FQFT and cohomology
An ordinary spectral triple is, discussed there, the abstract algebraic data characterizing supersymmetric quantum mechanics on a worldline and thereby spectrally encoding an effective (possibly non-commutative) target space geometry. Ordinary Riemannian geometry with spin structure is the special case of this where the Hilbert space in the spectral triple is that of square integrable sections of the spinor bundle and the operator acting on that is the standard Dirac operator, hence the “supercharge” of the worldline supersymmetry of the spinning particle.
In generalization of this, a “2-spectral triple” should be the analogous algebraic data that encodes the worldsheet theory of a superstring propagating on a target space geometry which is a generalization of Riemannian geometry with (twisted string structure) string structure.
Of course such data is just that of a 2d superconformal field theory, realized locally for instance by a vertex operator algebra or by a conformal net of local observables. But for emphasis it may be useful to speak of such data as constituting a “2-spectral triple”, for emphasizing more the important and intricate relation to the concept of spectral triples, which in much of the literature seems to be unduly ignored.
|quantum spinning particle||Dirac operator||spectral triple||operator algebra|
|quantum spinning string||Dirac-Ramond operator||2d SCFT||vertex operator algebra|
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 (Fröhlich-Gawędzki 93) 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 (Chamsedding 97), together with arguments that the associated spectral action indeed reproduces the action functional of the string’s target space effective supergravity theory. An exposition of this perspective is in (Fröhlich-Grandjean-Recknagel 97, section 7.2).
Later it was shown more formally (Roggenkamp-Wendland 03), reviewed in (Roggenkamp-Wendland 08), 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.
In (Soibelman 11) 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.
Now, since there is information lost in passing from a stringy “2-spectral triple” (a 2d SCFT) to its underlying point particle spectral triple, not all spectral triples are to be expected to have a lift to a 2-spectral triple (possibly corresponding to a UV-completion of the corresponding target space effective field theories).
The Connes-Lott-Chamseddine model is an encoding in a spectral triple of the standard model of particle physics coupled to gravity realized as a kind of spectral Kaluza-Klein compactification on an non-commutative fiber space down to ordinary 4d Minkowski spacetime (or possibly its Wick rotated Euclidean version). In order for this to work out, it turns out that the compactified non-commutative fiber space needs to have KO-dimension equal to . (Here the fiber space is classically just a (“non-commutative”) point, but it appears as the singular collapsing limit of a space of finite dimension. This actual dimension is the KO-dimension.)
Hence the claim of the Connes-Lott-Chamseddine model is that if the standard model is encoded as a singular limit of a Kaluza-Klein compactification modeled via a spectral triple then the dimensions of the KK-compactification are
with 4-dimensional base space and 6-dimensional fiber space, to a total of a 10-dimensional spacetime at high energy (after uncompactification of the fiber).
This, of course, is precisely the dimensionality of the target spacetime of the critical superstring. Algebraically, this arises from the fact that the BRST complex for the superstring worldsheet theory is consistent (has BRST differential squaring to 0) precisely if the corresponding 2d SCFT has conformal central charge 15, and each spacetime dimension contributes to this centralcharge (a contribution of 1 from each bosonic direction, and another for the corresponding fermionic contribution).
There is at least evidence that there is a continuous path in the space of 2-spectral triples that starts and ends at a point describing the ordinary geometry of a complex 3-dimensional Calabi-Yau space but passes in between through a 2-spectral triple/2d SCFT (a Gepner model) which is not the -model of an ordinary geometry, hence which describes “noncommutative 2-geometry” (to borrow that terminology from the situation of ordinary spectral triples). This is called the flop transition (alluding to the fact that the geometries at the start and end of this path have different topology). This was further expanded on and used for the mathematical study of the large volume limit of string theory vacua in (Soibelman 11).
(discussing aspects of homological mirror symmetry).
Exposition of these results is in
See also the references at geometric model for elliptic cohomology.