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 os 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.
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. Here the Dirac operator of the spectral triple is the Dirac-Ramond operator of the superstring, hence the operator whose index (in the large volume limit) is the Witten genus. A formalizaion of this process sending a super vertex operator algebra to a spectral triple was then described in (Roggenkamp-Wendland 03), reviewed in (Roggenkamp-Wendland 08)
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).
Similar considerations are in
See also the references at geometric model for elliptic cohomology.