Connes' spectral triples from 2d superconformal field theories
Promted by the discussion of the relation of Connes's "spectral triples" to what the physics community already knows and loves, from here:https://fmoldove.blogspot.de/2016/07/what-is-noncommutative-geometry.html?showComment=1468934322483#c6802080472818966598
I'll recall the following neat facts, which ought to be known much more widely:
Connes' spectral triples arise as the point particle limit of 2d (super-)conformal field theories. A commented list of references on this relation is here: ncatlab.org/nlab/show/2-spectral+triple#References
Under this relation, the way spectral triples encode an effective target space geometry as seen by a quantum particle is analogous to how a 2d (S)CFT encodes an effective target space geomtry as seen by a quantum (super-)string.
The first rigorous account of this is due to
D. Roggenkamp, K. Wendland, "Limits and Degenerations of Unitary Conformal Field Theories" arXiv:hep-th/0308143
D. Roggenkamp, K. Wendland, "Decoding the geometry of conformal field theories" arXiv:0803.0657
Yan Soibelman used this relation of 2d SCFT to Connes spectral triple in order to approach the analysis of aspects of the landscape of string vacua:
Y. Soibelman, "Collapsing CFTs, spaces with non-negative Ricci curvature and nc-geometry" , in H. Sati, U. Schreiber al. (eds.), "Mathematical Foundations of Quantum Field and Perturbative String Theory", Proceedings of Symposia in Pure Mathematics, AMS (2001)https://ncatlab.org/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory#ContributionSoibelman
For a kind of reverse construction, in
S. Carpi, R. Hillier, Y. Kawahigashi, R. Longo, "Spectral triples and the super-Virasoro algebra" (arXiv:0811.4128)
the authors realize 2d SCFTs essentially as local nets (in the sense of AQFT) of spectral triples.
Lifting a spectral triple to a 2d CFT means imposing stronger constriants since there is much more data in the 2d CFT than just its point particle limit reflected in the spectral triple.
This is a version of the fact that there are much stronger constraints on a string background to be consistent (anomaly free) than on a random QFT.
The article by Soibelman referenced above means to make use of this for saying something about the landscape of perturbative string vacua. A perturbative string vacuum is a 2d SCFT of central charge 15, which in addition satisfies modularity and sewing constraints. While the moduli space of all 2d SCFTs is hard to analyze, the shadow that it throws, via the point particle limit, in the space of spectral triples is more tractable. And Soibelman's article analyzes this shadow space.
In this context one cannot help but notice the following coincidence:
The spectral triples arising from 2d SCFTs of central charge 15 in string theory have, famously, KO-dimension 4+6 (mod 8). Now this is precisely the KO dimension that Connes claims in
A. Connes, "Noncommutative Geometry and the standard model with neutrino mixing", JHEP0611:081 (hep-th/0608226)
is necessary to get a viable standard model-like effective theory from a spectral triple.
Maybe it's a coincidence. Or maybe it points to something deep.