral triple in reply to David’s question.
Thanks. What about the “…and generalizes it to noncommutative geometry” part? Does that make any sense in the 2-spectral situation?
What about the “…and generalizes it to noncommutative geometry” part? Does that make any sense in the 2-spectral situation?
Yes, it makes sense. You need to find the characterization of the algebraic properties of those 2d SCFTs that come from your desired target space (Riemannian with string structure, say). Then you just declare that every SCFT with these abstract properties defines a “noncommutative string Riemannian manifold”, even if its space of states is not that of a string on a Riemannnian manifold.
A simple example of the general kind of approach here is spelled out in some detail in the article by Roggenkamp and Wendland that is cited in the entry. They show that the family of 2d CFTs that go by the name “minimal models” are “noncommutative 2-geometry” deformations of the standard interval $[0,1]$ equipped with a dilaton field. Whatever that means, think of it as analogous geometric data as saying “Riemannian manifold with string structure”.
So generally you can look at an abstract 2d (S)CFT $Z$ and ask what type of genuine $\sigma$-model SCFT it behaves like. Then you can say that $Z$ is a “noncommutative” generalization of such a type of geometry.
This is a standard and basic way to go about things in string theory, but there it is not usually formalized or thought of in quite the way we are after here. But for instance one of the early major results in string theory was the flop transition which is related to this phenomenon: what is called the flop transition is a continuous path in the space of 2-spectral triples=2d SCFTs which goes through points that are genuine geometries at the beginning and the end, but passes in between through a point which is not an ordinary geometry (the “Gepner model”). So it connects two types of ordinary (“commutative”) geometries that are not connected in the space of ordinary geometries. In fact they have different topology (that’s what the word “flop” alludes to).
at flop transition there is now a tiny little bit of literature on what I just mentioned
Sombody emailed me asking for and suggesting more accurate historical references at spectral triple and more emphasis on the essentially equivalent and older concept of unbounded Fredholm modules.
I have added those references, added the relevant cross-pointers, and highlighted the relation to Fredholm modules more.
Of course the entry still does not contain actual precise content. Maybe somebody with energy and time comes along and feels like adding something.
The webpage link at Henri Moscovici was dead. I changed it. Please check it is the one needed as it does not seem to lead anywhere!
added pointer to today’s
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